converge average compression geometry say finite net net n exist achievable scheme nz nj element true encoder operate indeed expect true net z b encoder remain step various definition triangle property two distance measurable number entropy instance euclidean space dominate constant f scheme encoder distribution finite description encoder unless reliably identify enough ability underlie encoder unknown away encoder description agent essentially le existence operational distortion infimum n define limit operational distortion state type ii operational f achievable encoder achieve infimum enumeration construct encoder learner nj x excess f nj p x nj p nj take achieve limit would express purely theoretic straightforward upon work communication immediate eq requirement list define distortion xy general omit code rp pp ix np ny p f rp triple random p rp pn result would constructing achieve rich slowly grow combine union code rate overhead devise difficulty distortion average f find infimum rate noisy sequence I unknown wish reconstruct distortion page within dirac concentrated leave general nature find bound variable goal performance setting description performance presence proof separation tailor joint rate constrain accurate predictor variable input focus access description context repeat agent agent inference rate channel trade present kind training limited noiseless digital channel part part scheme impose separation compression restriction encoder reliably tailor learn optimum encoder construct part namely design happen pair result theoretic constrain communication classifier also paper arise limitation structure motivation complementary regression allow knowledge predictor constrain one learner map z nz nz pz main depend training show probably pac excess achievable excess channel training learn set encoder fig observe noiseless digital bit per operating encoder f n nj nz nz type ii encoder output training fig perfect digital n nj ny ny shall
amongst posterior probability correct winning affect plausibility argument thing introduce large drastically particular thing however reason complicated sure thing reduce introduce take account datum take bayes updating subscript product usual marginal describe related data plan concerned knowledge distribution twice posterior thus stage however prior stage knowledge update think prefer update rather write stage probability use update behind split probability I lot relationship principle understand rule proposition build product whereas I proposition space stage joint calibrate typically often therefore however usually parameter constraint distribution distribution must satisfy constraint distribution lagrange multiplier marginal multiplied correspond I subsequently illustrate logic selection paragraph prior rescale measure closeness may break soon knowledge get specify specify yet must assign joint hypothesis incorporate zero reweighte exponential original entropy odd evidence win sharp example confident prior nature dark think responsible accelerated expansion universe survey dark energy exactly equal minus former indicate play really sure evolve least literature answer present contribution four equation varied scale factor constant evolve complex ia probe test forecast assign automatically evenly amongst predictive distribution prior solely confident course explicitly assign key check careful realistic predictive energy address physics california address ts maximum ease usual simple thing hypothesis situation alternative sure thing may number alternative formalism modify explicit enumeration outline resolve amongst dark way decide datum compete hypothesis possibly wish plausibility drop hereafter provide mean side posterior encode justify conclusion base regard primarily difficult hoc calculate carry comparison equation publish author community whole plausibility vice versa sure thing suppose
diabetes body diabetes indicator bernoulli random variable standard probit parameter illustration opt metropolis hasting posterior dimensional walk scale normalization body significant variate indicate end previous simple regression scale start diabetes row value additional control variate bring acknowledgment author grateful mcmc work team lead improved presentation support present rao accept reject metropolis high adopt completely universal scheme illustrate toy probit accept reject hasting generation acceptance precisely dominate metropolis hasting respect dominate measure hasting value uniformity directly take flow auxiliary preserving integrating size rao reject candidate part due rao argument variance result derivation asymptotic improvement toy metropolis approximation metropolis hasting construction reference markov distribute transition dy transition density integrate geometric time p balance reversible estimator accept weight estimate geometric obvious hasting small quantity estimator conditional pair expectation u therefore surely involve variability iteration require intermediate fix fortunately unbiased dy proposal computational cost additional compare metropolis weight get variable algebra rewrite depend expectation version bring estimation follow result denote reference behave following q pt directly denote I first imply q array show c g c ip f lebesgue without generality q denominator thus central numerator since prove n n eq fact side bound result arbitrarily go term hand side n f r q p ph go go let proof note universal control variate metropolis hasting unbiased estimator estimate setting exploit series toy target acceptance target estimate walk proposal rao repeating represent gain rao overlap source randomness addition gain illustrate replication proposal start function gain fact acceptance probability rejection improvement variance variability estimate additional previous pt additional require additional despite occurrence difficulty unconstraine metropolis hasting target proposal quite produce slightly indicate clearly gain widely target proposal proposition rao version importance illustrate table rao variance importance extreme ideal bring considerable replication
example lemma lx fy fx fx l lx fy fx lx fx lx fy dy dy dy let lx lx lx r lx r lx r lx lx e together derive bound function straightforward function depend poisson q eq inequality adapt proposition omit evaluate nf l use poisson lf lf u straightforward uniform drift moreover converge n view px show finite lemma eq exist finite martingale lf px cn rely measure measurable value measurable chapter proposition chapter measurable increasingly e vx kp ne let eq convergence let notation use next path go conclude q lx dy lx n lx lx lf theorem exist invariant follow exist check lemma e sample bind see g process covariance central limit uniformly ergodic chain check corollary suffice I trivially write h x u u nx u x nx p x additional straightforward simple nk n nk n sample define back partial k p negligible gx nk k gx nk nx nk converge almost follow nx arrive g nk k v almost every path use lemma n arrive converge xx central one xu p p u nk define x I j n nk nk x difference nk j nk nk dx ergodic term author grateful helpful discussion remark section chain method transition distribution design sampler adaptive mcmc paper chain substantial chain carlo carlo efficiency seminal let adaptively marginal behave way chain mention assumption invariant interest equal design limit otherwise resample central limit kernel variance always limit substantial illustrate literature law large number study also numerically equation develop interact ir mcmc prove ingredient martingale example section metropolis space equip borel measure measurable function l monte family order far measurable measurable kernel real nh throughout initial simplicity expectation k follow k l heuristic invariant lx way choose choice ergodic limit choice monte simulation define importance invariant ir probability resample lx lx form q impossible try limit hasting lx lx dy distribution always independently follow resample accept lx x sampler simplify version e ix ix partition get sampler limit metropolis hasting lx dy l partitioning pt lx dy l work practice allow jump state accept theoretical partitioning remain restrict ce moreover drift drift check metropolis langevin energy enough quantify kernel ergodic measurable immediate irreducible invariant distribution admit invariant distribution natural central center denote give surely central sampler restrict case compact equip precisely subset constant endow define eq simplify notation dependence sampler denote imply lemma comment let fs n introduce dy nk j nx field behavior eq kernel g see dx show sampler asymptotic chain arrive interact also estimate tend particularly initial enjoy whether unfortunately answer hold present often check p hasting metropolis hasting target resp acceptance resp ax ax dy away transform measurable function hold rely stochastic use long result still use compare walk rwm algorithm sampler limit ir algorithm
denote support sublinear visualize hull dual vector angle original visualize surface hull deduce property subdifferential hence functional differentiable iff subdifferential support achieve examine roughly serve hyperplane find differentiable subdifferential singleton criterion state picture implication easy verify strict convexity uniqueness minimizer hence curvature lead decay occur e suggest strongly norm state convex norm q suppose functional show minimizer proof fix let achieve theorem expression get subdifferential order expansion arrive result dual convexity follow assumption concluding intuition bound functional differentiable state term rademacher complexity bound prove actually strategy typically provide construction throughout refer rate slow rate notion rate minimizer arise slow differentiable derivative grow fast rate upon show regret always arise satisfie flat norm immediately strongly tight quadratic attain difference concavity write vanish proceed indeed eq bound average one notion rademacher rademacher uniform omit dependence sake average guarantee minimization rademacher average key sample rademacher averages bound average show eq minimizer particular suboptimal supremum q even whole indeed exchange go supremum tangent fix eq worst average rademacher average lipschitz complexity multiplied lipschitz loss hull finite vc dimension give minimax remark bound depend bound examine game norm dual boundary achieve take round distribution easy zero arise away adversary play maximization optimum develop describe upper quadratic later converse infinite curvature bound viewpoint point suggest minimizer translate flat singleton face face support hyperplane origin support normalize singleton face equivalent distinct discuss support singleton face convex minimizer sp low arise fluctuation refer suppose singleton I contain shift class center constant recall f pf gaussian absolute rearrange least remark case reduce discuss low bound bound enough describe primal dual average act primal game dual optimum true suppose put intersection problem conclude non lower far put round asymptotically surface ball regret grow adversary sequence distribution come within game case visualize low look expert suffer loss per action contain simplex shape pyramid I game times deviation minus ball random deviation let turn hull support uniform process verify index structure lemma adversary n maximum result therefore present address low whereas hope easily require I construct convenience shorthand choose expectation identical statement prove induction lead define induction follow since base conditional unnecessary conditional expression done follow nonempty respective function take subset nonempty eliminate let adjoint nonempty invertible transformation contain face transformation face e away regret modify condition transformation mapping last clear point last understand sum quantify variable adopt proposition swap expectation maximize replace supremum inside square distribution invoke q ranging depend maximize compare concavity easy distribution concavity first clearly linear concave strongly convex expectation rearrange lipschitz establish result curvature subdifferential q evident substituting choice thus verify corollary computer science division berkeley study von regret adversarial behavior minimization process distribution adversary difference sum loss geometric interpretation concave minimizer optimal strategy within theory topic gain popularity past adversarial manner work statistical find root theory argue online early recently argue attractive indeed problem middle adversarial quite analysis security spam largely responsible interest learn recent book similarity online algorithm link phenomenon bad strategy paper attempt build bridge von regret difference loss loss stochastic lead similarity game round round player predict convex determine point emphasize adversary note differ instance constitute adversary draw loss sequence joint face conditional pt sake clarity proof consider quantity clear optimize eq compact supremum observe appeal depend token influence see prove divergence bregman differentiable subgradient notion infinite instance define put subgradient subgradient fact generalize divergence focus omit definition immediately eq q since expectation useful word expectation obtain round roughly speak lemma say jensen inside suppose joint marginal easy
respectively odd together use observe follow gaussian z chi bound suffice condition together bind final expand bs b drop subscript get formula follow lead version appear bound except small whenever need know incoherent basis satisfy equal hand maximally coherence mi member mi centre f mail fr theorem proposition treat via coefficient nonconvex coefficient ensure identifiability derive dictionary assume bernoulli basis identifiable grow logarithmic I require compressed matrix identification optimisation component separation blind source efficiently know huge inverse blind separation representation exactly dictionary success heavily fit class dictionary dictionary wavelet stem tool harmonic mathematical require type dictionary meet treat know certain surprisingly work dictionary dictionary recently people start learn component ica identifiability available well geometric identifiability overcomplete condition size grow exponentially provably good identification identifiability outlier hand concerned relatively e availability resource provably combinatorial dictionary learn availability inspire give dictionary investigate detail minimum possibility replace learn efficient investigate minima surprising result sample draw generate incoherent matrix locally identifiable distribution consider basis essentially real identification identifiability initial optimisation perform optimisation matrix experiment indicate natural turn dimension gaussian imply vector complete matrix representation consist find sparse significantly select ideal amount pseudo number entry nonconvex nonsmooth hard solve turn strategy principle news admit relaxed related finding sense representation sparsity collect column fit total zero prescribe admissible consider representation give dictionary np hard try fortunately since stable respect suited goal criterion might success pursuit norm several unlike problem change still convex programming identify transmission observation dictionary channel channel dictionary convex seem behave criterion make amenable numerical need define admissible dictionary family library orthonormal basis cosine fast selection possible search parametric learn jointly common domain concentrate sparse assume reader inequality see constraint let aspect treat adopt ability dictionary ideal determine extent objective spirit dictionary uniqueness overcomplete dictionary specify advance optimisation want condition successfully ambiguity face consist usual ambiguity avoid ambiguity dp sign relax requirement ask condition indicate mean permutation like incoherent minima even find optimisation spurious minima independently raise complementary identifiability guarantee match permutation change concentrate minima carry certainly serve question contrast identification assumption ideally bernoulli sufficiently incoherent basis illustrate section make atom observe perfectly proportion htbp ny cloud figure angle dictionary minima locate sign ambiguity restrict moreover despite spurious minima none htbp infinity varied sample repeat display associate sparse black spurious outlier seem grey observe fully transition several possible orthogonality difficulty globally optima solution section provide tool progress even bernoulli model well account believe warm tool understand robust outlier b dictionary identification number fortunately mathematically initial reader local draw describe correspond relatively necessary satisfied consider check satisfied index indeed consider sufficient htbp bernoulli many sparse observe belong inside polytope radius necessarily orthogonal incoherent lie polytope conclude condition true word coherence polytope state radius large include provide dictionary incoherent sense incoherent admit strict compare assumption dictionary considerably quantity quantity specific value accord involve dictionary example substantial htbp draw shape coefficient highly outlier polytope seem shape cube constant function however heavily nature determine size bernoulli large shape essentially choice behaviour htbp draw gaussian almost shape ball derive need sufficiently constitute draw perspective would laplacian draw prior locate nonzero observe set compress ill solve allow previously take follow role strictly assumption gaussian mainly simplicity reason theory believe many certain concentration determine image large small probability ball e significantly exponentially use yield coherence constraint type eq wish basis identifiable answer question bernoulli describe section assume incoherent identifiable except failure rapidly soon scale need example basis resp maximally incoherent check bernoulli necessary algebraic local learn incoherent random sparse long signal dictionary learning question case assume local minima corresponding desirable converse direction basis generate investigate resemble current ideally could coherence extremely unlikely minimum hard overcomplete noisy overcomplete prevent intrinsic condition lemma theorem formulation contaminate dictionary close need introduce follow product operator convention proof eq similar relation follow decomposition matrix zero consider gram matrix nan dictionary subspace make abuse cover unit point eq point minima x yy equation notion square admissible ccc admissible completeness write admissible cc matrix pair tangent arbitrary c complete define local inequality local b may f x x x tangent yx yield admissible consist admissible cx therefore k equivalent row entry row denominator decompose decompose numerator sum observe match q numerator straightforward proof global diagonal nan x thank understand mean matrix characterization
incomplete robustness coefficient variation cluster largely context genomic rather mixture produce interpretable cluster immediately significant fluctuation order challenge simplified component reduce factorization elementary product probability limit development produce cluster anonymous note tend easily proportional provide advantageous gene weak assign gene make small feature gamma tune determined assignment cluster category substantial simpler analyze gamma small nothing temporal dependence seem involved line microarray impose complicated identify gene act act different another confident fit though cluster cluster seem associate correlate rich model dependent care need parameter intrinsic mechanism mixture support expression trait factor order probability develop ranking enable implementation flexibility shape method property explore distribution pe line move factor far mind evaluate z poisson probability indicate equivalence stem relationship process accumulate hence value basically eq shape proceed z p z probability highlight right precisely function gamma distribute gamma conjugacy page integral evaluate contribution possibly conclude token work force row table eliminate force variable great say second force seven repeat table repeat domain know term eliminate contribution degree complete replicate three balanced across replicate component observation simple exercise strict concavity likelihood invoke lagrange derivative calculus ij I constant determine quadratic form f ix jx ax regardless nonnegative concavity know force linear independence complete distribution imply every put linear verify linearly minus pick one nonzero impose drop drop let complete pt cd na parameter gamma gamma recall hence round calculation five order structure reduce set model one gene great mapping structure approximation ideal eliminate gene cluster affect cluster examine quantile relate sample coefficient note largely em cycle cycle estimate shape change receive computation cpu second per second affect gene analyze size acknowledge find report r associate comment improve development support grant gm mixture although challenge scope structure computation depend gamma variable attain binomial find dynamic programming accord among concavity beneficial method promise gene cell exhibit sort examine control post amount pattern investigate relatively gene identify hundred thousand pattern substantial multi many different ever gene share recent perspective method partition approach informative satisfactory like select subtle anonymous external pattern gene approximate contribute technical problem minimize narrow biological treat mixture assign probable procedure mixture popular page considerable cluster technical affect modal establish constraint identifiability switching somewhat group possibly call place gene anonymous component linearly computational characteristic structure record pattern specification density embed gamma extend poisson response characteristic gene measure next sequencing relatively vector represent cell chemical stage along course record say indicate group analyst intensity microarray adjust systematic relate advance allow explicit abundance mixture gamma gene datum treat finite discrete structure order latent specifically proportion modeling partition carry ordering subset structure denote group subset partition expect group structure number grow think structure correspondence allow tie pt sp number association example entail group single mean without amount order subset subset constitute replicate induce gene replicate differentially equal subset low e absence expression generally union replicate language assumption necessarily regardless probability replicate determine bayes alternatively though gene go gene characteristic discrete calculation integrate transformation rate reflect identically gamma ordering parameterization center nan specification assume measurement independently empirically stochastic population consideration move decrease represent depend choose normalization joint gamma integral variable precede arrange product multiply factor involve simplification follow mutually shape structure entail single exchangeable multivariate compound gamma variation scale inspection shape parameter parameter special report density evaluate contour display contour three mass one way constraint structure restrict way share something wherein model solve order event mutually gamma possibly special variable compute numerical beta distribution clearly indicate assume formula develop monte certainly fast accurate preferable efficiency target shape setting numerical computing shape positive collection process denote distribute gamma gap form marginally process dependent overlap point next equal variable appendix sum construct inner indeed recursion although version viterbi density seem linearly identifiability strict concavity establish identifiability key step denote real number structure finite special balanced replicate density proceed multivariate degree lead center rational factor rational rational require parameter different treat linearly independent structured admit unique maximizer sure expectation apply strict concavity multiple insensitive starting position page confirm implementation recover draw show mix proportion share prototype share simple mixture proportion gamma www mixing structure effect identify appendix assumption quantile coefficient observation component check comparison infer cluster pattern parameter guide reduce issue represent biology though examine example summarize identify large code infer gene standardized display raw cycle rule heart stress three replicate measure five hour stress several consider old process produce temporal fitting mixture structure work mean aspect diagnosis appendix possibility cluster gamma contain stress baseline significant example give different find choose www adjust rand measure dissimilarity gamma rank gamma worth investigate involve activity activity molecular complete picture biology apparent set rank datum repository set relevant exhibit case f gene gamma order cluster gene fact gamma smaller wide low number significantly way rank em gr km cc al al pilot al et empirical ranking often seem frequently category functional gene category gene biological calculation cluster across go proportion small comparison set rank biological green plotted proportion
initialization local optimum pca dim train input dim dim mnist letter breast less diabetes peak converge mnist letter breast cancer diabetes peak diabete parameter simply wise adaboost criterion constraint adaboost affected value uci mahalanobis computational running algorithm task ram operation corresponding time dimension artificial dimension keep triplet interior sdp solver scale instead combine sdp solver back cone need fig input comparable dimension become significantly involved randomly pair test local represent histogram visual subset image visual histogram subset classification carefully rate respectively slightly svm classifier right triplet trend triplet lead face google two object retrieval problem target class subset use face calculate retrieve image precision subset subset report precision codebook consistently attain high advantage triplet codebook test triplet face face new generalize sense algorithm show art exist currently handle theorem conjecture updating eq eq exactly adaboost solve z denote base optimization summarize triplet ss parameter si u ss v ss tasks right pca recover datum preserve visualization artificial toy dataset circle dataset circle eight noise fail first informative dimension lda center overlap informative indicate successfully eliminate speed handwritten face uci last randomly use
cca maximal correlation cca address cca semantic indexing cca provide capture space bag word sparse cca subset language interpretability identify document inner bag require document immediately improve retrieval computation bag word illustrate achieve retrieval specifically english model feature feature associate vocabulary indicate collect vector collect english feature vector matrix collect document respectively english document product datum cca perform across english efficient dc cca successive sparse stack subsequent component reader english convert project english one loading differently say onto span associated language select projection distance neighbor use sentence smaller align english rare english term computing generate english appropriate query retrieval test document canonical percentage zero loading dc cca dc cca sparse cca cca curve query rank project vector euclidean query performance example return suppose approximation sparse apply method program iterative computationally use instead convex sdp algorithm formulation paragraph support vector machine minimize loss support formulation well study show study sparse propose tight cardinality formulate optimization program behavior dc cca dc cca demonstrate pca cca application case sparse experimentally benchmark real life vary dc pca explain variance perform pca scalability relevance cca language vocabulary music retrieval priori guarantee level similar sdp well shoot although original propose solve quadratic use version acknowledge national foundation california micro appendix derivation idea deriving program derive bi lagrangian dual dual program convex consider relaxed lagrangian eq q lemma bi obtain result derivation alg alg differently apply idea program indicator eq check derivation alg alg maximizer lie boundary lagrangian give q equivalently alg wherein goal obtain sparse principal pca canonical correlation cca cardinality previous method context sparse tight log problem solve convex program minimization result exhibit initialization subsequence iterate point program performance demonstrate gene gene cca vocabulary selection retrieval component fisher minimization music language eigenvalue find identity fundamental area multivariate prominent deal high visualization positive problem maximum matrix pair call know widely specific pca classic analysis compression visualization maximal variance use wherein ambient dimensional significant variational covariance semidefinite multivariate canonical cca dimensionality however pca deal space multivariate space information reflect correlation maximally correlate represent rewrite write xy yx xx yy fisher discriminant find projection onto lead covariance variational give rewrite formulation multi lead discriminant simplicity popularity lack suffer disadvantage vector interpret different pca cca application coordinate axis biology might correspond specific loading moreover asset trading technique solution consequence imply transaction cca copy corpus write english extract multiple dimensional variation document language aid translation interpret well music annotation description review acoustic content music retrieval sparse summarize generally desirable aid understanding reduce economic denote cardinality problem cardinality one usual approximate early cardinality constraint solve iterative pca theory iterative subsequence iterate c like mention cca sparse scalable dc possible show dc cca deal document vocabulary music annotation document retrieval application different language say query string language language experimentally cca loading canonical cca select pruning vocabulary underlie audio generalize programming computationally intensive briefly section cca instance algorithm section algorithm respectively mean definite semidefinite absolute value denote zero x I principal generalize solve former let consider p concave constraint non computationally intractable quadratic objective homogeneous quadratic two reason objective solve briefly discuss convex could version program objective except follow sdp tractable large sdp expensive wherein instead cardinality constraint minimization formulation concave convex relaxation simplify cardinality constraint scalability explore oppose e regularize version penalization parameter equivalent limit equivalent q combinatorial program selection machine factorial bayesian improper show demonstrate validity expansion approximation machine tight end know would know maximization easy formulation derive sparse value two function I I g formulate program trivially choose introduce difference convex optimization branch cutting solve c program nonlinear solve quadratic present generalization know maximization em behind algorithm numerical appear place robust correspondence recovery mm algorithm reference general idea mm idea construct function update already stop jensen quadratic upper equality follow sign change min max refers call put thing perspective jensen construction refer refer negative name study sm stand surrogate stand call surrogate example construct derive let optimization convex differentiable solve note concave suppose strict achieve g helpful return program eq write idea derive satisfie result hand check sequence constrain quadratic program clear alg irrespective unique lie boundary alg also appendix detail define diagonal alg similar constraint interpretation l l compute approximation norm ellipsoid iterative therefore small weighting therefore toward discussion clear problem alg l choose setup unsupervise cca free tune desire noted sparsity reduce lead searching value say cardinality check present obtain remove entry pattern nevertheless replace p solution termination certainly discard loading obtain loading surely improve mention iterative converge converge address question like initialization behavior mention front convergence theory iterative show globally convergent sequence converge constrain tucker kkt condition kkt obtain apply alg carry globally convergent corollary problem derivation mention set assign power map set say tucker kkt case sense fact term global result provide convergence ix jx uv continuous dc solve point generate dc alg nonempty suitable dc l vx vx f lx l f vx alg set alg obtain apply uv alg correspond alg nx follow convergence algorithm since whether follow corollary algorithm match eigenvector converge algorithm solve l variable n imply multiply side complementary give eigen suppose result algorithm generalize follow converge alg constraint solve sparse consider address pca special dc covariance reduce correspond sparse involve complexity dc pca exhibit dc pca exhibit property suppose interest dc converge dc reduce power method suppose proceed far briefly discuss rotation component thresholding principal true framework lasso enforce pca bound non pca angle cardinality lead sdp complexity scale benchmark set scalable high even nesterov propose combinatorial method lead target sparsity numerical sparse pca mention knowledge art compare approximate let version program maximization apply solve dc lx guarantee dc pca ensure irrelevant cardinality dc reduce zero svd principal unit loading semidefinite regression interpret circular mean laplacian maximum posteriori penalization aforementione define improper prior replace dc solution note like unlike pose approximate program dc obtained obtain program ellipsoid result gradient scheme confirm dc performance mention unlike eigenvalue setting cca effectiveness small dc dc except scalability wherein greedy algorithm comparison comparable carry ghz ram motivated discussion pc dc become benchmark pc explanatory pca eigenvector mention table pc loading dc pcs capture cardinality loading pc non loading capture pattern non loading total loading capture cumulative cardinality explain loading addition c show dc similar variation pc compute dc plot summarize compute dc pca setup assume curve variance cardinality vs cardinality computational size explain various induce regularization case pca increase vector decrease display computation see dc pca perform perform well
equilibrium population consist therefore variant player one player player bit choice quantity length bit feasible interval quantity ensure nash equilibrium prove induction equilibrium quantity always value ensure homogeneity population choice make choice accordance size etc run game social number time simulation equilibrium game estimate lead estimate give ne strategy one trajectory market quantity calculate figure value respectively ne unbiased deviation evolution see estimator player c c c vi player individual parameter run parameter testing player correct play hypothesis accept quantity polynomial model see player quantity value individual player cm p cm individual co alg co establish ne play nash ne quantity subset play strategy co co mutation population ne within mutation population require ne state course population differ less bit nash matter high calculate unbiased ad expect state arrival ne two heuristic ne potential equilibrium cs state end identical equilibrium qualitatively potential game since ne attention game play order check quantity one quantity heuristic depend complexity investigate player quantity lead nash equilibrium quantity mutation number inside prove statistical heuristic equilibrium ne belong finally social version algorithm allow multi environment simulate bit length population bit encode feasible value therefore model especially ne finally could theoretic ct like thank la h evolutionary stability behavior iv rao statistical process economic potential stable genetic optimization read van company tc co evolutionary market behavioral stability genetic evolutionary games equilibrium optima http www individual learn co evolutionary genetic learn version co establish nash contrast version see player strategy nash outcome player canonical evolutionary methodology general statistical find social ne play chain individual case ne simulations ne ne large indeed nash equilibrium game theory nash equilibrium two quantity turn price quantity market evolutionary study classical genetic optimization evolutionary version genetic evolutionary player determine player choice evolve goal dynamic system consideration player represent population algorithm strategy fitness establish play define active quantity player market quantity total quantity lead player fitness dependent previous co establish fitness proportional player produce price determine quantity demand algorithm update game converge nash agent determine tend nash strategy co evolutionary price quantity market see nash multi article fitness calculate game population current population pick give round update usual genetic operator mutation regard equilibrium quantity market prove finally population single order current fitness choose mutation lead market ne identical function game study et al decrease game pseudo potential game genetic therefore discrete principle course dense strategy ne game ne coincide investigate nash pure strategy case give player linear cost q decrease finally demand represent alternative choice genetic algorithm individual lead ne consideration introduce characterized opponent generation population create algorithms version player take generation form pool usual genetic operator mutation generation aggregate finally player population idea player pseudo represent player ga mutation player create realize fitness correspond evolutionary programming population population update take pseudo follow player decide fitness repeat strategy assign apply selection mutation keep implementation use selection proportional fitness mutation mutation rate bit classify ensure nash equilibrium introduce social algorithm population transform randomly draw player ga single mutation consist union population copy correspond population social evolutionary initialize player strategy assign decide value strategy evolutionary operator mutation union player copy new generation repeat difference social aggregate generation form economic choice update strategy learn since economic genetic allow player opponent every population assume k ij nj condition selection fitness scale solely
various justify reversible chain subsample asymptotic efficiency behaviour central chain f construct mean argument interpolation precision decrease estimator median space estimation uniformly ergodic chain bound hoeffding available lead bound functional consider technique result rigorous result cl truncation directly bound analogous sequential identification time difficult implement approach know towards small integral unbounded section parameter drift parameter design total elementary median multiple run illustrative toy emphasis unbounde ergodic practically drift apply particular effect paper interest measurable state homogeneous kernel interest mcmc walk transition qx g e function define evaluate variation precisely norm sequel geometrically ergodic chain markov say chain ergodicity drift define drift towards small satisfying sequel drift type condition assume geometric ergodicity definition ergodicity allow relevant devote c total sequel make ergodicity convergence drift explicit convergence unique establish appendix explicit improve ergodicity establish nx inequality hence theorem define bind without effort error essential confidence main drift constant explicitly corollary particular motivated quickly inequality confidence lead trajectory take chebyshev term roughly autocorrelation bottleneck somewhat improvement e burn approach measure appropriate precision typical bound imply therefore chebyshev get define good minimize calculation complete trick complexity need general odd random p k algorithm average markov chain base run estimate addition cf illustrate chain parameter drift k dy indicate reversible reversible relationship point reversible formula confidence possibly unbounded mcmc example starting compute lemma c total one walk e e e p uniform ergodicity optimize one walk find total reason bottleneck short run significantly effort long mathematically tractable reality estimator phenomenon infer asymptotic function compute result available http www ac uk message chain reach relevance unbounded function priori often practice visual look bound burn use possible derive conservative total drift remain universal tool obtain markov bound tight difficult convenience reader repeat sequel term refer respectively eq unique lr markov reversible dx reversible chain say reversible thm proposition thm thm remark partially education uk drift
would inclusion order covering interval interval partial let show quantile partial acyclic direct direct illustrate acyclic quantile comparison relation relation arc surface statement section dimension partial index vc whose e condition drop finally uniform van theorem display index estimate quantile look quantile fig show light quantile slow bottom right corner correspond point comparison fall diagonal quantile evenly space instead near minimum grow large estimate quantile monotonicity expand estimate unit square computing interval quantile rate illustrate quantile fig violate coincide except modify eliminate monotonicity dispersion quantile region boundary dispersion region extend concept quantile involve detailed graphical quantitative summary information regard c collect department periodic survey use formulate education consumption define run daily et propose approach quantile focus separately two protein gram partial situation usual sense comparable recognize extremely undesirable yet partial positive correlation alignment dependence important designing moreover quantile preserve transformation multidimensional protein subset survey use diagram fig align scatter diagram monotonically increase expect comparable people emphasize nonetheless interpret univariate quantile derive policy program maker quantile reasonable comparison quantile quantile partial quantile quantile reflect intuitive interpret quantile multidimensional quantile table quantile whereas generate quantile c ptc ptc index level give estimate color partial quality comparative quantile surface level hand light think well surface comparative need stay quantile estimate partial protein boundary rough symmetry surface location partial quantile figure figure dispersion def end finance literature central return return asset asset arise exposure intercept adjust yield return zero finance see reference market return break negative capture note well capture partial fig partial previous align b ptc quantile comparison surface narrow probability quite everywhere quantile monotonic quantile monotonic note fall support datum partial quantile strong true interpret result choice dominate slight performance dominate datum explain target ideal trade near datum extent comparable project effect school social media intervention pt yes yes partial maker pass fail regardless cost political consideration social cc media intervention cc cc tv acyclic direct probability pass partial value quantile similar traditional quantile outcome tv make partial propose quantile multivariate order important partial might space include robustness preserve partial regard discuss particular important order order additional partial quantile crucially depend heavily concept link application achieve quantile desire partial probability partial partial etc type application concept tie partial order quantile instance partial possibility partial surface covariate surface concept censor wide attract quantile suitable apply censor multidimensional motivation connection axiom preference allow identification decision partial quantile strategy valuable although pursuit scope believe future event proposition mapping mx mx x mx px x partial quantile px px px px lemma establish compact probability associate measurable measure vc vc therefore measure support dx e van singleton argument vc ensure van building upon p fx pointing let k fx k cone interior vector positively therefore continuously differentiable implicit twice differentiable compact condition continuity continuous follow union condition cardinality take note trivially follow piecewise constant mapping jump include jump p h eq theorem proof proceed quantile derive step pt x x definition feasible identification optimality x p dx x nu yield leave hand side use ii relation dx corollary complete satisfied convenience w p x n p x x p p w get multidimensional central simplification x follow corollary build space p g pg px proof satisfy p n x x p note proof since every thus dx dx x combine differentiable since strict minimum interior every p pt proof transform indicator see proceed modification fw iw iw restriction analytic zero open nonzero borel contradiction vanish nonempty open bx convolution everywhere turn us contradiction proper cone proof generality proceed connect separately general already px yy terminate individually strategy previously similar x du du u u du proof theorem point jx j jx jx df jx j j optimal optimality feasibility j take note variance around z f p q vary nonempty interior strict trivially assume point concavity pr show show def constraints optimal contradict lemma x arbitrary integration log walk construct membership control oracle construct oracle thank comment thorough version grateful suggestion type style e e proposition focus generalize preserve outlier characterize distribution partial sufficiently rich generalize concept partial perspective partial quantile furthermore procedure might establish complexity concentration natural order finally discuss several impact policy concept quantile prove notion robustness quantile role interest counterpart attract survey recent focus develop measure usually suitable nest partial instead incorporation work focus fundamental difficulty reach agreement suitable quantile arguably lack multidimensional various quantile character quantile acquire loose usage page simple multivariate quantile quantile fail multivariate feature attempt feature influence exploit metric multivariate quantile notion gradient univariate quantile variable away quantile relate notion assume minimum key insight rely family induce lack distinguish partial analysis different detailed discussion section definition framework generalization quantile study quantile mapping instrumental case applicability relation convergence infinitely many quantile accommodate restrict identification difficulty convergence partial index subset index probability study probability distribution quantile due quantile could curves context partial lattice new point monotone upon estimation mild possible curse dispersion partial moreover interesting primitive finally illustrate application evaluate within quantile imply value probability arbitrary set point compare comparable define x xx derive relation order relation exposition imply iii binary partial comparison draw point comparable usefulness fact xx xx sensible involve probability draw precede order index partial quantile define associate quantile univariate simply quantile would univariate quantile representative quantile quantile use maximize comparable partial quantile partial complete exploit quantile surface geometric multivariate well quantile occur partial balance correct comparable good approximate maximizer probability consequently interpretation quantile allow partial traditional characterize overall minimum quantile partial quantile considerably note def write saddle move imply definition notable interesting mapping preserving imply invariance preserve value quantity quantile surface comparison quantile preserve transform partial common invariant translation require symmetry distribution every partial quantile automatically quantile relation interest explore quantile view counterpart empirical let na nb carry notation depend notation level imply primitive discuss impose regularity induce eq nx px condition partial behave requirement condition imply primitive several primitive technical three considerably weak however lead sharp treatment derive mind partial quantile contain positive quantile nr identification partially identify spirit quantile continuous singleton convex criterion estimator van functional mild index weakly directly mild verify main example cone order nonempty case setup dx probability map acyclic ex partial describe direct direct acyclic sampling hold hold example estimation partial quantile order estimator univariate lack restrict achieve result quantile drawing comparable quantile interest whole framework huber partial reasonable highlight difficulty quantile rare might completely eq create ambiguity choice quantile probability comparison quantile surface partial quantile uniformly uniform intuitively ensure likely point estimator slack go comment aim ensure nonempty associated pt x cone partial describe sufficiently general suffice simplify affect could quantile practical case underlie bring measure estimate might application metric rely avoid application moreover need need account underlie motivated develop monotonicity curve quantile greatest refer join meet close construction base partial monotone scheme cone x monotone previously derive improvement conditional quantile always monotone assumption side therefore corollary lack partial point quantile quantile reflect independence carry random variable independent let partial value variable quantile norm eq partial close univariate quantile quantile phenomenon decrease contrast extreme comparable advantage soon concentration curse dimensionality comparison positively less align order perfect correlation transformation positively correlate trivial univariate quantile surprising concentration measure exhibit concentration quantile value order quantile logistic zero mean extreme measure grow likely extreme quantile surface close connection mass corner corner notion quantile surface connect order definition surface allow generalize efficient random realization interpret quantile surface partially parametrize probability draw comparable interest partially efficient close partially efficient might quite appeal support univariate dispersion measure quantile traditionally dispersion median expand quantile interval eq dispersion variable I shift interval region moreover quantile specify quantile comparison quantile typical partial quantile quantile help characterize dispersion quantile contain surface constrain unbounded region complete coverage apply go whether computation efficiently literature mr tie regularity relevant object object could report alternatively distribution quantile evaluate might problematic cardinality moreover emphasize regard discretization suffer curse requirement large surprising case cube unknown px x exponentially ever cube extreme arguably explore regularity representation probability allow draw relevant regularity let concave every every cone nonempty interior particular concavity convex concave achieve representation carlo chains survey assume evaluate concave cover many case interest illustrate condition partial order practical equal cone equivalence considerable condition equivalent problem due concavity change convex programming membership available maximizer efficient anneal power objective rl step p iv x independent random run empirical maximizer membership approximate walk simulate use construct follow evaluate
consequence stage identically variable suggest epidemic branch al non negligible major number later limit epidemic individual individual refer arbitrary infection let period intensity contact contact occur define variable follow activate without activate period correspond contact period stop recover former happen person person integer imply person get infected period start contact activate already infect nothing happen contact activate contact process period select individual already infect go contact period stop finite probability straightforward check desire period individual rate homogeneous branching process constant birth variable whole branching agree point detailed branch branch individual apply epidemic infect except contact epidemic already infect consequence epidemic branching agree contact infect epidemic contact infect contact equal contact contact branching process epidemic agree contact contact epidemic branching agree precise ball epidemic branch approximation branching study e instance length birth quantity previously branch total ever branch branching grow beyond well branching epidemic coincide space branching go arbitrary special opposite branching individual go happen go denote initial birth event satisfy birth life length course duration poisson process must hold third use transform e initial epidemic approximate homogeneous birth life epidemic whereas approximation epidemic branch correspond epidemic epidemic initial treat period parameter e epidemic small mean major life total rely individual infect branching approximation show epidemic approximated branching individual goes never happen large enough branching grow beyond limit implicitly question course happen epidemic something outline elegant interested infect initially call infection pressure contact multiply period community individual infection pressure become infection uniformly thus increase accumulated infection pressure epidemic stop distribution lie infection accumulate embed process straight process make obey perhaps something expression final infected probability infection period equal approximation approximation fraction infect equation negligible minor unique plot else rigorously summarize minor therein von consider epidemic initial sample space square variation illustrate epidemic start initially look subsection conclude equal equal conclusion simulation minor see distinction minor major theoretical look course hard seem agree previous question occur interest epidemic eventually sketch minor focus major hence study time depend see epidemic branch approximately individual infect branching infect epidemic counter arbitrary fraction initially grow decrease epidemic already infect epidemic behave average follow branching duration plausible duration epidemic result technical many modelling disease model attention last decade attack way come severe measure restriction contact e particular disease somewhat change reduce make suppose available fraction individual initially contact equals want hence instead epidemic approximated branching birth mean life denote conclude major approximately unique central limit major total infect normally state apply point impossible fraction surely prevent example prevent whole far give parameter epidemic stage stage mean individual infect vast quantity define epidemic enable error illustrate epidemic infect individual let fraction satisfy normally distribute around standard deviation asymptotically normally deviation delta e cox asymptotic square root replace quantity variation impossible infer period proceed available example critical natural normally variance delta community infect upper estimator normally distribute make number epidemic continuously precise refer standard stochastic epidemic stochastic epidemic finite randomly time intensity choose despite model contain simplify behaviour spread quantitative epidemic individual period reality population heterogeneous social infect quite nearly age gender experience define epidemic individual individual close give population epidemic branching attention reason imply average individual individual secondly simple whereas occur threshold limit state number infect epidemic take involved proof desire refer perhaps term uniform epidemic previous include uniform sense assumption contact specific individual situation tend mix epidemic behaviour epidemic common epidemic individual group assume contact pair individual contact individual individual period rigorously et al infect infected treat approximated branching process refer biased infect major happen et number equation additional derive central another epidemic contact call specifie upon epidemic assume frequent cycle stochastic epidemic open problem solve influential disease highly disease edge correspond focus sir epidemic allow enter behaviour sensible approximation disease g certain disease country another potential disease brief outline refer reader study sir epidemic model dynamic individual rate exponentially state size markovian epidemic individual contact infect immediately recover become life individual irrespective infection status epidemic jump enough individual play roll question property large correspond get equilibrium show disease equilibrium basic number individual typical stage leave recover disease stable equilibrium call stochastic reach state fact state stationary individual epidemic question size go population influential important present epidemic reality complicate affect spread list effect play roll dynamic reproduce high weather perhaps social school start event effect al transmission school transmission school present spatial increase dramatically last spatial component modelling study disease always take et al individual decay main epidemic growth assume infection think product contact transmission contact period first infect often eventually activity start drop dynamic long epidemic growth period equal infection infection process activity length complete reality rarely response et et reduce compare effect infection higher critical true call rest something efficacy course enough make often clinical usually hard reduction since actual rarely book discussion trying include realistic model completely predict happen situation nearly people adapt behaviour disease say health measure reduce disease epidemic minor modification area spread g recently wide epidemic even use terminology disease spread give introduction big mathematical disease spread probably spread come epidemic classic book cover acknowledgement grateful financial cm plus minus survey paper model simple stochastic epidemic property model critical towards epidemic disease coverage epidemic epidemic epidemic threshold early disease aim evaluate model rigorous make contribution consider early question big fraction community epidemic fraction arrival model generalise way individual equally example spatial deterministic epidemic preferable study community aim epidemic intervention stochastic advantageous contact contain graph network say epidemic role present epidemic epidemic model close stage behaviour epidemic epidemic duration epidemic main summarize assume early epidemic branch birth branch epidemic branching grow beyond limit happen determine final infect divide three beginning fraction place end infected order answer study describe many epidemic realistic complete guide contribution epidemic define epidemic property insufficient end call sir epidemic approximation rely community generalization epidemic model one deterministic epidemic g individual individual get infected individual remainder assume period community consequence assumption move I say sir epidemic individual sis model infect becoming call call allow refer sir close recover effect receive contact infect rule individual remain become distribute also contact agree epidemic start epidemic evolve new individual infect community imply epidemic
multiscale interaction borel regular function measurable integrable area simply weakly road regular close also easy restriction configuration weak union weakly thus measurable argument weakly clearly integrable note x integrable derivative dominate integrable integrable strong generalise absolutely continuous respect poisson perfect feasible neither purely multiscale model correctness monte rarely simulation reach solve introduction perfect spatial example attractive rigorously check burn error estimate identically reduce unfortunately response require evaluation literature theory exception simple define process interaction mention capable modelling cluster regular suited whose vary multiscale area either demonstrate sample simulation coupling move dominate justify perfect area describe perfect simulate infer datum discuss suppose desirable finite go run return chain chain let minus infinity question two chain couple stochastically whenever stochastically maximal minimal chain bottom state ingredient coupling although continuous state space notably section still truly general high moderate dominate coupling coupling past soon type space poisson impose constraint sample poisson evolve birth death configuration step mark refer initial process evolve birth death happen accept point however happen remove event mark keeping involve expensive costly version calculate dominate partially unique partial replace preserve equation wish density monotonic respect induce intensity poisson dominate process mention modify step requirement point parameter compact reduce homogeneous randomness cluster unfortunately allow place sort distribution nuclear force particle law model follow type radius markov range scale scale van standard multiscale write functions intensity substitute find dominating process simulate evolve time point whenever dark either amount accept whereas right little add maximum intensity pseudo arbitrary subset observe integral side note quadrature weight use extend likelihood approximate q pattern log independent generalised order procedure must nuisance fit value narrow different moderate large brief look set consider scale seem scale rather dotted give envelope simulation model solid dash line simulation choose remarkably necessary small logarithmic carlo function fit thing firstly fit reasonably choose slightly would secondly seem scale
preserve neighbourhood essentially well singular say somewhat degenerate bilinear index dimension resp product euclidean compact maximal n k kk geometry special algebra n k complement n intersection know decomposition via element except possibly wish matrix project define totally real distinguished property totally plane plane well hamiltonian mechanic subspace subspace conjugate transpose euclidean transpose unitary act action real observe I n ng see admit factorization totally conversely totally admit degenerate hence r symmetric admit maps n c nk simultaneously unitary transformation validity position n h transpose homogeneous addition lagrangian natural embedding obtain eq define totally real arise mechanic compute neighbourhood diag k consequently sum one geometrically amount totally isotropic uniquely degenerate neighbourhood nj orientation preserve conjugate structure complex structure conjugacy ng nx thus neighbourhood unitary homogeneous space compact tr ad let lie one know neighbourhood formulate term ng one obtain decomposition map neighbourhood deal despite minimax geometry construction similar regressor manifold embed density unknown unknown say discrepancy function smooth iff hessian euclidean intrinsic riemannian one parameterize dimensional highlight result differential topology fr manifold fr manifold hausdorff topological coordinate fr canonical define compact manifold lie functional determine geodesic assume second way positive definite definite along subspace vanish riemannian smooth minimum orientation angle compute onto formally regressor regressor canonical particular vanishing know differentiable translate minimum satisfie differentiable set iff resp triangle sr euler regressor euler angle draw increment computation solution histogram normalise euler angle regressor report kolmogorov normality angles regressor euler ccc see text information htb normality introduction section fy quantitie q dependence omit notational therefore measurable smooth r let fix bayesian estimator satisfy proposition observe hand integrate value single embed norm q bayesian due embed leave prove euclidean space inspection hand restriction geometry semi definite let defined transformation thing prove non degenerate span scalar multiple identity matrix projection dim riemannian manifolds tangent shortest smooth bad lie neighbourhood uniquely map integrable define riemannian measure curve cc ts clear figure discussion exist proposition examine application unit orientation preserve normalise haar logarithm part integrate q equal investigate theorem joint vanishing introduce orthonormal one rotation plane vanish vanish multi euler angles maxima compute vanish euler tr mat mat else mat mat euler j theorem comment thm thm example question thm thm compact manifold euclidean space riemannian manifold white smooth smooth bayesian technique variety second equality problem orient manifold boundary map observe via white estimator map second asymptotic technique variety geometric tool early applicable one observe plus geometry state estimator example geometric naturally extend regression observe state map observe output attempt infer belong transpose regard observe state may evaluation commonly geometry situation real dimensional inner resp riemannian riemannian manifold embed inclusion infinitely summarize diagram open neighbourhood orthogonal projection basic geometry compact smooth map space vector transpose row e minimax map riemannian minimax one approach determine view risk state riemannian permit derivative hessian denote curvature manifold random assume conditional point interested admissible parameterize regression regression assume volume functional regressor examine prove special map determine structure determine riemannian induce special sphere group preserve linear detail section dimensional denote e es linearly basis orthogonal latter euclidean unit transformation space inner denote naturally pass tangent typically call lie algebra equip lie denote element observe g call adjoint know ad trace adjoint representation act subspace denote complement similarly bundle bundle g fact neighbourhood open neighbourhood see neighbourhood analytic whose suffice g g g neighbourhood intersect figure value neighbourhood globally lemma consequence neighbourhood many linear decomposition neighbourhood introduction transpose dual induced euclidean inclusion ef f v derivative impose act attain haar haar measure act isometry note independent product flat invariant flat dim us dimensional sphere orientation pt orthogonal v plane alpha alpha alpha alpha alpha curvature unit
dash generate segmentation affinity dash segmentation algorithm combine classifier pixel weight tend belong compute affinity edge patch remove connected affinity segment image segmentation algorithm misclassifie dramatically segmentation split merging optimize rather affinity sophisticated spectral cut simplicity direct supervise optimize segmentation graph partition possible sophisticated still prefer great learnable rather hand design clean affinity graph spirit large assumption sophisticated partitioning segmentation way classifier segmentation special cluster similarity clustering recently rand define segmentation assignment pixel belong pixel fraction rand fraction rand similarity dissimilarity rand segmentation truth function rand sensible incur huge truth classified lead pixel segmentation affinity rand index affinity rand index affinity pair binary correspond belong rand index incur pixel must connect vice segment incur penalty penalize work thresholded affinity let train classifier relate indicator classifier characterize whether two connected thresholded affinity graph affinity pair affinity affinity let graph path affinity affinity affinity path important pair pixel thresholde affinity exceed pair thresholded graph path affinity minimal affinity affinity consequence affinity connect connectivity q efficiently span maximum sign maximum span tree neighbor affinity affinity number grow share image perform time edge classifier rand q replace continuous cost suitable operation differentiable continuously gradient edge choose ji affinity neighbor pair near function pixel adjacent affinity speak cost similar affinity cause incorrectly classify gradient often affinity pair neighbor pixel randomly w pair near draw w w w learning pair pixel image gradient weight pick train affinity edge pick train affinity integration neighbor affinity graph connectivity decision distance train train decision truly learn superiority affinity compute local affinity train computational brain identify diagram brain piece brain cubic brain image advance making collect image remains require accuracy serial block volume voxel train affinity convolutional restrict convolutional network previously standard second map sigmoid lead affinity cubic patch classify function lx affinity slow affinity predict proxy pick overlap sub original significantly image train less pixel total iteration measure correctly pixel curve recall quantification classification segmentation classifier pixel connectivity rand classifiers spectrum lead threshold rand image pair connect pixel reflect rand index imbalance affinity comparison instead precision quantification imbalance observe affinity affinity performance connect component standard learn affinity poor segmentation mistake merge segment contrary properly thresholded follow connected image segment miss segment merge merge cross neighbor boundary affinity affinity graph result partitioned graph thresholde key segmentation cost function affinity segmentation fast contrast base segmentation dominate simple proportional graph number segment size time partitioning connection linkage span ultrametric partitioning minimum resemble segmentation part ultrametric map algorithm generate hierarchical identical vary threshold graph incorporate improved affinity acknowledgement medical foundation max medical research max h predict affinity reflect image partition segmentation learning affinity affinity relate ultimately present affinity graph produce directly rand well rand quantify pixel segmentation use graph component
density conjugate dirichlet bernoulli corresponding policy mdp dynamic first function mdp follow policy hoeffding bounding estimate iterate follow probability implicitly belief root whose high expand either branch take let discount cumulative reward finally leaf specific branch bound simply heuristic algorithm shall employ upper low bound expansion perform start node among sample leaf child serial nearly balanced leaf expand expand lead unbalanced tree lower expand expand currently high expand ci less branch long agreement appear aim interesting rl node heuristic expansion result difference complexity expansion try exploration apart perform sophisticated depth simply optimality importantly observation space continuous reward also include stochastic one mp upper another interesting perhaps line successful problem acknowledgment denote mdps specifically state pair dirichlet count transition simply product transition mdp pair transition transition statistic way exposition mdps produce employ optimality increase planning programming infinite state employ regret present strategy explore compare ucb recent probabilistic mdps observable decision exploration firstly compact current secondly augment mdp child possible subsequent belief large fast problematic grow concentrate expand tree whole trade optimal computationally recently extend propose special structure belief order node look method tree branch help idea multi armed optimal algorithm already introduce mdp formalism discuss tree expansion introduce evaluate development interest agent seek expected simply discount arise markov decision process define tuple comprise transition satisfy state shall define value infinite mdp uncertainty f essentially maintain order optimally select suggest bellman name mdp comprise solve shall call mdps analogously bayes adaptive density I belief mdp augment mdp measure reward jointly mdp transition component hyper abuse shall horizon future action eq hyper step clear continue expand belief observe bayesian utility current future expand tree exception wang sampling use thompson thompson expand expansion closely therein branch bind also upper propose structure belief upper value expansion property propose combine bound leaf experimental bandit believe important towards applicability look ahead exploration current suppose observation together mdp transition recursively belief unbalanced hope fully expand tree especially true continuous even far computationally horizon expansion largely use reduce approach search leaf leaf strategy child formally wish expand space reward deterministic deterministic enumeration reward
intersection correct invariance problem deterministic broad class program generator polytope combine construction generator explicit form intersection regular constant depend constant remark gaussian counting quasi algorithm counting program let program exist deterministic run estimate program program contingency table regular program within hypercube quasi polynomial obtain integer solution work count solution program algorithm give task integer multidimensional run difference give strong far state invariance whose bound hyperplane regular however polytope hyperplane symmetric space universal additionally invariance sphere modify intersection intersection spherical explicit uniform cut similar hyperplane randomize box specific study play reference generic tight recent develop lee theoretic notion cover connection also give generator oppose good give outline invariance principle proof proceed two first replacement hybrid literature intermediate prove invariance prove normal coefficient smooth fourth derivative notice polynomial univariate wise construct framework bound hybrid bad form intersection regular hybrid group block irrelevant order suffice hybrid argument intuition randomly proceed partitioning smoothing well roughly speak contrast argument small turn analysis block construction translate closeness smooth closeness smoothness become test multivariate version test et rather invariance hybrid fourth p q small fortunately construct final closeness closeness differ end surface well intersection proof invariance obtain fourth suffice smooth construct replacement sequence principle low polynomial al proof involve principle smooth cdf behave involve prove smooth supremum regularity influence hybrid argument expand error fourth step principle smooth approximation cdf smooth approximate fourth uniformly smooth cdf everywhere small obey anti state variable low polynomial obey anti complete proof try invariance characteristic polytope equivalently logical instead wise polynomially face ball lie polytope grow combine fashion derive dependence exponentially step ask dependence improve lead read principle reveal obtain invariance variable obtain derivative approximation quantity outline ip column define face polytope uniformly small polytope invariance might face polytope true improve w w ip back outline invariance principle one step irrelevant suffice replace choose block replacement block invariance ip ess quantitative central give invariance principle single ess multidimensional ess identity matrix deal implication history approximately count solution especially regard contingency however much term algorithm deterministic covering problem cover kind regard contingency give run quasi relative error approximation state explicitly machine contingency table time algorithm contingency case box contingency table intersection al give al generators intersection seed bound generators seed intersection least yield set bound regularity lemma regular use handle strong difficulty reduction case use face unless section principle functions smooth show closeness smooth anti variable imply closeness lemma anti concentration variable finally anti concentration gaussian concentration let eq follow function proceed therefore straightforwardly surface denote element polytope face tx prove explicit hash family set however wise analysis complicated family construct form lemma regular wise indicator moreover inequality moment equation therefore equation careful outline detail hyperplane orient surface hybrid hyperplane disjoint hyperplane conjunction series reader find smooth p taylor first function hybrid choose similarly view replace generality variable form lastly taylor q equation follow sum essentially regular rr fact regular note factor intersection w kf first bound volume neighborhood invariance bound boolean boundary theorem agnostic constant subgaussian constant follow say regular perturbation perturbation k w ip union eq get sufficiently perturbation random flip bit see anti regular follow directly imply k applying immediately result namely main generator result construction polynomial et reveal construct principle observation ask invariance regular different careful application begin used family hash family know avoid technical easily generate construction generator generator consider generator argue generator regular argument indeed independent rely hash independent block word generate involve consequence independent move closeness cdf argument equation enumeration possible seed immediately time small integer turn regular program broad dense cover contingency quasi polynomial time algorithm contingency nontrivial algorithm program class integer aware approximately program notable counting count contingency count dense count contingency table contingency r c c note integer program proper result bound generator appropriately generator ct ks string get deterministic run dense cover get count table cover program negative important integer program combinatorial standard set universe give dense least linear appear dense constraint continue regular seed generator dense set problem constraint universe approximate additive time long elaborate approximately contingency contingency positive integer wish solution whose matrix appropriately correspond lie notion dense instance count number contingency contingency discrete cover instance contingency contingency c c ks moreover proper contingency quasi polynomial set prove invariance unitary rotation spherical hyperplane prove rotation bound hyperplane tail require apply let normalize hadamard entry probability observation diagonal wise fix wise later l dx observe degree multilinear markov distribution constant generator recall invariance unitary rotation thus regular invariance principle use area polytope faces distribute fix later distribute apply distribute two follow prove later sufficiently follow equation claim generality odd acknowledgment thank integral david section corollary conjecture spherical gaussian possibly intersection e dependence least prove theorem elsewhere important application invariance boolean noise sensitivity intersection regular give agnostic learning intersection seed length hypercube construction obtain algorithm program dense cover contingency table computer science problem analytic spectra sensitivity fundamental tool prove hardness proving conjecture hardness problem notably max principle relate uniform gaussians invariance multilinear multivariate parameter depend small say cdf polynomial cdf polynomial coefficient invariance generalization powerful hardness hardness choice boolean principle widely understand cumulative invariance principle possibly unbounded intersection support refer regular regular invariance regularity threshold main invariance invariance principle generally
maximum extract instance build reader familiar security concept depth converge dominate attack rational select valuable optimal path construct eq algorithm construct budget single cut cut minimum small attack surface start vertex boundary organization internet interior many security depth depth model system attack version center network depict figure receive server receive front web server front server large attack surface database server server interface application whereas database budget try sensitive database e rational budget unity budget right database attack end achieve unbounded attack analyze security playing alternate select select vertex yet bound competitive good via literature literature include play boost machine heuristic although extension apply online management formalize game round choose attack budget sense security principle make accurately allocation exist knowledge black reveal attack edge post function sequence edge notice beyond round column attack algorithm edge path attack already reveal reveal edge notice vertex begin knowledge graph update point update attack ever sum unity round allocation move attack small budget appropriate strategy compare online relate ta surface theorem establish well carefully construct attack principled strategy attack appeal measure cost result competitive converge cumulative abuse slightly singleton e know entire remove know vertex competitive ratio optimal proof well perform every inaccurate far optimal reason instead learn observe reward without knowledge payoff difficulty limit star system equal attack surface reward little equally indistinguishable rational number leaf course ratio bad vertex rational learns attack another way mistake matter rational actual whereas variation system figure budget rational budget edge view edge near unity rational perfectly rational room situation attack might knowledge attack software security equilibrium al discuss pareto improve security accord al optimal security division loss due lack generalize modeling et highlight theoretic theoretic due adversarial setting applicable nash equilibrium security scenario arguably abstraction detailed adversary readily equilibria many security expert ignore principled adopt management competitive attack never actually occur although abstraction support make security employ monitoring tool analyze attack tool effort attack security build roll quickly detect exploit determine security recent attack discount attack exponentially security security ill suited attack round learn attack conclusion security always reader appropriate part strategy significantly like acknowledge support nsf dms program establish know second hypothesis know advance full specifically surface reward round make output ta strategie lemma regret run mp lemma matrix add entry game sequence obtain respectively produce allocation sequence game game thus b yield ta ta bt online assume row advance relax perhaps allocation subgraph induce reveal visible letting actually round output identical round unity correspondence ability ta ta consider edge budget allocate time budget allocate thus instantaneous definition optimize reveal ta te receive immediately enjoy complexity constant bound quantity large technical ta ta b feasible eq b prove allocation attack reduce attack es de maximization allocation ready main equivalence convert ta ta imply ta b ta ta ta ta yield ta ta b ta ta briefly graph e converge attack path past vertex vertex edge attack ratio attack select attack iid round attack every entire cost coin attack allocation well frequently well probabilistic must attack lemma ignore try easy algebra conclude generalize graph small thm thm science superior security security security learn past attack security bad act competitive ratio inspire unlike robust security security risk security last typically perform benefit identify risk constitute strategy conventional look risk examine exploit security security attack study efficacy economic security benefit trade economic act strategy security make knowledge require make conservative knowledge assume know perform single meaning attack consistent say business meaning without security team rule improve access control single act attack attack make choose study patch security might portion budget web token make patch interaction place patch viewpoint evaluate cumulative study metric seek attack instead receive performing say business show competitive prove theorem technique learner know let situation result clause allow although likely strategy strategy gradually edge construct effective implement capability less budget attack actually encourage management inherently arise naturally bound security security generalize clause relate related work conclude theoretic attack capture security sense situation include internet direct graph define graph represent represent induce another might machine two system second select begin result general clause discuss generalization think attack drive edge include server connectivity back form make spam reward derive drive
choose please bias column fix contrast choose unconstrained initialize draw htbp cc high htbp glm show simulated dm snr glm dm show data bar unit deviation attempt optimize emphasis relationship importance bias contrast leave choose potential dm augment dm unbiased sign change unbiased pure solution cc unconstraine dm dm snr nan dm dm unconstrained shape curve fractional bias gauss markov bias glm analysis optimize simulated dm glm dm simulation generate dm bar represent deviation variance block application consist shift response maximum user want bias entire paradigm weight dm design shift figure v htbp run optimize dm c example dm snr glm dm also glm snr illustration purpose plot bar standard deviation deviation quantify response impulse response voxel illustrate enable simultaneous capture plausible shape plausible shape shape half parameterization use parameterization give plausible shape generate uniform automatic initialization describe result capture glm analysis capturing enter dm glm capture underlie optimally unconstraine dotted optimize solid fractional r gauss variance optimization glm analysis set optimize dm example fit dm plot dm error bar represent quantify detail scan half mean hour terminal half al scan baseline minute ml rate control automatic collect parameter tr te flip angle slice slice volume acquisition acquire rapid gradient tr te ti fa slice slice library volume mr instability volume raw fmri correction mm automate perform map minimum identify pattern co probabilistic pursuit test analyze dimension reduction latent via decompose base core use constrain demonstrate subject htbp produce spatial drift find figure response covariate drift contrast multiple profile baseline could general response exact easily model expect brain structure seem delay response subsequent inspection stage capture represent pure drift purpose optimization potential dms capturing delay let dms dms dms dm process drift pure negative linear drift absence response drift conservative snr fix response htbp constrain leave automatic dotted bias example htbp glm optimize dm c roc curve dm bar quantify computation dm glm analysis represent derivative figure result validation maximum optimization size dm equal potential dms value unconstraine initialize maintain around contrast gauss monotonically imply gauss estimator model maintain curve automatically optimization column example compare contrast bias increase maximum monotonically imply maintain column unconstrained initialize uniform maintain decrease monotonically becomes indicate gauss maintain use unconstrained dms augment explicitly enable signal presence nan decrease monotonically four variance bias illustrate user want bias indicate primary primary reduce around increase gauss maintain capture enable explicitly indicate nan signal optimal dm find optimal dm unconstraine find contrast bias maintain mean maintain shape maintain signal fmri reveal profile take baseline initialize section column column estimate dms dms dms presence drift signal dm absolute model roc generate drift choose extreme produce absolute contrast rule roc cutoff residual dm specificity snr specificity real approximately primary response specificity detection correspond fmri illustration purpose sample detail fmri partial obtain illustration full partial fit temporal derivative covariate delayed response delay point delay find dm delay unbiased produce dms snr match size specificity sensitivity real fmri snr cutoff opt cc e figure dm dm cc st show partial fit figure dm dm htbp temporal derivative full interest figure optimal dm objective theoretical enable design fmri analysis framework allow capture potential specify weight occurrence various design associate validate numerical sophisticated test column design selection algorithm bias illustrate study control solution algorithmic fmri come model shape maintain bias variance capture amplitude optimize temporal derivative bias reduction optimize comparison temporal good next examine property glm design find variation basis recommend fmri shape relate notable shape allow presence include amplitude analysis amplitude never approach control capture example shape amplitude objective instead propose potential explicitly objective result capture amplitude pe glm optimize dm column rd column plot dm glm use analysis simultaneously signal glm optimize develop motivation fmri general variety quantity denote identity eq consideration small combine apply repeatedly term simplify r combine note compute problem constraint slack replace constraint slack equality solve lagrangian define eq subproblem tucker kkt optimality penalty update base feasibility monitor sufficient accuracy subproblem point choose try px k cx iteration augment sub use gradient quasi update recommend recommend convex switch newton framework trust update progress monitor b xx bfgs sr limited bfgs limited sr newton non projection vector elsewhere truncate newton cg cg inexact cholesky bfgs sr bfgs sr quasi newton even find describe plausible shape half david glm tool analyze functional fmri fmri univariate fashion analyze voxel limitation vary nature well potentially main develop set design validate numerical fmri implication ability match signal magnitude also size thereby specificity enable capture multiple profile interest oppose optimize enable pass estimate variance group fmri capture fmri fmri fmri quite wide applicability fmri analyse design dm glm stimulus paradigm explanatory variable dm pe fit define meaningful brain paradigm fmri voxel brain glm fmri univariate mean dm voxel pe glm gauss exact mechanism fmri fmri signal possible effect signal interest region profile correctly dm voxel fmri glm handle bias pe mis specify dm extremely ignore implication gauss markov theorem mis specify bias pe generalizing group subject correct add dm mostly heuristic still result pe approach flexibility response shape regressor fit use difficult amplitude pe bias trivial wish derive theoretical enable dm glm analysis practitioner develop simultaneous dm contrast dm controlling bias optimization optimize detect second specific task fmri experiment design fmri fmri response start generate section whiten whiten misspecification use analyze eq glm contrast interest show eq recover glm quantity define follow pe becomes normalize normalized change ideally tradeoff capture simultaneously interest residual function enable optimal ideally pe well residual compare performance capture enable optimal dm candidate dms expect contain regressor noisy contrast dm frequency matrix htbp cc first contrast unconstraine contrast find initial acknowledge initialization one initialization heuristic try find primary column singular leave singular matrix singular optimize column primary primary modify strategy many account via inverse framework allow inclusion datum simply noise maximize available level high number variable algorithmic user choose cutoff default cutoff dm z bc c column compute j est algorithm algorithm maximum reduction value please choose example variance curve local correct essentially initialization problem algorithmic initial choose cutoff default cutoff output number dm find cutoff slow tail sensible design motivate write
concern screen dimensionality empirical likelihood propose allow marginal moderate enhance multi procedure sis sis select variable simultaneously nevertheless marginal challenge jointly nonlinear sure natural effect family improvement sometimes class ordinary enter nonparametric closely signal converge extension adaptive high challenge penalize modeling fit nonparametric response covariate model goodness preserve non minimum regard nontrivial sis error modeling nonparametric depend function bring challenge extent iterative screening reduce computation two spline basis selection additive group variable page wavelet selection implication basis whose select simultaneously additive present demonstrate effectiveness conditional zero important curse dimensionality marginal nonparametric regression denote joint integrable onto rank utility select group marginal spline polynomial spline degree sup smoothness nonparametric projection approximate version express respect smooth rapidly np correspondingly define square regression nj j denote screen rank importance strength view rank nonparametric sense descent residual regression u residual sum fit screen possibly much whether active procedure sure property sure screen property rate screen additive assume admit identifiability whenever theoretical sure screening vanish follow simplicity support marginal belong rt n positive lemma nj consistency partial signal active separation sufficiently large set sup follow sure sure screening convergence exist addition condition f hold n nd eigenvalue design upper whereas f spline compare univariate marginal independence function small signal converge long lemma taking nonparametric property control selection rate basically false negative ideal nj active inactive variable nonparametric zero tend nj c ii achieve consistency select set show relate exist nd size whereas original np converge fast size fan case covariate diagonal hence naturally select additive model example additive active enhance employ term independence permutation inactive enter nan permutation nan permutation nj row quantile nj nj ny nj numerical use maximum far inside cross validation minimize component marginally pick determination except apply spam adaptive apply enhance positive stop variable study simplicity method selection show sure major difference unlike square remove make flexible particularly effective correlate tend high choose positive due screening improve chance select variable performance numerical experience outperform high positive rate illustrate datum analysis setting sure screen effectiveness also advantage method choose parameter example example I normally linear model vanish normally distribute ex sis summarize respectively screen fail contrast screening worth note sis particularly normally select whereas sis select nonlinear behave sis underlie newly propose simulation fold notation additive accord effect follow pn though model aim sparse show row computation report tp pe g greedy much false positive whereas scad important nonlinear example reason look signal contribution signal significantly introduce additive snr varie lot merely snr play measure difficulty scad nonlinear bad scad quite selection prediction subsection conduct estimator snr follow covariate take different make table positive poorly true active inactive one signal look table correlation competitive performance snr snr achieve sure screening current analyze illustrate week old microarray analyze rna contain probe genome intensity use probe interested heterogeneous system probe represent genome array whole follow probe express marginal implement analysis probe follow follow select pe evaluate performance validation compare prediction set observation observation select prediction set repeat deviation replication far fewer small biological probe selection additive use marginal marginal important correlation propose apply preserve sure screening selection modification propose deal marginally uncorrelated response appropriate additive approximated series expansion paper readily smoothing wavelet approximation smoothing spline proof property ef nj nj decomposition desire result condition together bernstein need reproduce bernstein exist c nd denote iy bernstein lemma tail inequality utilize bernstein ij lemma probability b j nd euclidean
analytic open definition cover grid covering analytic neighborhood cover next ease notation kb great cardinality easy bound convenient choice one subset let equal radius radius regression precision impossible assume calculation fix rd x pr r eq proposition trivially v closed suffice proposition xu u consequently probability q imply due x since complete main proposition immediately theorem shall facilitate subsequent discussion real function different suppose analytically xu v ki n formula side equal let xu thus therefore due fix p notation eq provide bind end cover k u entire connect finite tu w let therein eq series define let apply dominate cover satisfied compact covering grid lemma hence brevity first proof compact pz jj sl j de se mb sd fx mi db u mb sg sg complete proposition ft radius convergence assumption fix cauchy contour supremum get grid cover f z cx z contour regularize sparse underlie linear chi department statistics university ct email ex nonzero coordinate number parameter sufficient establish base error ls regression analytic rely expansion require take regression selection nonlinearity power analytic primary secondary support dms large machine ls error mean nonzero coordinate much nonlinear structure henceforth clearly criterion precision article sparse matrix shall despite conceptually besides regularization take advantage linear fail model prototype mind instead basic regularize yield estimation reduce result also collect establish model analytic analytic exponential much handle due explicit mle discussion establish series example corrupt vector impose vector remove unnormalized collection exactly observe denote seem eq general form regularize search mle minus likelihood typically position value mle ls precision proceed easy condition ease notation statement constant check mle form random letting meaningful need make sure try play role study main proposition condition due great establish rise satisfy fix later mild reasonable condition recall shall example suppose since aa rise say wide way correspond family counting hold result somewhat simplify k define find constant contain easy length mild proposition estimate reasonable order behave extra impose section devote establish outline easy conjugate relation u nonlinear exploit expansion expansion row transformation desirable bind work generally fall power series expansion may fall deal approach line connect account treatment whether use bound answer seem polynomial finite expansion general get guarantee bound hold simultaneously neighborhood unique open contain henceforth
also localization choose formulation locality constraint representation remove simplicity case origin appropriately shift invariance requirement put try optimize work exist machine nonlinear manifold laplacian method pre affinity graph compact handle unseen importantly coordinate direct sound unsupervised pre facilitate set model fix local kernel smoothing regard include widely machine non kernel kernel suffer high hand order overfitte dimensional learn locality compare coordinate balance balance coordinate code quantization widely process acoustic zero method relationship regard signal knowledge answer question space reveal important good code sparse code code locality property method learn th low order order yes code order various point claim particular code locality robustness first base roll primary goal demonstrate simple representation traditional code newly code formulate mainly point basis learn evaluate square rmse coding basis nonlinear behave look datum representation basis figure sparse basis encode nonzero coefficient code nearby basis get coefficient sparse data locality fail facilitate interesting coordinate encourage datum linearly enforce negativity remain interesting initialization unlabele base basis depict indicate coding remains code near point low coordinate code approach increase unitary variance smoothing consistent suffer cccc rmse rmse rmse rmse rmse b digit gray anchor code formulation anchor nonlinear enforce locality representation call svms code processing change sign anchor manifold anchor point obtain optimize basis compare smooth neighbor obtain various auto encoder manifold testing comparison code code raw local coordinate code across various basis check locality find unlike roll portion nonzero mostly distance work remove far locality table belief back network classification cc svm sparse code svm coordinate l rate raw smoothing code linear belief svm third classification patch background visual degree variation rotation code entire approach pool representation state method extraction sift grid sift descriptor pooling code classification examine replace code simple setup repeat sift extracted center sift descriptor partition block scale pool block average pooling try code block pool code local much accuracy rely nonlinear coordinate code locality ensure good code average average pooling svm code pooling pool max pooling nonlinear distribute see coordinate unsupervise local unseen also scheme locality coding depend handwritten digit object finding manifold valid manifold manifold coding theory apply mean worth many locality code local explain effectiveness code origin shift invariance bind definition requirement projection span orthogonality imply eq imply th follow stability lemma terminology hold obtain derivation convexity imply loss sum qx derivation third respect measurable independent definition ex introduce dimensional unsupervised basis learn basis provide anchor manifold approximate nearby anchor coordinate approximate global quality code nonlinear global learn drawback inspire use since sufficiently locality possibly suboptimal empirically verify synthetic handwritten digit unknown underlie distribution dimensionality traditional predict curse dimensionality argument become large distance large real dimensionality although new learn high idea point manifold respect observation show effectively localization dimensional extensively dimensional method interested function define let although restrict specific often norm curse sample require observe intrinsic depend intrinsic manifold use typical take manifold intrinsic dimensionality q statistical dimensionality involve cover manifold intrinsic coordinate code set anchor lipschitz accuracy manifold code data point anchor
convenience horizon eq modify discount agent assume know predictive improve gain experience take environment assign rule posterior experience implicit within predict setup agent definition identity section next allow environment model whenever environment step context describe mixture environment environment predict next equation model prediction posterior model give equation maintain maintain likelihood good adaptation agent entropy difference n arbitrary supremum finite fast predict rapid weak statement tell good motivate agent class observe action include computable environment computable function seek computationally replacement ideally bias place suitable candidate limit resource compute action computational use constitute approximation na I take tree use horizon mdps extend algorithm domain deal problem space pair produce state construct node give time converge agent history next reflect environment directly agent planning process true ignore uncertainty recommend carlo search technique construct search tree compose decision chance node history chance end reward contain child conceptual phase iteration phase root exist chance expansion decision child environment root finally trajectory four conceptual limit select child exploration gradually estimate reward cause towards high predict reward future choose heuristic horizon focus exploration practice allow build chance width search sparse stochastic branching chance search sized chance node star circle line star node indicate expand policy proceed flow back detail search always child represent history pose classic exploration exploitation child node like armed action logarithmic carry ucb domain playing ensure node gets select sample visit take horizon instantaneous history action pick ucb positive exploitation choose ucb reward follow immediately decision decision also reward action see decision tree remain require future reward sampling repeat natural baseline choose step tend structure search long limit full determine overall performance unless state complete root tree leaf maintain history follow q increment visit counter chance receive history construct tree estimate available retrieve importantly computational resource horizon simplicity exposition search carry obtain end experience keep subtree root tree use routine routine routine pick accord policy horizon accumulate reward trajectory value estimate update per history argument tree node create reward reward reward th action choose ucb policy child explore add search parameter value create deep selective high short tree ucb automatically focus explore alternate action eventually exploration exploitation uniformly node remain search generate estimate state general horizon mdp main consistency pick monte carlo tree search routine easily main invoke routine providing mechanism interior node scope note applicable program environment problem experience slow mixture environment construct context weight weight efficient summation tree huge covering I computation require outline way generalise compute action brief kt bernoulli one kt via put uninformative jeffreys beta probability string one q next form markov model work tree right identify node child set string stre pair binary call accordance terminology map intend example learn model tree learn learn kt depth aside bit see repeat long need describe history sequence binary predict bit furthermore achieve without kt idea convenience loss l reward symbol symbol action leaf node aside initial cycle variable long mm mm bit bit prediction action reward reward grouping node observe deal action specify prior code work give depth pre traversal perform encountered depth leaf otherwise nothing model length code tree describe impose like penalty structure ingredient depth internal estimate context tree history observe bit update maintain probability understand depth reason binary node probability binary node empty history aside middle processing tree fourth bit practice course count instead complete henceforth tree important depth tree treat leaf block node law highly context induction true leaf depth statement consider depth tree leave right assign simply aside sufficiently mm mm select next bit probability new drawback action potential history string may predict domain choose infer tree remove restriction arbitrary one would multi alphabet consist exploit space note difference former latter work symbol property helpful deal large fortunately describe technique incorporate level action precisely history bit string bit bit factor assume induce computable lemma modify predict follow scheme maintain mm create receive bit history factor conditional well environment markov environment reinforcement learn recent environment markov say ax markov represent converge model update kt see updating produce markov meaningful environment bound environment mixture imply cumulative difference also square zero rapid environment way computable environment first present improve automatically extensive clarity also environment move relationship equation optimal agent contrast reproduce kolmogorov expression share describe scale factored prediction tree model gain property environment result combine lemma adaptation negative depth life planning maximum planning instantaneous policy define j ax max apply absolute distance divergence give x precede b b w b km drop fix square difference average sufficiently importantly horizon environment ergodic intuitively mistake long thus environment mistake average markov say ergodic every occur infinitely agent sequence say self policy long environment policy exist restrict ergodic environment ergodicity early agent stationarity behave learn stationary ergodic environment model ergodic mdp ergodic mdp function action arbitrary apply countable set ergodic environment sequence produce self sequence policy choose sufficiently agent environment agent usage kt class number conclusion entirely justification mixture mixture form node interpretation kt would feasible result grow dynamically considerably tree process piece reasonable cycle furthermore exact suffer issue early additional needed set history length context tree length context recover reverse perform operation bit write phase context discard belief uncertainty lead high e situation complicate issue recommend horizon computationally force exploration intuition domain good achieve use exploratory action within planning amount exploration help avoid set belief underlie u routine exploitation softmax environment pair agent incorporate cycle agent environment across perspective noise partial test domain domain uninformative yes yes yes yes biased yes yes partially observable agent begin location action agent effect reach third receive reward reward problem regardless agent location inside piece agent choose four action move receive find movement depict equivalent bit observation receive exhibit observation familiar environment start action agent suffer receive begin stand right successfully open two stand action agent plan solution slightly simple agent gold grid corner receive reward remain bottom receive attempt remain large scale part correct step move domain opponent randomly agent reward reward top move end reward repeatedly play opponent opponent play cycle play pick uniformly opponent receive domain involve opponent play nash zero slow playing remain strong round pass player player put put opponent pass subsequently otherwise occur high round opponent pass either pass pass immediately opponent agent winner receive number remove play another begin play nash second player partially classic game agent exact bit indicating except indicate within location another indicate single indicate whether effect power every empty receive movement action collect e episode representation domain domain fundamentally put compare compete reinforcement literature page find slightly previous agent environment cycle begin agent bit encode table symbol interpret integer negative reward handle non remove grid observable bias partial agent break learn exploratory various time agent report reward cycle per reward receive cycle phase reduce early exploratory performance core intel ghz parameter phase recent capability expense model phase explore recorded amount experience active discount exploration exploitation discount greedy exploration small gave slightly reproduce experimental result report phase complicated tree general completely specify due absence publicly reference splitting criterion apply exploitation policy design decision list mm split step resource choice allow cycle domain split try recent distant exploration tune separately domain help fair tune exploration extend implementation agent performance exceed domain active experience learn overhead split limited period tree advantage cycle symbol require enumeration cycle large domain illustrate u observation vary domain learn cycle experience except cycle normalise domain significant planning extended effort good performance c grid biased agent domain resource near report day induce benefit dramatically context reason exhibit slow reason ensure dominate structure decision provide search action although practice least per cycle compare tree act motivate active guarantee u tree heuristic criterion never configuration dramatically somewhat applicability learn suggest framework along adaptation history action return sample remain complete unlike build tree highly stochastic planning improve exist reward differ total horizon overhead account simulate trajectory evaluate planning combination grid versus plan importantly performance bold perform multi plan particular region set match believe important challenge environment plan difficult choose action explore reward per cycle greedy selection time point vary show cycle effort use affected behavior inferior learn visual inspection agent play perfectly concept know know seek limited provide sensor know away learn red become nearby visible reasonably exhibit optimistic several attempt study fast asymptotically parallel pick fast program enumeration run cycle pick well provable agent g b domain universal feasible force expert game exhibit monte algorithm universal early influential partition leaf tree partition tree incremental fashion leaf begin leaf split history fall leaf show exhibit deal effectively environment algorithm attempt discover raw stream tree discriminate allows effectively ignore irrelevant observation represent sequence experience around kolmogorov heuristic try place learn state action selection combine scheme produce optimal build distinct accumulate statistic estimate refined time symbol algorithm probable prediction much distinction usage prior predictive representation maintain probability experience experience experience markov observable environment complete dynamic unfortunately impractical form improve algorithm currently area representation approximate abstract prediction future prediction network give current update promise recent network give parameter maintain contrast bayesian variable share similarity estimate ucb limitation perform environment bound depth prohibitive amount experience need cope limitation unless planning solution exploitation intractable need augment
bind satisfie second rhs rhs proceed consider sequence introduce partial proof deal omit let inequality eq inequality proposition proposition b ax ax check x hence martingale get u bound interval mm boundedness get mean covariance grateful r n last ac h n see stationary show lyapunov trace handle nonempty empty interior include ac assume tailed class compact convex measurable finite weakly characteristic variance take ac study related adaptive mala mala drift truncate cm remark fr central drive geometrically stochastic asymptotic adjust heavy tailed subject include theorem drive geometrically ergodic sub many markov kernel geometrically example target interest tail langevin mala kernel drive enjoy uniformly central stability chain irreducible v martingale proof proof study law adaptive paper mention review development organize section adaptive drive approximation section illustrate theory adaptive langevin tailed transition measurable stand dirac act function nf dy dx nx space norm resp endow open borel markov kernel measurable invariant nonempty compact subspace measurable ax practice dx x main paper order paper without far chain chain describe n initial state arbitrary systematically write instead control vary compact develop take aa commonly chain easy markov eq expectation natural convenience notation compact convention strategy strategy former main projection law large hold measurable nf probability section strong number hold imply law measurable assumption notation usual define ax show admit martingale let take let martingale hold free adaptive array value random refer path random weakly characteristic establish weak law restrictive recurrent markov suppose satisfie let simple checking condition exist hold take ii eq establish drift uniformly polynomial v v b geometrically uniformly exist measure explicit ergodic hold also find get assume whether check drive condition indeed difficult typically cx check drive let convenience write eq h ny random continuous continuously w dy x x magnitude approximation notice projection key sa framework proof line enough hold exist assume b suppose langevin mala density lebesgue mala metropolis whose langevin dimensional brownian mala work give mala practice depend regularity fine properly transition mala obviously many mala target acceptance generate x make cosine frobenius compact smoothness bound section variable characteristic x recurrent satisfying consequence hold imply consequence easy conclude corollary asymptotic proof follow weak law theorem basic serve allow markov constant compact notation keep give family depend notation resp approximate well satisfy eq xx dx aa let ii direct write q integral sign assume rhs rhs p x f v q yield part ii write gx ax ax ax ii conclusion compact integer x dx expectation consequence exist next proposition general bind inequality martingale fix initial kernel p measurable nf nf n x p g p notice eq consider k martingale array choice converge r k q rhs since p ax ax central theorem take ax kn n kn sequence x x k assume k n n p n deduce k obtain na n converge n g ax p v probability give non markov allow transfer limit adaptive equality follow one w sequence variable nc jt equality law number triangular array non x kn idea proposition write write k ns ks n w ns get
derivative relax shall apply several rate normality throughout value choose constant namely theorem type student pearson corollary specific random quite explicit finite bound depend long explicit denote application complicated state reader constant subscript pre bind remain moment pre self normalize pearson correlation constant simple bound corollary corollary place appendix hilbert let satisfy counterpart number hold exist constant depend bound explicit expression uniform essential appendix satisfy enough mention remain case small bind probability bind conduct constant theorem complicate corollary simple explicit constant size especially unbounded improved factor able obtain theorem natural number continuously twice smoothness denote upon specify r theorem explicit first involve particularly nonlinear statistic investigate f c f coincide quadratic statistic respectively note without generality adjust use z cf conjunction l point work well nonlinearity method tailor specific specific commonly student ty ia let generality let easy condition inequality satisfy distribution bernoulli concern appear show distribution degree concern degeneracy equivalent appear cf imposed achieve expansion distribution order central nan discuss remark end hold therein remark self close self nz cf uniform form depend namely show great necessarily distribute obtain also I type bound structure bound state number mistake kind triple several choose range parameter thus rather slightly great also advance concern central make normal wang wang probability moderate wang concern logarithm wang bound concern central normalize somewhat early central exist significantly stochastic central heuristic reflect well triple appear competitive remark detail corollary involve theorem triple triple try weight constant bad assumption eq eq triple either ht statistic bind reflect small maximum attain whereas rather put perspective recall even simpler identically explicit similarly proof corollary demonstrate obtain variety constant numerous accurately idea hand complicated suppose somewhat constant proof ht well moment order contrast high moment instance degree freedom symmetric vary one tail absolute bind triple check monotonicity namely rx reasoning dt dd nontrivial standardize skew enough nontrivial note mean h blue student freedom dot ht red green center pareto potential cause eliminate truncation truncation truncation moment kind look comparison truncation sensitive sense knowledge needed compute another corollary eliminate product entail accuracy accurate complicated appearance z either triple proof degree center pareto shape numerically also minimize significant tail heavy e find truncate truncate small small follow large consideration somewhat surprising bound develop self normalize compare tailor sum also want case statistic behavior particular take continuity expression view standardized freedom standardize skewed enough py pp get use directly short dotted precede imply e picture ratio approximately hand contrast paper certain various choose asymptotic behavior bind provide though still assume stand coincide wider taken grow slowly hand moment specific pearson correlation coefficient eq denominator invariant transformation r standardize x smoothness x immediately satisfy exist point union straight origin understand axis equation root vice versa lie union line origin r cx cx c p standardize uniform third rely let value vector let accordance type derive cf statistic linear particularly satisfie condition proof lemma defer subsection q p complete recall condition place provide hold consequence write satisfied definition let expression positive q recall view obtains replace inequality display combine accordance type respectively prove easily verify inequality inequality respectively first old inequality definition reason q well truncation definition xy xy satisfied assume definition follow since take recall accordance notation use eq remark concern ic replace one q z concern pre constant choose n correspond theorem normalize q accordingly place allow recall specifically ab triple rational expression ccccc lemma expression tail need wherein expression several q q restrict real abuse symbol represent expression take express use also indeed result establish immediately theorem constraint remark improve pre inspection pre complicated appendix constant assume theorem hold take condition g xt cf lemma k w jj satisfied definition combine definition lemma also accordance z use use definition imply trivial convention together definition upon case remain yield well prove bind bind hence factor triple denote inequality exist depend chebyshev chebyshev hold satisfy smoothness modification fail extra contradiction triple rest far w q enough contradict statement prove modification argument easy verify previously corollary computation arise proof course calculation could principle aid computer r v arbitrary far recall smoothness whenever norm increase specific rational number whenever happen necessarily satisfied positive q take u try weight expression substitute parameter give interpret exact rational triple ccc adopt notation use corollary recall satisfying shall specify proof eq cf triple shall specific triple use decrease inequality inequality inequality fail eq definition definition upon list accordance definition w restriction recall definition one concern table necessity perform round intermediate exact rational within prescribed precision take expression algebraic calculate triple calculation replace absolute triple list contain general number use attain paragraph formula mention root value list place one prescribe remark value table c cc cc cc cc whenever supremum attain finite positive function define x hx expression table lemma trivial verify statement suffice statement iii iv decrease since change hence statement easily finish make hence statement positive critical rational attain checking checking complete case case old q take condition cf discussion line use instance expression desire expression listed verify recall identity denote open origin u namely far bound bound q combine program algebraic number rational particular statistic pearson define rr population normality rapidly zero come normal david suggest close normality approximate distribution close normality closeness variance therefore whether nice transform moderate non population carlo skewness tail normal expression kf rf asymptotic bivariate easily consideration briefly statistic correspond aside application namely upon let g moreover view whether population despite appear g particular worse view pearson yield bound thank remark part nsf grant dms uniform order closeness normality abstract statistic bound delta pearson student kind study central student previously method stein chen er transform constant pearson independent identically property relative efficiency pearson correlation well g test close normality neighborhood closeness bound correlation statistic pearson surprising consider bind somewhat perhaps somewhat simple student identically distribute I student statistic normal establish e method linearization chebyshev inequalities form statistic function delta origin since linear demonstrate main sum assumption moment norm obtain comparable factor infinity extra logarithmic factor far interested soon remarkable chen make offer relatively refer chen pearson deal define variable take multiplicative allow moment chen er yet modification level bind group closeness abstract may upper tw present fs z delta effort finally delta method statistic one bound appear know central student obtain turn one delta section know identically distribute general replace necessarily identically organize tw mention state vector delta bound fs ls z commonly namely student pearson proof section defer proof statement appearance bind prove even relaxed practical make computer preferable pearson statistic value measurable borel brevity stand borel assume q note hand surely independent let r q simplicity replace validity replacement show theorem upper behave chen modification make define simply paper would place instead allow possibly happen third improve constant application state counterpart minimum let p even state independent mean obtain g center g shall upper application real bound possess monotonicity clearly follow relaxed condition true e consider paper expression cause moment bb root exact low bl c j independent borel
figure case loop head relation polynomial chebyshev modification assignment graph concern cycle belief assignment theorem hand positively proportional comment sum equal proportional assignment regard bethe loop series expansion run loop sum loop interest bivariate integer cycle q assume side bind loop generalize loop generalize degree equality factor model pairwise straightforwardly hypergraph hypergraph type example hypergraph mrf node satisfy message rule bethe partition convention connect modify sketch choose loop sum call bethe topology strength formula technique polynomial since contraction contraction relation polynomial edge loop end formula essential proof contraction loop edge classify include interesting divide loop loop connect omit theorem match matching match introduce index word denote restriction principal minor summation run cycle regular connect denote omit polynomial assertion acknowledgment grant aid aid scientific field joint set variable dependence joint often give form normalization mrf mrf computationally require approximation attract computer equivalent successfully image algorithm compute surprisingly cycle behavior marginal assignment know hand many little topological structure underlie give loop expansion term series bethe loops bethe correct expansion highlight pass diagram expand easy derivation bivariate ratio bethe approximated investigation understanding graph pass message initialize message approximated formula bethe partition ambiguity update specify depend choice normalize b ix x marginal prove play polynomial transformation chebyshev polynomial induced edge hand bethe rewrite q bethe
consequence shall far respectively proceed subject follow multipli direct application lagrange multipli conditional test let exist satisfy inequality quite straightforward eq mm test test type probability obvious assertion hold inequality stop rule let fulfil almost everywhere condition proof sequential test original characterize natural truncate fulfilled stop stop rule eq rule coincide eq finally attain everywhere proof characterize minimize stop truncate result precede apply particular basically proof lemma exist lebesgue monotone pass side leave need stop stop need side stop rule conduct follow fulfilled rule condition satisfy satisfie find generally violate eq suppose normally mean also stop take eq remarkable finite process definition justify fulfil lagrange truncate may optimal truncate stop additionally g condition formalize let call test condition fulfil everywhere immediate theorem satisfy regular locally test vs strict least locally see remark similarly locally irrespective note regular need additional fulfilled arbitrary regular problem vs locally powerful vs strict strict analogously fulfil strict inequality show much also sequential exist another test kk number happen thus follow vs interesting example powerful average irrespective family I symmetric theorem test class I sample exceed theorem space integrable equality everywhere lemma side substitute right hand equal respective hand everywhere proof lemma major let integrable equality everywhere k x easily eq thus obviously equality inequality happen fix limit virtue particular equality integral finite fulfil almost part everywhere us b l r check infimum family side tail converge series second tends third start everywhere schwarz let virtue everywhere kx k almost everywhere everywhere almost everywhere theorem eq follow eq let let therefore analogy show satisfie essential non decrease obvious claim side virtue lebesgue monotone go non go possess thing property wise similarly pass thus lemma tend jensen addition non tend reach easy convex prove theorem generate see follow author work partially cb c subset testing hypothesis versus goal article characterize powerful sequential rule maximize sequential suppose structure discrete optimal test sequential testing locally let real testing fixed structure powerful sequential definition well interpretation characteristic see also hypothesis suppose measurable value probability make experiment observation stop suppose stop stage interpret reject hypothesis generate paper process
cpu three ht edge mb efficient perform study variable span forest complex forest complexity cpu grow linearly vertex cpu grow ht curve reflect short much start figure consume large iteration analyse package adjacency matrix list model likelihood aic find vertex cycle graph return connection return perfect free correspond neighbourhood find find perfect residual subgraph base list subgraph give ci degree degree vertex neighbourhood pos label false vertex col c label vertice col restrict neighbourhood subgraph neighbourhood radius neighbourhood actually general close high degree aic homogeneous heterogeneous could information false clique return list strongly decomposable list return short edge return inf far vertex distance diameter plot plot default circle continuous grey circle define shape place iterative force place consume clear iteration plot allow appearance plot label vertex shape black triangle vs r c r vs pos symbol code isolate displayed figure r r col ed col col ht vertex package package package way class forest model convert package limitation numerical acknowledgement stage research european support project life science innovation david package high package decomposable criterion aic bic graph package graphical involve useful wide application g application science internet model whose independence encode vertex correspond conditionally represent indicate independence two contingency linear variable modern account graphical model variable difficulty model package model central search forest decomposable typically term package distinct basic definition notation present cover discrete show profile site non moderate sample total site profile evaluate genome array probe independent process use compose characterize gene expression patient international project genetic single nucleotide four different basis g occur position dna individual base occur allele snps select snps snps snps structure minor frequency website www allele individual allele dna reference allele copy allele decomposable representation datum know record consider different give sketch complete vertex connect conditionally independent give graph present conditionally conditionally independent ht relationship conditionally give b add non loop allow conditionally independent subgraph every edge subgraph maximally addition would subgraph incomplete clique graph say distinct vertex vertex example great since separate path connect adjacent discrete describe graphical search http r package selection discrete row vertex use lr aic lr aic bic name name name original datum vector homogeneous case first last edge index besides true na core package aic bic forest tree search minimize criterion specify calculation use format storage type discrete discrete first represent refer index representation format index r frame level factor indicate type name length length attribute object refer column number dataset variable connect width edge attribute consist aic use repeatedly add optimize edge preserve cycle continue edge measure computed step reduce bic exist bic bic addition edge return add edge return decomposable forest summary span find
specie classify specie apply example class normalize mean near leave validation I mistake test error mistake order explanation result window classifier choose one cross explanation vector live input visualize initially show dot explanation class group dot dot feature roughly different dimension part distinguish scatter explanation class blue versus digit follow digit kernel width training approximate window example part example display explanation digit end explanation interpret look top example eight part vector dark add digit classify eight lack digit classified explanation red adding white two mistake explanation part change classify probably inside cloud explanation correctly classify focus explanation make digit weight classification similarly explanation mainly suggest remove dark part add light part look overall finding vector example digit label reason classify problem fed determine predict chemical cause requirement investigation drug discovery chemical modeling machine class randomly compound class split investigate individually confirm result example compound represent molecular software manual normalize mean rbf confirm figure remainder section evaluation feature histogram local indicate examine cause set certain seem mostly model outcome modify predict probability indeed conclusion explanation agree knowledge exist could explanation question rank trend exception rank consequently random use establish methodology example compound global explanation need identify relevant display kolmogorov ks relative frequency leibler prevent zero lead add bin generally figure almost positive gradient global chemical explanation model influence seem knowledge yet discuss literature conclusion learn explanation make potentially feature overall notion explanation error label decide way sensitivity area science outli sensitivity case remove estimate remove line influential point actually change regressor whose input explanation vector view thus influential vector sensitive influential decade explanation expert topic ai especially context sensitivity use remove explanation variable affect target connect explanation value decision train knn svm omit difference odd difference probability prediction without save combinatorial consider calculate difference layer measure importance change variable input turn interaction principal framework framework start binary discretization ii combination approach feature estimating discuss situation panel middle e project representative slice explanation middle maximal finding influential g x py derivative expansion start quadratic explanation second interesting direction eigenvalue instead meaningful mention practically estimator coarse structure far g extension demand useful classifier gradient explanation precisely classify explain hand estimate correct agree exactly boundary train area boundary space explanation away data area side corner away vector pose locally influential investigate high respective explanation gradient datum local gradient predictive model assume stationarity explanation reflect g explanation assume deal set taken stationary shift book paper propose box explain decision arbitrary possibly classification characterize point change predict information approximate validate conclusion various fisher different identify digit distinguish challenge drug agree exist available chemical space discover tool practitioner would biology decision make experiment promise approach prediction acknowledgment fp european mu thank follow illustrative explain respective examine exhibit kernel introduce locally ht effect toy gradient explain situation fail misspecification reflect rational quadratic kernel able linear separation illustrative purpose obtain local perturbation trend class clear previously observe feature interact triangle class ht two rbf kernel capture affect depend parameter item local gradient explain explanation extract local gradient outli near outlier reflect explanation feature reality nevertheless model place histogram figure distribution affect single outlier local region slide thus gradient hypercube center appropriate averaged appropriately window locally boundary point accurately follow circle shape range towards instance introduce small reflect gradient elaborate explanation fit window derivation estimation gradient window calculate approximation svm use rbf window minimize absolute predict local boundary resemble accurately choose width pointing width practically useful region width fail obtain gradient leave blue bottom tu tu de tu tu modern unseen answer question answer influential currently base explain decision automatic powerful tool learning classify unseen label nevertheless explain decision essential give answer typically jointly relevance determination salient coarse still provide instance conclusion equally classification view combination influential see answer corner contribute jointly solely ensemble pruning provide individual tree propose framework explanation vector order result explanation organize explanation gradient apply learn distinguish estimate explanation vector classic classifier explain digit distinguished digit discuss world scenario capability human decide capable modification calculate gp posterior analytically model use
normalize numerator side complete define notion hypercube prove thm area noise show sensitivity noise quantity asymptotic surface furthermore asymptotically tight distinct homogeneous area agnostic polynomial threshold learnable along prove sensitivity relate essentially also boolean interface origin recently multilinear polynomial point uniformly hypercube always note similar thought first prove conjecture related surface area set volume metric open area furthermore surface probability two define degree follow theorem q threshold threshold devote begin let gaussian define eq gaussians integer function equation sign interval note periodic sign change sign limit threshold show sign polynomial note sign happen ignore noted root way change sign circle number sign spaced distinct homogeneous occur asymptotically need slight sensitivity threshold begin prove follow variable q lie dimensional change bind probability least one change suffice correct degree distribution q
natural latent consider e ip probit give bernoulli rv distribution probit suggest ng corresponding complete latent deterministic call conditional gibb complete conditional indeed posterior intractable implement conditional new estimate probit population test diabetes health organization collect national diabete disease supervise tolerance pressure mm diabetes accord criteria goal diabetes explanatory bp analysis median max std dispersion degree aic scoring relevance reproduce perspective bayes probit covariate probit testing nest probit covariate intercept already factor hypothesis nuisance improper obviously variance elementary approximation ratio standard correspond prior prior eq estimator force distribution prior monte evaluation evidence inefficient produce integrable infinite require effort efficiency monte usually figure survey obviously integrable simulation simulation define support two offer importance function possible approximation sample candidate approximation hard use importance distribution distribution estimate provide general specific probit since replication methodology figure table require compute simulation sampling simulation prior simulation bridge factor rely representation bayes sample distribution perspective bridge parameter common poor possibly variance nothing importance function integral bridge representation posterior apply long exist choice poor performance connection harmonic quasi optimum se approximate base posterior approximation alternative difficulty difficulty derivation restrict embed correspond advanced bridge appear space joint distribution clearly approximation two generate depend completion form consider posterior bridge sampling handle implementation pseudo technical device bring bridge close cross alternative complete globally link posterior green jump mcmc cross model seem randomness pick step useful infinity average form asymptotic ml quasi optimal average gaussian suboptimal result replication methodology hand bridge variation excellent sampling considerable upon initial monte side superior accounting example iteration leave right generic harmonic matter representation remarkable direct processing mcmc variability estimator importance tail instance harmonic approximation infinite discuss opposite support like hull simulation correspond region importance approximation mean ml estimate replication methodology simulation importance fast compute due gibbs necessarily account dataset harmonic versus estimate approximate bayes q rhs say select harmonic unlikely framework use preliminary special model distribution probit approximation particularly attractive sampler rao available probit replication simulation approximation reliable dominate asymptotic particular case dominate harmonic estimate harmonic importance distribution bridge harmonic median estimation sampling
contain previously estimator differential ordinary widely physics biology usually involve parameter dynamical seek concerned variable ode use complicated thus analytical besides estimate covariate vary assume know generality investigate likelihood optimization require eul principle exploration since require solution seem ignore nonparametric smooth spline smoother find side estimate covariate simple implement take recent besides spline method work ode coefficient minimize functional reflect satisfy ode numerical ode take asymptotic vary tx dt extension multiple straightforward non covariate regard simple case involve dimensional functional coefficient associate error author analyze difficulty problem extra dl lt lt simplicity derivative bandwidth localization local result dl though ht h differential try around approximate together derivative identity note order even kernel avoid discussion issue complicated seem choice affect except multiplicative always fix go infinity support well derivative zero measurement finite denote independent distribute support main result concern lt p diag I I I lt lt know state completeness q eq general appearance probably entry r iii dl rx nh ft lt dl ta vi dl ta rx nh expression dense time consider besides appear calculation involve slightly rw ft nt uk uk equation bias component ta l dl ta expansion conditional ta r ii rd dl ta ta dl ta
object use database instead link member subsample object since always empirical matrix link maximum leave present application sensible basically query dependent since useful measure extended relationship relationship measure occurrence frequency word metric continuous however substitute regression web page collection relation web come recover page relationship asymmetric web imply page page simplify I web page unique web page later bayesian standard bernoulli merge link page evaluating serve purpose treat probability methodology object indicate word document avoid introduce extra approximation original obtaining measure b I intercept logistic cosine measure practical advantage linearly adopt feature make comparison suitable relationship web page symmetric relationship reflect pair subsampling computer evaluation item measure define setup query web page label fourth page class return pair criterion reasonable objective page possibility hard particular demand way omit page return however page page query four four standard binary standard original cosine page combine cosine document relationship demonstrate superior equal precision perform ask retrieve fall detect link close evident adopt pair retrieve notice hard instance group mostly short basically member web page recall intersect summarize performance bold indicate pt model molecular biology protein interact protein physical carry cell resource collect protein bind protein role source protein protein localize degree hybrid collection total analyze aspect large interaction network include de prediction consider categorization protein evaluation individual annotation protein sequence gene collection protein encode collection bind experimentally protein collection mode interaction bind place context pathway functional annotation go combine functional annotation say ii random replacement iv interact comparison purpose approach process large pair query vi calculate comparative summary genetic localization bind protein generate attribute process attribute correspond expression different attribute experimental measure perform well metric batch protein interaction select query link pair network satisfied category rank pair connect protein path reason filter pair undesirable filtering correct match trivial query pair interaction generate ranking respective auc replication last smoothed version replication obtain replication top give cb b c c provide include indicator however notice considerably reason degradation change method increase slightly method analyze criterion random add tie smoothed count method winner obtain relational bayesian winner rest pairwise comparison often method another besides proportion among categorization categorization explain pair top rank intend well link database link protein link categorization instead perform query gene category select category query replication challenge scenario optimize able pairwise top filter link protein coverage category pair categorization include neighborhood proportion histogram search coverage categorization perfect valid pair gain positive rank considerable across particularly experimental setup protein categorization select filter candidate pair protein link protein query include category replication category link summarize evident pairwise automate criterion believe much reflect high coverage top rank table success room artificial intelligence cluster reduction classical planning exploit derive would graphical incorporate existence relational use step probabilistic motivate retrieve query reasoning discuss interpret link choice future extension consist discover latent formal discover latent relationship inductive programming inverse overview relational particularly active lie analysis discover gene retrieval text index method answer idea give treatment simplicity task conditional compare predictive framework similarity remain appear international conference artificial intelligence formulation calculating compare relational size graph kernel provide metric property membership object cluster role framework indicator conditionally available relational reasoning application exploratory anonymous several suggestion presentation additional reference reasoning formulation support nsf grant gm foundation fundamentally develop relation question retrieval analogous object way object combine require specify relationship illustrate text application discover protein work even american assessment record include reasoning reasoning implicit although relation implicit nontrivial measuring similarity way discover extensively discuss cognitive analogy isolate atomic discover relationship paper concern implicitly tool exploratory protein protein cell cycle functional interact protein molecular experimentally protein bind clear interaction molecular protein particular generate list prediction expression encode protein localization relationship protein biological know interaction role aim detailed protein pair correspond analogy example section present role protein possible pair want match possible metric protein interaction section perform query ranking molecular general exploratory link relational database link way retrieval text illustrative web page page link relate search analogous analogous wikipedia evaluation criterion review tailor analyze base corpus characterize relevant set similarity unlike feature need define similarity avoid mistake compare relation similarity paper initially group use multidimensional pair represent connect similarity pair compare operate object instead distance solely complex approach logical reasoning seen reason latent relationship relational approach discuss however reasoning tackle planning create exploratory text way exactly rank respect query query retrieve exploit one retrieval query mass rank equivalent expression collect advance query give estimate general query instead define use class datum art ranking inspire event model model hand provide interpretation compare compare define hyperparameter query generate generate next biological modification analogy object aspect link feature object query pair illustration pair good query reasonable would match nevertheless infer similarity feature kind world hypothesis need object relational reasoning similarity meaningful feature represent say interaction b similarity query correspond pair directly object similarity capture classify want quantify link unobserved integrate bayes explain latent dimensional logistic parameter particular pt mapping attribute potentially predict link model bayesian methodology explain object parameter compare link indicate link relevance set feature integrate prior bayes give compare hyperparameter generative point represent rectangle conditioning set
score scale lack small turn rademacher ensemble hadamard rademacher hadamard develop fit display rademacher typical already hypothesis mostly shift meaning realization rademacher ensemble typical less hadamard success ensemble mostly slope score sign success treat random varie realization expectation fit without display output symbol denote interaction median coefficient std multiple adjust df score toward residual score close hinge term increase significance scale call residual min coefficient pr intercept de degree freedom square f residual ensemble particular analysis rigorous quite belong particularly unclear hadamard display score fit show describe ensemble transition develop entirely display present early describe ensemble comparison demonstrate rademacher validation ensemble early explain gain develop ensemble mean decay ordinary ensemble existence seem confirm extensive experiment success programming recover million cover ensemble limit diagram success success tends measure sparsity ensemble location closely match behave survival width transition unlikely tend broadly ensemble locate advanced maintain conclusion use ensemble counterpart level score support fitting statistically nonzero fraction mean exhibit trend ensemble weak finite accounting mean score ensemble phase match case far current author software great deal describe resource compute dms jt partially support transition occur dimensional processing sense reconstruction model threshold combinatorial geometry transition vary location appropriate subject area model hard throughput hard limit degree break compressed sensing define sparsity tradeoff theorem derivation transition combinatorial assume underlie identically distribute required experiment inferential analysis underlie ensemble run million span several ensemble phase transition agree asymptotic careful explain decay experimental large ensemble reject throughput measurement dataset sense transition phase amount rapid shift critical threshold concrete example surprising processing cloud intuition surface hull intuition interior segment intersect expectation even intersect interior human hard visualize phenomenon phase appear continue bit small threshold bit intuition work tuple indeed intersect phenomenon one consequence field select model sample design imaging device consequence nothing really hour medical necessary help reader transition occur choice observe phase transition evidence million random transition iid formally geometric open contain standard ensemble class broad view attractive feature tendency ever observe pixel patient technology put throughput measurement ever detailed protein expression whole fluctuation activity ever st mark observational hundred throughput measurement device fundamental obtain observational face observe distant galaxy many field observational perhaps hundred set abundance characteristic modern develop new tool comprise activity institute gauss variable measurement expect measurement observational datum allow set gauss throughput batch potential automatically know useful project throughput popular believe fraction among unfortunately throughput everything together batch stage expand conduct stop choose denote predictor regression record square observation world observational unit fall slightly increase position curve derive combinatorial notion fractional axis vertical observation curve combinatorial rapid fall match curve field geometric think unlike normal variable contain know design partial hadamard column hadamard matrix design choose generator either large negative well relationship try standard outlier quantify create range digit agree record outlier six digit attribute six digit panel axis vertical leave panel use expert agreement change behaviour fraction observational unit fit large contamination break n derive curve coincide axis look interpretation fit design experiment robust certain critical outlier calibrate match critical fraction match geometric analysis decide acquire measurement imaging signal actually kind observe although degree freedom attempt measurement experiment choose horizontal axis measure vertical axis contour success roughly fraction image accurate digits horizontal axis fraction axis fraction interpretation usual limit twice already arise recall geometric draw denote hull random polytope suppose figure simplex interpretation tuple vertex line polytope dimensional second black see curve define dimensional hull origin typical every segment polytope dimensional face involve positivity recover describe derive lack formal connection establish ball observe frequency visible accurately situation present believe kind limit formalize geometry polytope hull polytope point project face projection family call regular simplex analogue polytope analogue hypercube polytope available reader object think however face count reveal system frequently responsible three lp course fraction lp solution face simplex tell lp correctly consider polytope optimization instance equation infinite general position sparse solution underlie solution q face count polytope cross polytope tell short simplex cross rather tool polytope theory rigorously existence threshold face count iid triple count display curve polytope year run experiment generate million various specific whether polytope accurately matrix involve gaussian ensemble detail material supplement l name bernoulli iid equally likely fourier row dct fourier dct element equally likely iid equally iid iid iid hadamard row hadamard matrix hadamard special binary rademacher equally likely equally likely specify pattern ensemble column fraction varied sparsity range lp solve exact reconstruction correspond polytope face face success frequency success systematically sample success show success ensemble nonnegative nine curve present solve note use sign excellent agreement ensemble qualitative et al show qualitative agreement transition theory surprising merely prove polytope accurately ensemble ensemble accurately suggest phase transition behave prove behave sized asymptotic writing ensemble know know discover phenomenon identify stage identify comparable identify call phase transition phase diagram ensemble physics mean something different instead phase underlie structural strong transition conduct like instance specify factor coefficient ensemble coefficient indicate draw uniformly sign ensemble indicate visit triple triple give run observable take value within accuracy otherwise aim rather exploratory inferential carefully explain apparent attention scale separate span situation many available require cpu different table exploratory study day comparison calibration vast majority experimental run compare popular consistent rather inferential comparison say ensemble may gaussian everything else ensemble realization ensemble success ensemble hypothesis really assertion equality standard compare proportion comparison experiment non gaussian standard merely consideration score formalize nan asymptotic seem priori consider universal correct behaviour size asymmetric occur analogously systematic score arise binomial proportion verify show describe exhibit ensemble obey slight shift away negligible shift cause lack small ensemble ensemble complete use validation validity lack good hypothesis device list table vector either underlie process optimizer underlying sign process solve optimizer optimizer desire solution digits replication coefficient replication specifie rademacher type type presentation coordinate shape success dataset slice hold vary success fraction monotone consider compare baseline varied systematically grid range trial region report happen analysis conduct hadamard rademacher ensemble rademacher hadamard restrict question transition profile function reduce manner empirical point appropriate place complete rigorous asymptotic exist either width rigorous rigorously scale rigorous phase constrain death probability problem slice gaussian monotone increase failure function success three probit response complementary distribution present checking identify choose match display curve residual column datum model column generalize linear model binomial response take eq success obeys call logit probit links fit call fact achieve among probit residual probit zero thin deviation logistic link good residual probit adequate ordinary statistically sensitive residual fit inferential test failure estimate way find large solve small spline difference estimate limit l phase transition quantify show approach limit roughly transition response present nonparametric analog base spline location normalize distance probit nm methodology score ground difference asymptotic independent compare setting sure procedure compare score present present plot versus fraction score exactly plot quantitative ensemble either sign score absolute line assume distribution comparison value much line thus behave observation need check score construct know normal figure score compare ensemble ensemble fit trend show obtain case table l methodology detect difference compare reflect curve reflect plot identity also display truly normal every deviation apparent dependency force lose polytope decrease effect conclusion main strict asymptotic report focus exclude discuss fit consider first report result later restriction display restrict symbol symbol denote list en sum de en z median max std e na na de de de de residual freedom df value model significant roughly proper reduction sum square nan residuals min coefficient estimate std pr intercept e de de e de de de de de de code degree square fit associated contrast mostly worse adjusted restrict df df pr exceed significance word vary sample root function motivation fix success ensemble asymptotic transition c theory width describe success indicator perhaps interpretation rigorous provide surprising underlie indicator variable correspond one common score gaussian ensemble obtain collection intercept fit table l l l intercept slope exponent case easy find software exponent fit two ensemble include term vanish joint eq term exception effect significant fit give include although tendency vanish joint intercept term exception treat two give degree tendency vanish exclude figure panel score versus differ score overall get evident plot dramatically apparent mean adequate provide case residual versus mild nonlinearity model model preferred scale q de residual median std pr na na de code degree square square statistic substantial residual dramatically give residual min std error pr e na code freedom multiple square adjust square f e individual coefficient compare statistic df df df sum pr fit term fit score contain immediately residual pr intercept de e de de residual degree statistic df error allow extra explanatory drop scale model adjust bad scaling remain cause remain error everything satisfactory way summarize residual group summary deviation median absolute deviation fairly residual deviation
dag matrix every connection element denote dag later early influence influence imply influence define dag identifiable strength discovery algorithm define reject evaluate reliable return bootstrappe reliability th create replacement bootstrap eq q ij sign important way divide would make boundary region close solely multiscale multiscale mb mb replicate generate bootstrap replicate denote hypothesis compute multiscale specifically multiscale bootstrap ica point change estimate ica step practice maxima test equal imply accept model violate could might give order case tell order create two boundary create laplace size mb select replicate bp mb mb mb bootstrap bp bottom multiscale bootstrap mb give line bootstrap bootstrap six compute bootstrap histograms multiscale imply rejection probability ordinary bootstrap versus level six case equal level plot bootstrap bootstrap give reject nominal level multiscale much show multiscale bootstrap provide propose statistical value variable estimate tell investigate sensitive smoothness although might problematic support continuous recently gaussian call model various direction affect random conduct ordinary hypothesis propose procedure advance call multiscale multiscale asymptotic artificial utility widely causal recently call variable datum alone order causal relation variable infinitely bioinformatics tree replicate bayesian probability
summarize split right stand perform get conclusion clearly small show cloud mask analyze multivariate estimation dimensionality independent factor latent mixing use mirror average aggregation density classifier achieve excess support sense outperform several intensive efficient algorithm develop mirror note imply transform dp x x p jk bias h variance w suppose canonical denote al diagonal convolution kernel brevity term statistic far since uniformly inequality hoeffde straightforward imply nh h nh standard lemma apply inequality l imply j b get constant follow corollary expectation aggregate estimator first subsample suppose take inequality respect subsample outside hand entire recalling obtain rank brevity denote j g j proof proposition ni ni g systematically k ai f fm condition ng cf complete classifier r bt bt j I I f plug brevity j prove statistical berkeley berkeley california xu comparative model journal l minimax ph thesis plug asymptotically estimation infinitely censor institute mathematics integral representation theorem g machine reduction regression probabilistic recognition york fan inequality completely continuous k n n density statistic negativity estimator journal multivariate scoring association journal american new york new york mirror average mirror average new component mathematic family use principal discussion journal american association statistic example projection discriminant science estimator advance modeling inference world fourier framework introduction york nonparametric adaptation freedom cc aggregate ks ratio size noisy black area dark gray clear white classified figure figure cm cm multivariate form noisy model either component estimate mirror aggregation achieve estimator density construct classifier logarithmic dimension experiment promise factor aggregation complex lie multidimensional occurrence science source kind include biology genetic network gene molecular imaging internet phone management important challenge visualize well even moderately motivate de li al paper generalize ordinary ica recover hide ordinary assume source unknown ica equal mixture without source situation involve source factor ica ica concentrate source xu xu mixture model present serve reduction suppose noiseless estimate know section recently supervise plug classifier label achieve excess condition classifier penalize empirical risk difficulty plug moderately region purpose gr suggest overcome outperform quadratic poor early discrimination orient projection pursuit produce quite high procedure appear model source distribute report promise result section give excess plug plug achieve logarithmic bayes describe implement report application consider deterministic loading zero mean normal mean covariance identity ica widely blind source separation source basic ica assume et unlike literature goal serve different gaussian technique column know propose factor extra arbitrary rotation factor allow throughout assume orthonormal convolution convolution matter whether mix kernel hard rate give moderately example show source orthogonality eliminate dimension specify none quantity orthonormal substitution expression q q denote th density estimate density convolution smoothness property estimator bandwidth use la integrable several follow suggest outside nonnegative density q discussion give know estimator integrate note distribution identifiability factor allow since section still outline strict factor example noise imply ii sample arrange decrease consistency root consistency matrix fan slow conditional estimator bind plug nearly excess size density union sample one problem consist predict assign membership explanatory associate denote population borel misclassification exist sample misclassification goal construct classifier possible estimator relate excess plug follow noisy let mirror denote excess numerical aspect dimensional bandwidth singular let svd center sort use cf form density feasible fast huge procedure control implementation average computation integral realize mean form node associate integral involve multidimensional domain prohibitive exponentially realistic alternative monte integration sample monte carlo several consider fast one realization density array dimension end estimate kernel estimator algorithm estimate independent compute kernel iy diagonal output predefine generate cumulative pre linear noisy extensive source include distribution well unimodal multimodal dimensional run generate signal ratio snr process q density estimation ks implement ks package implement matlab request ks effectively contrast integral compute necessarily lattice impose conduct numerical mix matrix gram schmidt monte replication correspond noisy brevity representative figure display present snr chi superiority aggregated ks show analogous case factor keep
selection base cluster necessarily fall impose fig cluster plot red bar member bar field galaxy red symbol weight weight square color error galaxy comparison width ordinary gmm panel show measure ordinary strong nearly clearly power without contamination location mean scatter strong scatter gmm mostly member point relation galaxy member generally respect lack galaxy galaxy extend us slope cluster measurement red follow process removal determination member galaxy measure slope slightly method directly likelihood red galaxy section assign membership difference galaxy within fit member galaxy color cluster background likelihood identification galaxy straight measurement call tb tb point dot bar every strong despite slope see mean slope bin mean place measurement deviation apparent observe slope statistically slope evolution slope elsewhere slope red cluster strong measurement determination galaxy red galaxy cluster contaminate foreground reject via galaxy address foreground result preserve galaxy galaxy choose galaxy extension slope field fair bin galaxy bin velocity slice km galaxy separate galaxy sequence galaxy big galaxy sample fashion one galaxy galaxy band come color inverse error record bin illustrate slope slope panel high environment correction perform scatter increase magnitude scatter evolve red frame galaxy find intrinsic rest increase frame reveal scatter show statistically significant iv qualitative agreement trend slope kind scatter trend slope cluster lack color fraction galaxy star formation increase contribution come choice importance use place galaxy motivate choice derive band indicate intrinsic precede intrinsic evolution color toward member paper purely sample variation location scatter width correct measure improved cluster galaxy apply method recover namely variation slope scatter sequence slope trend observational correct k corrections measurement individual attention formation measurement check acknowledgment lee helpful tm acknowledge grant de er would thank physics nsf theoretical national science u energy site http www institute group university national institute university gmm introduce brevity represent mixture color galaxy color member galaxy sample component density color color extend parameter could relate eq iteratively parameter maximize likelihood lagrange multipli multipli q arrive similarly within analytic iterative major iteration relation fine eq back arrive relation round ignore measurement relation easily simply datum variance repeat upon request p david center laboratory il department ann mi department university il department university ann mi california b center university department il institute particle physics stanford stanford university stanford department physics il laboratory red cluster optical measurement scatter red sequence red sequence galaxy new correct gaussian mixture galaxy use technique remove effect intrinsic select galaxy sequence measurement red galaxy find scatter increase slope observe slope galaxy observational check galaxy trend scatter intrinsic evolution present base galaxy large universe abundance growth expansion history universe feasibility parameter demonstrate author galaxy recently red evolve core varied least mean cluster part red galaxy population galaxy dominate color old star observe color smoothly optical cluster application yield proxy precision measurement extent exploit accurately characteristic measure cluster red play important complex physical galaxy include galaxy red environment relation identify galaxy high environment galaxy red sequence galaxy population rich cluster scenario summarize fill formation formation magnitude space slope scatter red scatter sequence effect age sequence measure color allow galaxy well refined cluster measurement image red measure various individual turn considerable digital survey sequence elliptical galaxy various environment study identify red galaxy constrain galaxy relevant galaxy sample robust red using scatter systematic slope scatter handling error correct gaussian reliably recover property measurement relevance cluster trend red emission early galaxy dominate old rise remarkably galaxy color addition galaxy color filter band long therefore informative color detect member galaxy field galaxy galaxy member whose color cluster narrow width galaxy color broader separate component represent galaxy double adequate model gmm suit gmm negligible measurement galaxy measure intrinsic scatter cluster contamination galaxy without account scatter large intrinsic color scatter break toward band intrinsic measurement error maximization em fit clearly good sense decide free bayesian bic component corresponding small intrinsic scatter measurement method gmm fit subscript cycle cycle denote location width point gaussian brevity parameter give q maximize expectation way introduce tell case negligible arrive lead maximum component fail initial suppose point repeat process resample set datum get estimate beyond good result take real monte whether reliably identify cluster component reliably input parameter respect cluster member color cl galaxy galaxy color cl generate allow cl vary keep choose color member background galaxy field note color generate noise noise add plot cluster
gradient descent work magnitude distance lipschitz continuous denote first reduce claim follow opposite arrive parametrize curve equal distance path note derivative path expect assume loss vanish fast exclusive integrate yield q side switch prove note proof monotonic information part generate consider path optimal particular reach define eventually restrict always align derivative choose loss generality weakly monotonically increase point definition integrable integrate multiply side become expect vanish smoothly vanish control theorem assume gradient risk bound quite gradient optimality observe move make oppose mean reasoning extract bound l delay expectation feasible independent upper bind appeal monotonically stepsize small wish rate risk delay small last bound bound likewise divergence lastly integral term period cover segment guarantee plug collect yield divide govern regime initially quite increasingly delay essentially dramatically affect parallelism step average dominant parallelization set conclude set strongly smooth occur logistic surprising ratio eigenvalue theorem rate provide second inequality rate decrease combine obtain also simplify rhs divide dependency factor fully make generalize bregman begin convexity moreover convex whenever finally function scalar delay convex obtain algorithm unnormalize delay identical strongly update bind key constitute yield exploit continuity transform obtain tight easy example considerably scenario feature bins bin come canonical distortion hash dimensionality pick try delay goodness check system delay secondly whether scale well delay update upon hash regularization e code machine gb core parallelization divide piece give piece piece master piece master master add piece together proportion magnitude quickly multiple dot dot maximum would simply first run delay observe delay example bad run delay delay try turn handle slightly one find parallelization dramatically show parallelization trying delay intuitively delay theoretically example effect small three simulate delay secondly hard problem small prevent prove delay parallel paradigm choice large framework stochastic online excellent tool address current algorithms process receive instance make update word entirely process modern machine graphic core core disk processor speed typical network interface throughput mb disk array reach size whenever amount cpu distribute bottleneck propose evidence work guarantee variant core gradient sharing update accelerate intensive problem whenever gradient computation subsequently update occur delay core available parallelization synchronization consequently comprehensive cloud home core computer processor execute piece code processor easy exploit affinity core share architecture processing graphic tend element execute piece synchronization kernel process synchronization mechanism undesirable come expense significant memory resource availability graphic mb high speed ram per communication nontrivial bandwidth computer communication communication equivalent cycle tend slow server configuration communication unable directly disk network storage transfer moreover typically second reduce processing stage play critical analysis exclude parallel suit problem banach family category vector variant game communication within team adversary response whenever induce loss goal cumulative loss minimize abuse loss achievable radius anneal schedule compute update gradient anneal entirely delay current gradient previously extend extend implicit update divergence section lead parallel modify bound plan delay function measure bregman define need auxiliary lemma instantaneous divergence decompose expand product delay gradient distinguish difference protocol yield plug show project decomposition key characterize successive gradient bad cost constant prove briefly identity inequality lipschitz gradient via tackle term diameter last sum discard contribution negative become lipschitz property decrease hence gradient q plugging rhs processor claim converge bad adversary may algorithm fast result practice bad assume online regard least old construction function chance every consequently instance guarantee delay could set strongly difference correlation eq monotonically increase pay delay version minimize objective typically small function combination successive stochastic gradient descent
collaborative reconstruct considerable collaborative ranking subset movie rating factor contribute user preference infer dimensional incomplete distance wireless recent algorithm low guarantee successful high recovered rank entry np adapt compress sense incoherence correctly recover recently bound sense without impossible fix due entry per row value suboptimal appear manuscript guarantee relaxation incoherence original relaxation recover non rank completion unique introduce randomize complete specific furthermore minimum theoretical focus prove completion low approximate provide rmse plan generalization relaxation rmse analogous side directly solve grow proportional year solve thresholde atomic trying solve describe solve problem minimize singular matching problem nuclear pursuit iterative approach hard thresholding procedure estimation incremental broad performance guarantee estimate original reveal turn broad show underlie condition carry reconstruction algorithm applicable organization relaxation mostly introduce efficient modification discuss numerical assume dimension entry let matrix original matrix rank solve recover operator matrix match observe notice doubly especially matrix sense convex relaxation equivalently recovery completion nuclear nuclear I singular lagrangian rank namely performance compete minimizing provide excellent high initialization add description represent estimate rank input initial condition represent twice entry row analogously column input represent grow spurious dominate respectively singular provide unobserved entry degree throughout singular following base minimizer idea reveal clear separation reveal matrix reconstruct spurious one describe guarantee reconstruct appendix bound rank probability consist perform rescale singular appropriately svd projection notice require available form involve define estimate compare eq allow suitable proving throughout paper fx p complementary set paper explicitly x definition generate interpretation justify manifold descent r manifold k left matrix start numerical good stop p f basic criterion also author provide initial tolerance iteration size w em condition novel case reconstruct ill far discrepancy start first singular next incremental output projection e x yx k em demonstrate incremental bring gain implement test computer gb use modification simulation section different scenario completion generate sample independently reveal reveal notable use stop criterion generate corrupt additive identically follow subsection reveal independently probability entry stop matrix illustrate convergence matrix decay iteration close validity criterion next reconstruct reconstruction fraction curve prove plot plot rate extra come surprisingly low one solution lower display rank proved figure plot use plot rank rate sharp threshold location surprisingly close bind admit compete algorithm rank plot relaxation solve algorithm low rate algorithm consistent various present correspond hence outperform algorithm error time per c time entry per column high novel robustness incremental exact completion generate simulation ill condition let orthonormal respectively diagonal entry linearly form criterion incremental improve different mean square error define comparison start take entry entry standard relaxation performance relaxation performance oracle comparison root one small square error become indistinguishable low compare average error root illustrate different rank c time observation corrupt ratio row column illustrate performance change entry gaussian distribution unless independently add noise generate matrix observation missing estimate variance entry depend completion scenario reason error suit ensure matrix almost evenly distribute performance change reveal per row scenario distribute equal accuracy measure rmse result rmse show rank bad gaussian coincide rmse close oracle implement performance observation curve reason rmse decrease return return performance bind factor projection reason rmse snr less good rather get rmse close correctly localization sensor observation assume formulae choose note case entry rmse multiplicative rmse correspond display figure respect noise difficult distinguish motion position capture location failure corrupt outlier rl accord target independent entry noise outlier value affect figure clearly noise norm error however standard quantization regular near choose carefully quantization bad multiplicative entry whereas show
pass form considerably consume expansion phase slow indirect diagram align force see eq series drive force implementation start amplitude unit svd transfer successive assume series drive vary smoothly considerably slow closely work drive allow absence driving consider delay define scalar odd even index center alignment driving visually drive bring drive alignment offset sign define align constant indicator low value slow signal signal dimension unit fulfil e extension result embed logistic completely corrupted number expand eigenvalue svd mean order magnitude drive fast eigenvalue blue dot red break whenever negative complex eigenvalue experiment implementation fail expand become indicate expand space routine generalize eigenvalue singular fact eigenvalue occur matrix svd matrix regularity time happen short logistic move singular observe low high natural noise unlikely singular perhaps reason algorithm circumstance frequently application happen svd occur r r signal amplitude case solution rank might correct exception r r e way deal rank embed thus svd extent try parallel way signal constraint cutoff algorithm one rely eigenvalue secondly usually condition test usual result phase dependency large least show eigenvalue circumstance wrong term slow circumstance svd closely approach stable matrix implementation available algorithm original execution since consume part expand datum rank span number dimension expand cutoff threshold insensitive span decade reach noise amount noise dimension drawback carefully tune new drive force plan always look covariance sometimes worth modify accordingly review regard give define covariance diagonal dependency dimension carry amount whiten search zero transformation eigenvalue contain eigenvector row eigenvalue usually singular easily verify course run close become infinity noise error multiply trick svd deal singular replace effectively remove become row invertible subspace zero eigenvalue eigenvalue investigation numerically contrast eigenvalue eigenvalue package feature maintain namely algorithm look main modification v svd interface line execute code signal new pattern goal minimize pairwise difference new diagnostic drive force experiment line handwritten benchmark uci repository full modification available package grateful work university science grant cm cm cm http implementation http www de analysis extract slowly vary multidimensional signal easily circumstance expand decomposition free handling multidimensional signal modify datum tune slow ok approach ok ok eigenvalue ok svd ok improve ok broken slow ok svd ok discussion ok remark ok ok zero ok dependent cutoff ok conclusion ok appendix ok ok deal ok ok ok ok slow processing slowly multidimensional series already successfully numerous reproduce complex cell primary formation handwritten digit extract drive force role datum understand various temperature drift vary heart parameter refer drive force dynamic smoothly slow rarely e eeg electrical particularly drive force observe aspect clear convenient sphere expand transform basis accordingly signal singular however eigenvalue pairwise eigenvector define desire obvious direct fourth eigenvalue sequence signal signal zero eigenvalue equation q sec multiply line four expand possibly nonlinear ii sphere expand obtain component zero mean derivative expand matrix
full stop count wrong approach conservative expect batch draw batch batch tend ip batch ip batch number calculate analogously substitute voting possibility batch batch contrast bound ip batch bit draw strong single correct conversely five zero confirm low counting draw expect number test three independently simultaneous procedure chance hand incorrect outcome incorrect maintain keep split across chance least count count count incorrect outcome total column design control draws expect distinct batch vote draw distinct distinct vote row batch far work vote independent control expect batch analogously expect batch number independent control effort vote associate batch count work batch high independent control far low pairwise batch incorrect outcome compare independently batch vote en n en na na nb nb nc tc nc nc nc en na nc nb single random relative margin limit entire build sampling scheme control chance fail full count per batch keyword size sequential simultaneous post full count pilot california risk conduct collection arbitrarily count appear build state california sample vote pairwise margin extension margin winner margin candidate error batch summarize pairwise margin exist incorrect instance limit chance go set application use chance hand wrong wrong margin winner suppose together represent batch every candidate total take candidate candidate lose apparent vote candidate batch report apparent apparent actual vote candidate vote would include actual vote apparent must relative pairwise margin maximum pairwise margin apparent outcome hand think incorrect family strong apparent outcome hypothesis batch rp batch margin batch need otherwise cause wrong must spread batch even batch proportional error observe compound apparent outcome differ outcome calculation procedure outcome test hypothesis outcome keep incorrectly conclude outcome batch convenient batch draw apparent winner apparent b involve half apparent winner winner batch either mail batch illustration batch margin batch rr ip batch winner winner winner entire include include b batch cast ip cast mail
take program reference list least sum compute program uniformly requirement upper version depend every list neither logarithmic additive logarithmic string denote word xy x ne yx list element additive constant short short program program mapping element program addition nothing string show origin fundamental string theorem list string length extend finite list triangle property divide improper normalization list reduce restricted lead proposal improper symmetry violate equal hold kx inequality divide divide equal triangle inequality divide divide element correspond list change equal th dividing divide inequality contain useful kolmogorov call symmetry hold logarithmic short universal function additive kolmogorov string logarithm result short code great section proposition remark centre science address email support parameter free extend pair study kolmogorov purpose approximate version use real world pattern kolmogorov mining information information objective object object object multiple object classical kolmogorov objective information pair practical focused arise normalize kolmogorov real alignment measure impact decade conclusion classification represent file weather forecast music bioinformatics internet abstract information engine produce aggregate phrase relative semantic run semantic question answer reference many interested object example customer review article occurrence comprehensive specialized thus go object extract essence example list internet review tv list binary string string order list express string universal machine convenience string distance string term state interpret length comprehensive interpret specialized object similar information list practically theoretically promising case imply constant overlap take correspond state argue prove minimum normalize list element distance v necessity idea general case pairwise kolmogorov information subject informally kolmogorov complexity string universal constitute program technical reason choose read right without computation take upon initial call set program free reference definition kolmogorov shall simply kolmogorov formally kolmogorov short reference input output unconditional kolmogorov consist nonempty string present appendix list list string may order abuse conditional kolmogorov list length short machine list kolmogorov put law term list x overlap go single string length everything additive logarithmic length bit possibly suffice element finite length string k put edge next care trivially color length color appear since color always know reconstruct know appropriate length length suffice select element take encode vice versa string length possibly program encode string side shortest assume interpret kolmogorov comprehensive fact choose go list program overlap quantity short list string order increase list finite distance list order length distance metric symmetry permutation triangle inequality
sr r prior actually loading ibp prior prior synthetic dataset propose nonparametric gene connectivity synthetic sample gene underlie ground factor efficacy factor loading bind site also expression breast gene prominent figure actual infer factor recover loading bind ground approach loading loading permutation follow spurious factor hierarchy fast configuration never std mse response compare variant approach fit separate discover see phenotype binary figure real hold predict reconstruction random initialization variance suggest fairly w initialization nonparametric factor ibp power ease integration specific gene improve output factor interesting open ibp model cs edu account relationship factor variant couple base apply data task solely achieve benefit discover underlie predictive compact motivated feature greatly potentially overfitte approach stem reconstruct gene expression datum pathway contribution parallel need gene pathway couple predictive instead model fundamentally treat relationship propose variant ibp design account ibp explain pathway fundamentally involved synthesis nonparametric ibp distribution infinite motivation bioinformatic alternative sample movie versus action movie spurious process e pathway relationship pathway matrix originally motivated observation analogy customer enter restaurant infinite customer incoming select select customer select easily precisely customer select stochastic thus infinite binary turn stochastic limit exchangeable kp km z ibp nice possible ibp second parameter control factor parent easily individual limit exactly continuous process singleton evolve leave event pair binary topology infinitely exchangeable therefore limit markov evolve brownian dimension covariance non node gaussian pass al propose agglomerative approximately maximize propagation associate message current child recall factor consist feature factor variation treat factor purpose simple begin factor ibp ibp hierarchical infer presentation mechanism ibp factor ibp apply nonparametric past ibp places ibp feature gene factor factor loading context gene usually small factor ibp prior number model unbounde expression binary factor loading instead use hadamard analysis I ibp prior hence inverse gamma thousand pathway factor analogy enter restaurant spurious effectively ibp sparsity fundamentally conventional ibp ibp ibp rich get rich get truly whether likelihood correspond ibp ibp exception customer gene bernoulli basic fact hierarchy ibp describe exchangeable mean efficient ht consist b depicted aspect ibp prior follow example component nonparametric factor analysis response treat factor binary extra probit predict binary response summarize propose set pz ik pz ik ik simultaneously proposal acceptance fast show figure propose new leaf new find tree node tree accord prior proposal uniform predictive newly pass pass variable indicate
minimum arbitrary direction often example strictly hessian low family behave quantifie almost strong let analytic standardize eq q mle phase initially somewhat newton quantify enter arbitrarily burn central theorem key idea central shorthand moment eq hold key characterize expansion attempt use moment expansion argument e comparable selection small subset away whose support mostly see fisher eigenvalue complement quantify substantially weak subset previous different need replace small regularize optimization expectation reduce set regularization noise state deterministic e free appropriate quantify satisfie analytic optimization fisher bound intuitively expect think dimension hence favorable condition quantify distributional assumption actually relaxed sub eq bound sub gaussian unbounded long estimate linear sparsity general characterization level solution two support first proof generate analytic well expansion proof specify analytic prove core theorem furthermore convexity prove consider jensen know fourth standardized proceed analytic moment second claim max argument claim follow lemma case ready claim us solve prove prove claim assumption second use triangle proof support divide add ready prove theorem theorem note satisfie observe eq use restrict use conclude showed care standardize specify clear exposition bound leave coordinate plug choose obtain theorem applicable complete second case q us simplify conclude q q simplifying claim bernoulli thank email existence exist classical follow easily completeness work polynomial root claim root root gauss root derivative hull real root extra claim express effective dimension sparsity characterize convexity ability quantify show exponential discrete optimization generalization issue ambient much large size special high body characterize rate tackle challenging grow model family hold though modern problem rapidly asymptotically case quantify relevant aspect family throughout agnostic necessarily generate analyze log nature convex asymptotic limit log gaussian information quantifying occur particular rather natural standardized standardized recall standardize moment th grow similar study tail growth rate characterize rate exponential draw newton burn behave locally quantify strong eigenvalue design show family enjoy rate condition optimal incoherence condition provide essentially family relate final selection low drawback mutual incoherence permit perfect feature price sparse eigenvalue multiplicative level recover exponential family merely nearly mild risk result rather mild favorable lie statistic finite exponential general though keep mind variable covariate point loss eq natural space later eq consider expect fisher minimizer main quantify family family also property regularization find standardize satisfied family prediction behave quadratic analogous exponential tail standardize th power normalization deviation standardize moment use term reflect analytic standardized denominator analytic respect subspace direction univariate mild use obtain sharp bernstein th analytic tail standardized generating th neither analogously th standardized deviation quantity use certain setting natural standardize univariate bound hold denominator analytic standardized subspace bind interior analytic standardized finite analytic going issue mind quantify
decrease eq pareto admit arise mix derivative q whenever derive tend tend toward apply easy whenever law although may theorem explanation build process see model prove right think scatter characteristic link multiplicative scale invariance real set generalize version replace indeed idea regular mod preserve scatter scatter closely u henceforth theorem continuous law thank implication lead digit general apply six mathematical sequence slowly perfectly last experiment l kolmogorov term exception display kolmogorov arrange speed fast go fast understand integer kolmogorov odd perfect six uniformity allowing theorems v except everywhere absolute distribution digit uniform logarithmic function tend toward tend uniformity root concave exp j j dt expression toward complete mean l yes yes exp last kolmogorov apply read noticed root rather exactly rare roughly approach law directly relate multiplicative formalize general regular intuitive idea regularity actually thus explanation course argue fact study ii apply may argue mixture density regular multiplication lead density explanation simpler arguably good argument favor invariance relate easily historical implication digit proof remark mod approximate version phenomenon receive depend assumption often link author implicitly characteristic variable come regularity part prove intuition regular simple corollary result law link log view law scatter digit code law sequence number mod random uniform stand logarithm recently many vast fit law pass seminal paper test population half law r binomial array tend toward law law detect anomaly pricing report finance indeed one put forward appearance particular variable rule special show invariance imply limit multiplicative datum look truly explanation notice likely law order cover magnitude normal invariance assumption view mathematical random law link circular express transform signal expert would smooth implication explicitly actually explanation relate r formalize probability henceforth example scatter idea follow scatter imply irrespective surprising explanation several far simple property normality admit r uniform law special law datum various explanation uniform mod soon regular precisely regularity scatter
preliminary remark well approximated eqs analogously use inequality course imply f b v point bound bound proof eqs cn notice begin four form b bound c r c c us thesis b three one n observe f small hypothesis us u ad u degree eqs simplicity case treat manner side ib desire bound j j ji analysis exist generalization strong matrix discrepancy hold bipartite show subset discrepancy assume thesis remark let principal angle span thesis inequality x write x tu u f u tu therefore thesis dx perform explicitly hand f whereby thesis plus pt minus pt minus proposition corollary remark sufficiently guarantee reconstruction complexity massive proving statement spectrum matrix imagine customer available dataset customer movie pair rating rating miss suggestion make community predict rating error know one problem solve particularly important massive actual mathematical rating rank assign movie dimension dimension retrieval refer large limit away factor notion formalize es incoherence satisfy uniformly alternatively incoherence bound rating reveal decomposition iy singular rescale matrix rescale poorly contain column singular concentrated respectively singular operation column reveal diag entry per heavy column amount rank apparent important reveal heavy following let project residual eliminate small particularly consists locally minimize quadratic low furth column manifold well understand definite descent search practice loop cost instance collaborative sparse matrix achieve small entry frobenius matrix correspond usual section top value efficiently iteration iteration operation prove one exponentially calculation mention systematically assume procedure prove basic close approximated quadratic observation due necessary contain one entry consequence bound suboptimal collaborative use machine processing satisfy set heart compress collaborative matrix match entry entry thus incoherent prove correctly purely consideration reconstruct point counting treat realistic semidefinite program pose important provide convex science important fast rank hold short international completion prove substantially different point simulation completion fast degree indicate characterize point theoretical recent one study collaborative far uniqueness solution completion fast rank incoherent order formalize incoherence factor shall say satisfy apart difference normalization coincide assumption entry whenever depend reveal entry reveal probability guarantee performance well vanish notice transpose denote denote vector matrix sometimes first integer explain crucial consider matrix let rescale understood eigenvalue spectrum rank probability theorem inequality q use lemma want show basic belonging apply discretize I heavy discrepancy challenge bad enough definition light heavy z notice relate original discretize x max would apply contribution j subsection prove remark prove thesis subset column j entry rough size column event belong want proceed bound tail estimate l enough pz lm bound proof whose potential rl ij max mn z e follow thesis follow pz lm bound analogously finish contribution bind mh matrix entry satisfy mn iy notice bipartite set result mn ij iy n result adjacency bipartite hold inequality heavy recall singular understand max analogously minimize naturally view manifold important fact geometry calculation defer section recall numerical etc denote matrix orthogonal manifold subspace easy depend matrix mm gm n manifold reconstruct give correspond usual scalar w distance geodesic arc arc principal angle column useful fact propose admit computable form arc length express singular decomposition time vector curve one cost condition fully factor regularize row force remain incoherent take appendix compute kt kt kt x kt x kt kt way consequence c theorem discuss gradient pose indeed need repeat hessian lemma function numerical happen c stationary define lemma claim c gx triangular x claim observation gradient descent exact unique stationary point time write recall degree order term numerical thesis contain set neighborhood obviously equivalence canonical remark
indicate ref expect resample replica sophisticated resample section backward unable discrepancy run back particle c component focus particle thus cost complexity path main gaussian condition publish search support office science u department contract de ac national foundation dms tu department mathematics university california berkeley berkeley laboratory berkeley particle filter datum assimilation sequence function pdfs significant number maintain offer detail keyword particle normalization many must consist sde brownian scalar diagonal identity motion equation assume random simplicity evolve state nonlinear independent linear solution kalman distribution approximate filter posterior information past avoid identical particle expensive backward markov monte see alternative approach sample bayes backward monte carlo density parametrize construction idea sequentially nest conditionally subset sde explain already interpolation mesh use f nx nx eq function see solution mcmc increment know explicitly hand pdf pdfs check get pick obtain similarly fix normalization repeat sample previous obtain want pick draw remain iteration guess increment long case increment repeat start iterate vector leave become different also use strategy readily work satisfied iteration establish suitable course choose result variable context problem discuss demonstrate capability filter set point plane variable previous observer make measurement quantity scalar denote letter reconstruct position follow particle section show result datum discuss section increment known sde come normalization increment position reference particle pick independent reference present calculation unchanged increment normalize call say increment assume value brevity start explain observation like evaluate taylor series around define dy j dx dy normalization variable variance orthogonal connect find begin complete phase converge time short iteration create particle sample gaussian vary one number distribution density bayesian get probability sample respect phase cumulative weight exceed summation resample terminology divide size create great diversity strategy step create accuracy refinement future past tag probable observation make observation sampling often solely diversity particle boost problem sake completeness set step partial history particle occur history share among particle know last member project quantity remain deal slight increment increment slightly usual go intermediate correction observation half way reach connect replace accordingly look dy dy gaussian equality wish define forward impose observation time equality parameter increment separate equation motion converge approximate direction approximate multiply pdfs variance time create dx dy phase iteration phase forward step one
mixture copula universal approximation contrary dataset copula poorly model normal component conversely normal normal weak finding normal density factor marginal approximated show normal marginally adapt mixture normal expect mixture normal approximated adaptation concept outside normal mixture normal simple behave fast approximation copula multivariate copula cube marginal correspond transform popular copula implicitly transformation multivariate cdf marginal cdf transformation imply copula implicitly take normal challenging fit separate treat copula follow distribution marginal demanding give estimate function suit incorporate outline copula define normal copula mixture normal number choose normal implicitly copula component pose evaluate account distribution normal implicitly mixture normal section empirically copula normal skew estimator main I normal copula perform copula moderate need hold mixture normal copula bad normal depend simulation marginal mixture normal normal copula stand alone never simulation estimator appendix divergence estimate compare performance estimator kullback loss q similarly simulation number nc freedom tc normal frank copulas normal mn skew st aim capture describe briefly copula normal comprehensive set extended simulation normal report divergence multiply percentage increase copula ratio logarithm ratio ratio loss median report mixture copula generate confirm simulation normal copula perform similarly criterion almost always select component normal copula normal copula copula negligible copula despite loss normal estimator skew substantial simulation datum mixture normal perform copula generate one normal outperform depend density two reason normal fit well bad normal direct marginal indirect joint transform normal difficult normal conceptually report indirect density joint misspecification mild misspecification parameterization normal define impose normal parameter normal copula normal univariate density normal normal become highly parametrize get large normal copula marginal component medium large normal copula confirm analysis ability normal process bad rather likely group normal poor criterion generate perform multivariate easy cluster evident bivariate normal variance show marginal distribution copula normal component confirm need separate copula datum normal normal poorly poor normal copula improve copula still loss compare normal section highlight place strong conversely focus hard effectively idea fit datum fit marginally close make precise dimensional marginal I f iy h iy iy give sufficient condition define suppose multivariate density marginal lemma everywhere note give know suppose know note condition verify know density estimator marginal necessarily note complex multivariate distribution degree normal nonparametric density complex assume marginal specify normal normal therefore straightforward marginally adjust normal normal poor normal general normalize slightly prevent bad ratio iterate adaptation need example component normal considerably experience help tail capture marginal marginally metropolis proposal efficiency marginally normal normal marginally mixture normal estimate generate mixture normal mixture normal normal normal multivariate estimation estimator datum multivariate normal normal copula mixture well component three fourth component relate efficiency loss normal marginally adjust mixture normal marginally adjust normal mixture normal normals copula pure regression simple regressor unknown straightforward financial display moreover portfolio use practitioner excess asset health website http page ten validation copula skew mixture normal choose ten subset mixture normal three factor assume insufficient representation well model volatility finance decade construction treatment volatility distributional dynamic find exhibit long memory skewed logarithm realize display approximate study economic realize volatility report pay capture gain volatility volatility relative return daily realize period year daily return day bivariate vector lag specification capture cross validate sp explain normal apparent marginally normal copula disease cause concern molecular great develop anti expression level gene hour cycle expression process cluster number multivariate degree freedom estimate result normal bic ten component model none eight cross subsample marginally normal average copula mixture normal multivariate identify modification simultaneously well marginal challenge able perform thank pt pt axiom conjecture
sampler metropolis hasting yield algorithm include ease mcmc implementation empirical full associated wise markov particular connection strategy ensure existence mcmc confident identically sample hierarchical one maximum estimation mix fundamental carlo metropolis update proposal acceptance create state chain proposal particularly lead optimal kernel update variable sub block choice unclear advantageous component correlate target distribute version optimal hand also show metropolis langevin updating use consideration rate borel x chain value quantity value whose simulation markov approximate average n gx justify ergodic along condition existence ensure central limit existence sufficient batch mean method strongly ensure monte least tool confident independent much establish ergodicity metropolis full dimensional study establish yield geometrically hasting chain include well none full ergodic many updating example ergodic com establish ergodicity scan metropolis walk fix walk step perform gibbs sampler geometrically scan sequence sampler probability ensure uniform ergodicity strategy despite none metropolis hasting sampler especially two fix state particular connect scan sampler scan random scan develop condition ergodicity component metropolis practically relevant gibbs likelihood empirical sampler counterpart component wise sampler support wise practical technical fundamental combine kernel mix markov mix eq kernel preserve invariance ib b dx iy satisfy conditional invariant correspond elementary correspond hasting dirac delta wise irreducible combine scan eq admit q another wise update easy order composition use combine order clearly special employ satisfy com mcmc focus deterministic scan sampler goal rate sampler mix brief description establish existence chapter borel p nx j j ergodicity small ergodicity begin sampler component ergodic establish ergodicity ergodic ergodic wise ergodicity follow case ergodic ergodic probability component chain random walk chain block expect produce markov hand full hasting ergodic component I algorithm independent proposal case sampler uniformly typical accept reject truly think extend proposal update existence ergodicity argument case able give pair together ergodicity describe position markov additionally nan result define suppose exist eq ergodic ergodic selection follow corollary indicate verify proposal ix ix dp condition amount require jointly almost observable th k normally analytically challenge monte carlo monte maximum monte unknown require chain independence sampler proposal full proposal marginal invariant ergodic performance geometrically ergodic walk sampler concrete suppose geometrically nonnegative call metropolis gibbs markovian invariant se markovian let nx ny geometrically ergodic geometrically connect straightforward ergodic geometrically sampler condition yy lx effect know know nx p satisfy density hyperparameter posterior observe generic conditional report gamma gibbs sample gibbs sampler update conditional markov say theorem establish ergodicity row n chain gs gs ergodic geometrically ergodic gs consider performance full efficiency chain geometrically ergodic central limit valid confidence interval quantile chain sample integrate autocorrelation carlo random provide size quality estimate assess mean replication quantity efficiency estimating chain move square jump square successive denote jump chain summary graphical summary trace take picture examine consider subject subject full coefficient q next definite matrix model gs geometrically gs comparison focus rw use normal equal rw acceptance rate know rw geometrically simulate value markov prior k kk geometric gs variance asymptotic run rw substantially sampler rw rw gs mixed middle show half width integrated act relative gs rw gs four simulation reflect half act size rw half act rw effective carlo replication rw gs take obtain rw ratio gs give standard notice ratio great rw sampler rw gs consistent discussion sampler explore sampler gs mixed independently distribute introduce current condition four sampler ergodic rw sampler normally distribute proposal metropolis sampler proposal ergodic compare implement carlo call q likelihood effect set table chain density define entire markov asymptotic denote consistently mean generate rw logit section implement skip report implementation simulate chain metropolis jump proposal determine minimize autocorrelation result yield plot update panel analogous appear panel four result rw significant autocorrelation panel mix rw panel appear sampler suggest conclusion entirely closeness sufficiently conditional worth recall theorem chain illustrate implementation time act panel estimate replication standard
approach wide sm sm c sm outperform et al et relatively presentation label solve box constrained mean method iterate regularize constrained programming appear problem apply instead matlab code terminate use absolute approximate end cycle consist coordinate descent code terminate accuracy see pp experience criterion terminate relatively set obviously thus fair termination detail duality word cycle consist terminate gap since inverse need reasonable duality gap iteration iteration intel ghz machine instance present table sample four eight cpu column substantially also outperform small termination hour code number five maximization hour except scale one hour well objective instance scale increase digit rr rr rr time c smooth concave maximization approach variant show substantially latter study et et subsection impose associate view iterate would nesterov technique compare though outperform optimization preliminary occur highly analyze behavior sequence hence k completely open smooth heuristic nesterov general max written matlab www code suitably plan extend eigenvalue like nesterov scheme author two greatly improve pt theorem remark author support discovery grant strictly concave maximization nesterov iteration primal sparse approximately solve outperform compare approach namely nesterov block method sparse covariance smooth outperform method variant c k nesterov convex problem close method final proximal optimization concave admit smooth convex dual counterpart result find dual problem one give impose estimation apply determine simultaneously discover despite popularity numerous real reference therein combinatorial technique lin show solve norm penalize author efficient order nesterov approximation scheme show iterate quadratic theoretically release paper variant coordinate square programming appear paper attractive optimal substantially outperform compare randomly instance show nesterov substantially mention smooth maximization interested propose smooth briefly solve penalize maximum estimation optimization compare smooth optimization randomly finally conclude remark matrix semidefinite write semidefinite otherwise frobenius identity entry determinant eigenvalue symmetric denote space endow denote functional endow dual operator concave non maximization strictly every conclude convex give endow arbitrary norm continuous assumption suitably solve prox strongly modulus generality nesterov approach smooth concave sd fu u du fu fu kk algorithm nesterov sd k minimization ready maximization special smooth algorithm termination apply minimization algorithm exceed q function xu kk imply fu fu u xu fu fu fu u gx conclusion mention enjoy addition propose give nesterov smooth except former prox subproblem every prox subproblem cost smooth subsection denote clearly know prove strictly indeed since continuous therefore strictly saddle together strictly unique sequence minimization statement compact suffice convergent subsequence convergent subsequence otherwise one convergent u f desire immediately covariance selection subsection solve relation together et bound derive handle eq expression derivation scalar one show discussion observe rewrite define therefore complexity interior method al nesterov scheme lie endow norm concave modulus conclude q denote decomposition show problem therefore suitably give proximal modulus solve ease aforementione smooth minimization algorithm problem smooth xu sd fu fu fu u kk fu gx complexity algorithm solve hold easily follow fu case iteration algorithm smooth accuracy non smooth approximated smooth continuous nesterov apply perturb problem problem iteration problem et block coordinate iterate box mention rate global theoretically moreover reformulate suitably ip nesterov bad initial newton optimal eigenvalue multiplication cost find superior ip small subsection nice theoretical ip nesterov block performance attractive section computational concern indeed computation new termination use know use termination moreover termination one despite advantage shall mention complexity termination useful termination criterion usually fairly know iteration solve complexity drawback dynamically update easily observe unique problem ideally unknown generate asymptotically view know asymptotically approach generate introduce notation active inactive let give fix iterate find inactive kx kx kp k recursively generate inactive fact imply termination replace accordingly termination criterion aforementione convenience presentation omit subscript em covariance choose
series different curse consider determinant cover determinant correlation amount uncertainty curse matrix cholesky eq innovation uncorrelated turn former strong location nonzero recall triangular great eq recall mild average say easier uncorrelated apply hc power apply argument fix positive integer nk plan polynomial interesting phenomenon hold sequence tend sequence apply standard hc yield result learn apply hc yield hc case higher powerful begin fix diagonal zero elsewhere vector coordinate elsewhere compare nonzero coordinate entry coordinate approximately function singleton hc approach strong hc sensitive signal expect great write normalize call hc comment calculation nb n n nb mean stronger stronger hard selecting must mention case decay diagonal nonzero large strength significantly strong preferred modified turn behavior hypothesis large bandwidth suppose reject cut value fast sec suggest select summary lower upper reasonably investigate range dependence strong effect estimate relate still could comment represent series characteristic datum long period record use deduce evidence early detection communication maximum great constant multiple uniformly equal p conventional converge zero consider signal amount acquire increase would involve genomic difficulty information dependence genomic quite argue correlation decay base et figure upper tailed genomic possible effectively expression genomic case signal readily independent randomly distribute note follow problem among location integer distribute distribute among integer pre dependence place assume negligible toeplitz truncate toeplitz k nf first value definite putting see toeplitz matrix convenient asymptotic suppose constant known inverse asymptotically toeplitz generate known result compare combine direct thm toeplitz symmetric satisfie hypothesis asymptotically error infinity bandwidth converge power curve plane uncorrelated region current view corresponding region uncorrelate region rectangular region see enough stand correspond uncorrelated cluster signal generate randomly probability section investigate appear whose location strength right follow shift position backward add signal comprise cluster consecutive g toeplitz density consider eq equivalently view entry expect symmetric hypothesis merge nan converge zero weakly investigate variate equal specify location without display slowly decay calibrate place first boundary turn definition assumption secondly significantly see hc open detection adapt hc optimal key idea correlation three relatively next toeplitz lower diagonal elsewhere additionally entry let equivalently asymptotically toeplitz detail introduce converge coordinate therefore expect consider model display strength merge error converge bandwidth power scale compare hc bandwidth denote hc hc hc hc correspondingly include fix generate randomly otherwise explore parameter setting describe take detection cf datum apply hc hc score nan hc way report type ii error possible cut second percentile hc score hc correspond exceed threshold display cutoff save threshold report power cut asymptotic moderately recommend instead leave three blue display hc hc hc outperform outperform hc large mean strong correlation detection increasingly increasingly hc score small hc hc mainly definition hc hc bottom axis dash display hc hc hc bottom cut percentile hc correspond panel dash green display hc hc hc choice cut hc consistently outperform hc hc consistently outperform hc cut percentile percentage bottom display display correspond hc hc hc toeplitz matrix generate experiment sum report hc outperforms hc hc hc investigate hc hc hc take type hc hc hc improve investigation case omit discussion extend hc hc play curve correlation toeplitz upper wiener interpolation therefore plane hc full fall interior neither call optimally hc performance hc explore hc paper relate mixture work relate low focused proportion hc situation relatively feature useful contribute weakly relate hc classification work focus unknown available even stay current progress show matrix estimate situation approach perform well error experiment hc hc interesting explore correlation challenge improve recent aforementioned b derive way incorporate raise leave proof theorem omit subscript confusion independently show monotonicity hellinger function short hellinger write old integrable dx hellinger combine theorem hellinger distance infinity equivalently eq compare model establish model x tends prove note define observe generic constant c symmetric great norm view greater write converge inter nonzero disjoint simplify hellinger summarize lemma lb state theorem put reduce proof location draw replacement way hc establish hc short n distribution monotone family fix x x consequently tends zero infinity put let survival r n q use proof ns n follow remain detailed state inspection matrix theorem diagonal entry decay condition cholesky factorization cholesky triangular entry lemma diagonal decay decay remain proof hellinger tends infinity small g eigenvalue hellinger suffices hellinger infinity g hellinger uncorrelated converge remove negligible hellinger distance combine hellinger denote divide sec cluster nonzero equal recall cf r nb matrix bandwidth show negligible except lemma last claim cholesky constant continue hold side proof take operation fixing define constant form row direct calculation basic algebra combine give k follow algebra give sequence infinity location signal arrange asymptotically vanish tend fast event define uk calculus last uniformly infinity since negligible inter less decay small therefore calculation n ks small moreover inequality independent sufficiently derive constant page use deduce bandwidth except probability note assume wise bandwidth k thm jt argument algebra independent therefore statement infinity establish note sub close th b norm norm tend zero paragraph proof theorem lb verify location triangular u note hellinger law introduce index distinct np apply restrict old inequality desired consider combine claim follow fix next distance write k k great follow hellinger bivariate satisfy dr proceed derivation inter q combine shall side near lemma first near two set index first one candidate outside generality independence pair q third equality independence use cardinality elementary show q recall deduce last exceed dr n cdf survival claim show write unit give old recalling direct calculation c view dr statement right side converge infinity need tend infinity model nonzero strength randomly draw replacement write direct q chebyshev identity deduce compare statement sufficient definite autoregressive structure exist clearly show equal j kt kt know suffice j jt er k jt j prove discuss separately j k k ts tt zero claim follow directly derive kf q er x x strictly combine acknowledgment would thank extensive van help proof also thank david gr theorem section nsf award dms high detecting signal noise reasonable setting nature exploit correlation indeed accurate level advantage decay rate toeplitz class introduction hc could hc capable optimally signal weak estimate signal al relate white make
row x iy iy apart normalization well regularization introduce technique one regard justified differentiable riemannian mm gm gradient descent generality shall maximum far uniformly subset correspond revealed column work model reveal independently since model allow vanish shift universal numerical projection map subspace frobenius norm sometimes indicate integer guarantee appropriate incoherence present provide reader interested go generality rank define represent rank reveal let max cc number entry degree contribution effect miss stress crucial achieving guarantee key satisfie matrix generality jk r incoherent rank hand discuss rather couple matrix variable zero gaussian latter basic main norm model far elementary column represent high regime independent cr c minimization theoretic analogous apply require point mainly indeed rather figure average rank rank letting noise take take relaxation bind iteration small root indistinguishable rank theoretic root manifold easily generate trace rmse evolution many error two rmse later error decay exponentially converge perfect reconstruction still complete include real next main guarantee stress strong incoherence concerned semidefinite front small improve several observe entry norm frobenius uniformly random entry provide subspace correct lie precisely dimension constraint project least error estimator show side gaussian oracle semidefinite finally deduce e optimal study review recent application collaborative study introduce restrict boltzmann machines rbm learn intractable performance approximate use argue collaborative consider descent residual justify lead factor gradient also record line quadratic factor basic sum square residual stochastic obtain convex provide nuclear singular regularization quadratic factor review norm regard convex surrogate prove strong implication immediate present correct quadratic rank try establish implication promise counter intuitive seem describe fact actually full exploit treat column probably row stick reveal non binomial different column large positive made want index test computing conclude singular crucial prove max n value matrix quite large singular summarize bad normalize phenomenon explain crucial consider decomposition apart trivial rescale matrix great max understand exist large cm e max max proves function riemannian controlling point reconstruct two manifold tx manifold optimization du f minimum two bound constant happen c hypothese e far follow enough verified sequence generate appear make modify constant appear statement corollary eqs eqs x x together x max term upper bound get claim constraint rescaling remark reader next k non increase thesis n x u enough f x twice converge x eqs imply z corollary noiseless reader convenience simply triangular proceed get cn cn replace e cn get desire corollary putting c take bx triangular instance u analogous bind trivial velocity tangent x expression obtained x get absence left upper use proceed analogously u inequality claim achieve e maximize correspond analogously generality observation differ side eq imply also tight bound norm random I function lipschitz inequality square function zero gaussian tail z e possibly variable jensen inequality matrix thus enough symmetric variable I column positive bind sub result independent variable choice tail apply chernoff analogous substituting z get
realization plain evidence exponent available provide consist hypothesis composite sum composite nuisance integrate away nothing uninformative uniform change condition hypothesis eq calculate analytical limiting exactly time distribution gradually transform limit illustrate find kullback leibler attain sum order divergence scale sum transform simple result upper log n log sum amount always log likelihood hypothesis favor latter nan desirable sum value sufficient exponent simple way value assume uninformative give expression maxima maximum assume minus reciprocal evaluate find likelihood permit quality state nothing bad bad detect fs middle focus include rescaled range local respective variant ssc mild accurate term decrease estimator online quality generate middle question arise ever fs fs corrupt importance short give fs mask away contamination contaminate autoregressive move average drive sample plot series presence go plain leave different proportion justify claim series fully adjust main summarize stationary anomalous evy type additionally distribute estimator enable portion two make valid interest motion preprocesse scale universit exponent assessment analytical drawing inference fs technique exploit result compute accurate characterization support scale regime close outperform time scale fs fs nature conclusion domain assess exhibit fs address exponent often verification look establish fs fair see heuristic estimate constitute assessment contrast discussion distinguish hypothesis rely series think fluctuation perspective generate stationary process normally series one walk associate exponent normal gaussian noise sample order reference diffusion scaling priori
ask nh nonnegative node single ensemble constraint optimization small node pick select ensemble receive zero sparsity zero ignore root always include equal set solution nonempty solution solution minimal amount add form simply weighted point fall fall new denominator define contain demand root nodes choose converge nonempty region unlikely node node root response reasonable observation fall select node nh initial turn insensitive long order node follow essential maximal interaction minimal advantageous competitive predictive fact maximal show empirically node solve get clearly costly hundred example calculate practically overfitte first attempt maximal node keep interpretable maximal factor choose assign impose could choice yet result across remarkably generate generate alternatively fit seem insensitive choice require node predictive initially forest rf ensemble fit rather order minimal add node choose one state qp qp solver efficient suppose root clearly fulfil actual nontrivial node satisfy many nontrivial basis arbitrarily choose orthonormal guarantee uniqueness term constraint price pay computation svd however remain qp implement language explicit take second ghz processor ram nh adaptive smoothing doubly symmetric value doubly fit training g g hence put define ij remain sum follow g q complete ensure irrespective size mean fit node weight hold convex real n minimal node size follow ij j sum nonnegative less positive strictly strict lemma last obtain positive replace pair observation member nh node might greatly exceed substantial forest build idea interpret forest ensemble nh since explicitly average bag possible possibly hundred consist turn variable involve measure ensemble despite computational nh interesting stacking weight classifier minimize weight stack tree nh trees spirit algorithm interpretability indicator variable prediction combination exactly define style put coefficient variable nh nh essential nh inherent nh select weight whereas particular node magnitude response nh breast patient fall nh response easy relate fall group group power ensemble seem well nh experience high nh cope complementary ensemble make rule currently build either node exist tree ensemble forest nod various center observation improve nh outline nh imputation sample fraction number sample row equality stem within surprisingly inverse thus fraction least extreme constraint effect fraction vector node beneficial unless look predictive accuracy nh sensitivity size fit validate choice penalty fit two poor structure default contour contour line fit tree forest fit predictive contour noisy variability default throughout node pick forest fit clean contour plot forest contrast tree across shaped fall broadly speak uncertainty environment precise behavior thousand perturbation well sample relevant section parameter gaussian note outcome hence explanation response temperature change period scenario area node weight node mean observation axis subset node fall annotate simply x annotate type contain receive axis training simply fall area sample happen fall mean pt four coefficient parameter belong node impose ignore new figure simply annotate nh though nh least tree interaction nh get interaction breast clinical variable tumor apply nh rf people tumor position sample patient fall one patient average patient patient assessment example characteristic nh breast select patient within patient tumor horizontal patient vertical plot nh belong annotated node main factor interaction select patient tumor tumor patient weight across people group train nh compare validate seem low label forest drop nh maintain completely variable concentration know diabetes median house price census measurement uci machine radial velocity galaxy contain gene product measure production gene essentially genome gene relevant gene nh could deal time part half nh employ cart ensemble nh select forest without tune node cv tuning remark nh forest fine tune know nearly nh rf boost tree depth weak optimize test datum record split available diabetes galaxy regularize estimator fraction well ptc diabetes machine galaxy solution table unconstrained estimator always solution average node apply improve advantage unconstraine regularization nh data estimator good additional desire extremely aim nh ensemble forest nh share ease interpretability simplicity node response nh overlap observation member predictive often nh interaction tree lack tuning thus nh forest seem comparable ratio seem nh drop nh simplicity arguably interpretability nh mix monotone nh deal without imputation censor forest forest associate stein helpful comment suitable predictor classical classification tree understand tree yet aim interpretability combine extremely initial generate randomly response observation identical new fall role nod weight optimal handle interpret predictive observation covariate trying predict covariate tree attractive understand partition simplicity notion tree subspace identical node rectangular eq subset support leaf intersection leaf correspond tree vector prediction observation error loss try partitioning equivalently minimize empirical complexity typically tree improve bagging popular ensemble average allow observation forest tree ensemble
satisfy property test datum gaussian mean alternate correlation simulation dataset perform error depict roc curves vary correlation calculate roc curve correlation datum experiment power method mining row randomize margin dataset threshold use list mean deviation dataset mean test combination dataset value large store property depict pattern control level calculation limit intuitive conclude reasonable topology randomization extend graph individually dataset calculate depicted ccc number frequent similar value plot level dotted line randomization especially test test generic mining dependency mining mining scenario make adversarial toy consistently argue serious limitation significance study scenario control fdr exploratory looking would detailed test fdr real also powerful control desire avoid negative randomization hypothesis hypothesis simultaneously mining assess frequent entail type generic algorithm control performance show control maintain power keyword test mining address statistical significance produce mining method whether nan randomization sample specify original dataset discard recently randomization mining margin matrix preserve matrix traditional interpret discover frequent define fraction dataset significance know advance hypothesis frequent false positive call significance choice collection frequency user specify apply significance pattern though evaluation testing simultaneously inference exist statistic tackle correction assumption dependency structure within common test likely algorithm mining exponential attribute pattern separate hypothesis naive would pattern due overcome example one frequent another specific association rule split fold half half significance pattern hypothesis work partial feasible find frequent subgraph completely association bootstrap limitation association bootstrapping dependency course sense mining contexts contribution randomize multiple testing suitable verification validity power provide section contribution proof summary method read paper end pattern set universe still define randomization one nan hypothesis intuition extreme dataset definition define datum mining pattern apply frequent mining subgraph mining general unlike restrict frequent dataset could could could frequency margin nan sample swap randomization decide frequent output statistically test shape follow within discuss detailed derivation experimental first base value randomize equally nan pattern return sample pool p dataset let return define pool eq nan extreme become equally control conversely treat denote sort value I error formal obtain adjust read section check calculated case property satisfied testing correctness result remainder ignore pattern theory testing reference test simultaneously know advance respectively unknown statistic statistic level statistical hypothesis nan hypothesis reject sample call negative call acceptable multiple hypothesis hypothesis significant incorrectly count correspond error positive significant significant hypothesis multiple observe count I way acceptable rate even type fdr control fdr control depend application would example fdr hypothesis multiple method adjust test value control testing control understand implement adjusted hypothesis use powerful slightly neither dependency control absence false nan hypothesis eq property test different encounter pattern risk pattern extreme statistic value possibly pattern might satisfy property vary visually plot never diagonal admit property mining constant pattern value prove property identically h fy return consider fy fx simple output number property restrictive mining pattern assume sample number uniform interval elsewhere occur output pattern probability denote happen output property assess randomization analytically property violate illustrate plot show empirical nan different however method may ignore theorem testing test arguably multiple testing mining output pattern pattern control significant small satisfie since q framework contribution broad zhang attribute calculate transaction attribute dataset break store dataset finally value show calculate use check significant rule et rule type error conversely strong rule might association variance association test random original dataset pattern dataset sort error level generalization control al define bootstrappe bootstrappe multiple directly assumption original statistic replacement difference sort decrease store contrast reasonable rule group transaction group disjoint calculate respective contingency contingency adequate
collect take used view member approximation simply contribute pl propagate gp classification essentially aggregate incorporate hide analogy px I necessary propagation upon information sampling may via setup prefer local pl propagate follow use mh approach class underlie prior drop index extend may accept ratio mh acceptance comprise acceptance acceptance nothing way loop obtain illustration real concavity third sign gps pair initialize round minute take hour less last minute pl exponential three gray open circle location circle posterior predictive surface test leave ix input circle misclassifie solid red one near efficiency pl less multimodal logit probit subtle versus computational offer comparison obtain every pair pl pl class pair run ht right progress pl section initialize size track additional point optima alternative right track maximum ei decrease except magnitude spike good diagnostic pl search take minute identical figure useful heuristic boundary exploration irrelevant influence base undesirable way versus heuristic mean pl class al entropy design figure design pl particle pre misclassifie point compare static section run long e candidate greedy near explore decrease nearby location result show smc pl contexts argue also well suit arrive component significant aspect subsequently context mcmc ill suited online acquisition arrive pl propagate maintain another relevant independently contrast mcmc inferential proceed serial get smc pl approach careful asynchronous implementation propagate particle resample evaluate package parallelism loop calculate propagate promising business method exploit sequential monte produce relative alternative ideal iterate new point attractive smc approach optimize boundary online key monte nonparametric sequential highly flexible nonlinear regression classification gps build point gp fit goal keep save gp via design utility design criterion alm gps task ei stationary mcmc determine circuit device explore label information new drawback tailor nature guide early iteration may carlo pl analytically rao quick fit heuristic calculate particle approximation smc pl gps establish right together sequential design remainder paper review pl strength pl pl fast al software implement specific illustrative covariance cx yx yx separate work parameter inferential thus problem likelihood response vector row covariance clutter drop subscript context mle via profile infer numerically proceed specify mix hasting mh multimodal hard crucially pl response degree classification use collection latent particular variable determine independence assumption softmax add degree expand proper linear take gp via scheme predictive pl algorithm irrespective detail classification hard practice fix say introduce undesirable simplify notation shall throughout implementation monte smc design inference smc sufficient information uncertainty time approximate sufficient smc update particle prefer gp pl derive decomposition suggest particle index pz ps ps core pl propagation resample propagation statistic ahead pl accumulation however setup extent firstly design keep possible gps poorly regardless smc mcmc etc drastically recommend secondly analytically integrate pl class smc note initialize explanation implement pl sufficient particle propagate classification smc use number maintain quantity store sufficient comprise define necessary prefer presentation efficient implementation initialization initialization initialize particle sample take proper word must start later work I much large calculate thereby obtain improper parameter sensible exist weight dynamic conditional probability z I propagate propagate update correlation pl directly counterpart propagate prefer propagate propagate liu gp represent equilibrium sensible tune mh proposal likely acceptance far decrease mh propagate whereas resample predictive global method follow delta introduce smooth encoding process noise parameter upon input one plausible greatly restrict exp mh proposal uniform slide center around work baseline may locality q capture low fidelity noise cosine enough fidelity pl particle improper mh round take second implementation pl mh take minute take take second fast round update exploit drastically line average right particle term credible student surface show fidelity surface find
exactly tp method base small storage comparison isometry iteration solution noiseless noisy case since method tune method large involve coefficient partial fourier art algorithm regime algorithm regime significantly outperform furthermore snr reweighte ii element converge fast present optimistic since algorithm portion support contain element also quantity snr mean criterion square absolute coefficient raw coefficient measurement use zero converge run noise measurement noise run run ii package package use fourier randomly l describe output square pursuit iteration iteration solution converge result quality db lasso pursuit knowledge outperform value quality produce reweighte knowledge also term visual keep thus actually perform mean observe higher choose fast much keep performance sparse affect test construction algorithm improve shannon remain unchanged set general construct small storage member frame reconstruction matrix decode art interest outperform fouri various research area pass schedule message pass amenable another may degree adaptive decode area finally tree decode may change accordingly improve schedule pass check code schedule neighboring node pass neighbor free posteriori schedule serial node message neighbor propose schedule base intuition message vice versa schedule mark edge connect inactive receive mark edge message information active index income message node active valuable reliable calculate edge schedule tend perform thm j school sciences edu compressive signal matrix decode presence additive noise frame matrix storage frame demonstrate significantly outperform art db range low frame sum expectation mixture said equation interested come estimate observe perfectly recover linearly uniquely np criterion noiseless various belong contaminate hamming theory exploit ensemble decode minimization complexity regularization satisfy property isometry program eq parameter selector limitation improve expansion represent estimate work matching pursuit variant subspace pursuit measurement rip noiseless also offer similar algorithm multiplication ensemble signal noise yet direction compressive apply compressive performance study dimension decode generalization various idea outline next introduce property introduce concept decode criterion finally conclusion future frame form frame formally non element redundancy restrict one dimension consider code bipartite sense graph show variable factorization undirected factor factor depict sum product exact computational increase issue one treat look subtree connect represent subtree associate summation yield message node node connect include go node factor start leaf whereas leaf description parent expression way calculate write structure factor modify every twice many calculation calculate marginal case could posterior marginal code row relationship code bipartite disjoint vertex contain node satisfy node zero bipartite graph graphical call connected sum purpose decode represent node code without tree product prior frequently ignore create interval compare accord jeffreys peak even demonstrate highly origin coordinate axis far enhance sparsity coordinate commonly expectation em find hidden parametrize em initial distinguish argument maximization respect random maximize em algorithm posteriori factor decode vector I use algorithm em em stage q px underlie density maximize summarize pass schedule really behavior alternative pass schedule maximum pass schedule attain schedule indicate fix note exactly construct soft stop schedule initial index message come check message incoming k criterion node enforce ii algorithm message likely zero denote develop compressive pursuit measurement stop criterion message schedule large index step absolute indexing vertex message decide decide make decision vertex calculate index index intuitively reliability maintain list large merge element decision update force e keep
differentiable map real number nonnegative nonsmooth subdifferential alternate penalize situation subspace alternate kullback proximal proof adopt space precisely rr set next write q alternate lasso proximal follow cluster kullback interior subset tucker point hold assumption belong without mixture check likelihood fact inside orthogonal verify restrict iv follow p ik probability let clearly differentiable around satisfie follow set number tucker optimality result tucker satisfy cluster et definition pt pt many problem set self regression penalization component mixture alternate penalize prove present obtain see maximum keyword finite regression widely great field pattern recognition biology finance mixture model comprehensive application mixture model traditionally associate notion mixture model independent identically multinomial random index latent estimating maximize log eq semidefinite case many practitioner notice maximizer log function viewpoint procedure mixture mixture available package instance question right local optimizer sufficiently many model biological propose estimation size small aim certain robustness spirit idea express simply restrict span obtain impose simplify instance likelihood condition whole formally em step encourage monte showing assume bayesian study chain I mixture approach regression simple easy strategy give involve stay small formalize subsection simple idea variance ahead notion follow instead stay impose combination sparse difficulty approach vector seem hard rely concern simple regression formalism estimate enforce unobserved cluster proportion maximize eq parameter obtain need accelerate methodology average iterate algorithm trajectory follow restrict multiple matrix detail method summarize convergence analysis alternate input problem cyclic iteration formula index address question testing simulate alternate em simulate datum generate permutation index recovery present carlo scheme expectation uniformly run code going obtain correctly obtain alternate uniformly cube example close correctly obtain cube box vector uniformly c cube cube correctly recover index similar initial choose standard gaussian mixture c recovered monte show increase space correctly recover standard size recover index output classification expectation increase recovery good dimension quite likelihood gaussian especially classify penalize compare investigation detail context goal propose base regression satisfy
exploit estimation follow prediction idea provide account similarity describe illustrate ridge along contribution adapt lasso penalize least study aspect aim lasso probability estimator important task contrary ridge pick provide way choose penalty selection organize procedure explicit form predictor present discuss generalization procedure illustrate performance sparse briefly book prediction predict label pair observation describe augment sample relative notion quantity lie associated explicit score adapt connection testing consider hypothesis permit py recall concept search ny new py yx new yx confident prediction reason measure procedure validity issue power simple eq note perspective asymptotic validity cumulative rather cumulative provide accuracy adapt shall point case sparse estimator namely nonzero originally linear label observation estimation estimation produce large sparse equal naturally define set lasso modification lar approximation lasso transition write sparsity obviously estimator cardinality sign evaluate finally variable invertible characteristic detail e begin end ordinary least square ol mention remain interval highlight lasso piecewise lar help regularization linearity lar variable index linearity lasso interesting encourage use sequel lar mind analogy lar tuning decrease reflect lar algorithm select yy estimator augment k n k sign equal purpose observation correspond let real k k make loose generality multiply whole collection propose choose particular study result augment satisfie actually exchangeability fulfil element depend believe predictor well present propose driven predictor explore methodology drive base technique attractive practical discuss validity select small measure validity validity validity union bind consider suggest validity construct modification lar kk associate thank use predictor fact piecewise form consist interval belong lar lasso predictor I kb nb I n km u nj b predictor w lebesgue say lebesgue construct valid selection construct value bring illustrated predictor true suitable predictor small predictor criterion add illustrated hard lar tuning aspect considerably reduce version could lasso lar th value correspond vertical line separation unstable generalize selection net linearity response score piecewise parameter computational effort lar estimator vector let sign base dataset soon construct estimator consider obviously elastic net predictor net modification lar predictor correspond elastic net lar k replace component make obtain dependency note lar lar computation least one test way grid window lar permit considerably tune point lar construction predictor treat turn sparse predictor reduce present performance accuracy observe predictor embed varie see validity elastic select introduce section net predictor last ridge select modification lar predictor level consider specify correlate moreover describe example sparse highly zero separately accuracy validity log top iteration blue predictor mark bottom show iteration modification lar illustrate predictor follow iteration appear predictor drastically even length predictor time unstable happen iteration let aspect valid variation predictor observe lc example stop validity cf show table observe variation validity good expect point part gap way early soon predictor least ii previous enforce early stop rule small consider
strong experience process sde wireless communications rao implementation consist static length series case mcmc proposal tune accord pre chain keep adaptive proposal adaptive metropolis issue relevant pg potentially space function elsewhere pg sample strong linearity particle likely preserve smc leave ti adaptive comment relate particle equation rely nx approximate mode path space variance study involve state degeneracy filter mutation kernel regard rao acceptance use mixture expert adapt distinct region separate softmax partition htbp static g htbp r particle sir rao effect particularly visible mix high total proposal note vertical unlikely prevent mmse converge iteration begin concentrate path around close mode satisfactory iteration intractable smc abc algorithm tolerance simulate observation degeneracy path control see due restriction htbp require replace smc compute incremental weight online sample k ks incremental q n comment approximate abc sake brevity set
literature look distribution minimal model investigate early possible adjacency matrix reference lead yet calculation hasting conditional census degree mutual paper explore conditioning likelihood exponential largely mechanism avoid unconditional something dynamic minimal offer suggest proper generate exact distribution matter discrete family utilize appropriate basis explicitly unclear proposal literature reach associate generate graph explicitly make early paper notion focus characteristic ensemble estimation assessment search optimal node block term stochastic go equivalence rise see g determine idea random could predefine discussion blockmodel involve discovery attempt framework generalization focus issue restrict blockmodel comprehensive treatment analyze interaction far traditional detail science statistical blockmodel basic algorithmic detection heavily display connection block maximize statistical link maximize hoc related link block blockmodel rely intuitive equivalence equivalence imagine super mind adjacency belong connectivity run index run suitable rely pre blockmodel social stable protein block leverage equivalence consider green definition although among tight direct connectivity blockmodel diagonal block equal technical blockmodel blockmodel block connection direct block node summarize specify mapping array specify interaction interaction nod blockmodel explain asymmetric block pattern mix explain connectivity pattern blockmodel identifiability beyond characterize concrete example stochastic blockmodel process membership context dependent different membership interact statistically p membership equal characterize network may symmetric interaction blockmodel integral evaluate analytically simplicity exact option thing nod social nest variational variable contribution bring approach discovery blockmodel community modularity biased modularity discovery incorrect favorable community substantial compose likelihood correct exchangeability develop pair intuition low adjacency distance measure individually first treatment cluster heterogeneity node probability adjacency position low pair relevant representation pair covariate odd general formalism generate quantitative edge weight example link negative integer node inverse may explicit distance separate position reference suggest latent likely distance euclidean distance carry scalability address network analyze latent project invert practice often interest identify group protein identify cluster position infer allow joint cluster introduce gaussians approach come blockmodel membership node binary relationship per condition blockmodel allow hierarchical distribution belong position group uncertainty cluster membership mixed membership carry latent proportion former infer variability membership posterior matrix million node interesting computational covariate argue blockmodel mixed membership concept extract ultimately new hypothesis mix blockmodel bic fit major benefit mixed could formation confirm longitudinal certain specification refer specification latent latent nice singular connectivity latent number eigenvector interpretation among capture connectivity eigenvector interpret latent interpret tight micro community interpret enyi way phrase os r enyi branch intuitively branch start node branching keep grow intersect node formal proof pick node work pick node place list take list neighbor already consider distribution add chernoff connect belong detail please p chernoff binomial line chernoff process carry analysis transition mathematically model static translate birth death old node drop due link partly partly study gain popularity begin increase dataset long span rich mind chapter begin os r generalization time dynamic recently os r static model static snapshot network record different step process link though view pseudo dynamic discuss enyi view os process start convention extend assume get rich description discrete branching process particular representation explore enyi issue dynamic major center produce network result claim exhibit subsequent generalization os enyi construct degree dynamic pa design generate scale subsequent node pick accord multinomial undirected much early intend grow get page likely page oppose little result law empirically whereas os enyi allow flexible modification dependence create decaying lead law appear mat forest name empirical describe infect main network certain network network often point contrary lot attention distribution company plot log fit straight visually careful case suggests usually justify law ordinary degree except cutoff degree adjustment search cutoff effort law g example careful often linearity give metric whether graph link description generative fall study fitting mcmc framework notable kronecker multiplication started turn analyze fitting real principled thought sense world model os enyi produce previously edge node drastically begin probability construction move toward os r small network dynamic evolve world variation finite grid connect depend greedy path number work attempt perform greedy within search reach walk show converge size different end goal amenable randomly perform probability range result chain stationarity span range link optimum search series study discuss link typical small involve aggregate formal examine small assess fit originally world model snapshot web static direct however dynamic demonstrate newly add web web web page direct edge web hyper page regardless match content web link web page closely match content specification et prototype among link th choose follow prototype possible since generate particular derive prototype node goal remain appear model protein interaction mixture assess evolutionary dynamic protein routine posterior statistic review dynamic evolution protein interaction recursive enable principled markov process network first shall become markov tie family evolution variant specification change party change modification begin provide quick notation outcome conditioning conditioning determine future distribution know network outcome possible configuration network configuration take node flip opposite specifie arc employ simple model derive close maximum model reciprocal q currently edge one direct exist reciprocal add direct edge exist either complicated popularity change edge node dynamic orient orient intensity factor component control specify two edge orient interpret configuration difference orient statistic oriented moreover choice flip two formulation orient write general orient edge edge combination statistic look familiar indeed orient equivalent update arcs number arc ji arcs target ji arcs jk statistic assume correspond undirected undirected reciprocal oriented suffer degeneracy triplets networks degeneracy longitudinal distant defines intensity q oppose edge seek neighborhood would suggest potential oriented modify flip configuration allow treat detail please proposal dynamic network operating domain markov factor simple version unlike potential consecutive configuration q list ij ij ij ij may attribute distribution may snapshot dependency pair extended estimate hessian covariance pair sampling well behave static degeneracy recall link dynamic position distribute observation euclidean influence node likely edge citation author co ensure radius one noise radius follow position author propose latent position base multidimensional transform observe ambiguity align position evolution rich previous ability word author mode kalman dynamic procedure line offer explanation step enable state network dynamic collection explicit behind network another citation physics community citation acyclic node show completely unsupervised manner opinion reference model reveal something new modularity variable modularity validate deterministic centrality discover reveal deterministic show several significant drop age mean gradually complement contextual represent aspect life meet interact context time strength social interaction people interact choose accord node appear update weight pair meet coin individual weight update birth death dynamic capture basic formalize context context hyperparameter idea weight show various relation brief relationship capable term past cost right represent line quadratic dataset weight contain email aggregate simulate relationship cf per represent strength take article drawback lack form frequently come weight life rich mechanism step realistic ultimately especially additional context individual model analysis section quality ease inference rise social visualization automate degree popular package longitudinal platform technique effectively combine visualization kind review want tool network estimation computation network g need newly package program longitudinal package capable learn truth really million estimate drawback sensitivity point really take come size dense contain assumption important focus asymptotic network serious problem confidence estimate behave asymptotic problem growth comment briefly asymptotic lack assess address assess especially involve could form assessment effect associate problem asymptotic exploit useful broad number entire subgraph even random condition bring parameter cf bias early random subgraph exploit statistical frank focus binomial size question hoc relevance sampling network address adapt account sampling design date work sensitivity sampling share divergence topological property expect relevance consequence parameter estimate along question non survey exclude consider directly empirical survey implication subject justify assumption interesting open datum mechanism specification inclusion actor response censor vertex scientific treat task treat estimate prediction next work evaluate dynamic paper relational develop prediction www literature predict biological work discover link evaluation usually validation know link interest distinguish utilize dynamic fit within paradigm paper distant future dynamic model implication epoch time oppose refer cumulative circumstance care actual realization markov process social process finance address identifiability refer lead procedure mixture solution various different g solution blockmodel pre identify reference especially arise context link example mail cascade full author attempt model message text membership membership way kind combine evolve diverse social biology computer science economic subject trend model proposal point visual diagram influence pointing influence pseudo dynamic green dynamic indicate influence motivation primarily literature science concern addition main lack model statistical lack degeneracy early broad dynamic clearly static snapshot network continuous hand clearly dynamic seek evolve os ultimately snapshot usually either pseudo within category main direction network equivalence existence understanding model evolve accord markov intensity dynamic observe various time network latent dynamic advance modeling decade issue model create model realistic mechanism great acknowledgment united national institute medical grant gm foundation grant health gm national grant grant dms office contract computer science institute thank anonymous valuable comment helpful correction citation list thank correction wish give original life major interest model date back social early active substantial effort literature past decade literature physics computer online community facebook specialized community begin overview historical example discussion focus prominent static emphasize description interpretation description challenge scientific field network interaction pattern much book decade facebook work selective statistical social computer biology book conference publish survey would impossible far attempt chart year major gap effort overview deduce promise complementary overview exist organize axis article static concentrate snapshot dynamic concerned mechanism change network early single static static year recent interest brief overview give comparative select approach statistically literature derive seminal last study mathematics os enyi random paper one impact network develop mathematical incidence structure small world phenomenon connection shortest people complete majority experiment provide title play movie degree ignore due censor formal chain like analysis early description variation mail chain network build upon effort os enyi os enyi along papers os work number fix choose description might sequentially version enyi associate value connect trees component literature extend os popularity effect allow estimate contingency formulation generalization multidimensional paper demonstrate lead spread structure cumulative link full maximum approach appear example social network point report network could truly evolution relatively reflect computation assess fit form statistic area increase section truly evolve ask continuous many work network model macro description physics think os enyi probability intend give law rich rich back world model back distinct adjacent world utilize law include statistical physics detect phenomenon counterpart social link pick idea epidemic variation network book length process description dynamic exploit extensive literature already existence complementary property problem diverse notable assess physics consequently attention noisy network often extraction graphical law centrality either sufficient descriptor dynamic free probability send delay mail nature activity demonstrate solely bayes representative mail poor fit law show parameter key structural comprehensive primary example author input mutation model seven mutation well several decade grid degree degree requirement treat model combine blockmodel idea membership generative model way sized mixed membership resemble dirichlet allocation offer kind briefly model truly extensive theorem proof largely present literature mathematic network substantial science deal short diameter connectivity centrality cluster volume issue strategy modify beneficial search rare population detail neural connection recently computational tool pattern model relatively area study economic book excellent technical concept structure relational area probabilistic uncertainty net etc different meaning review give arise property representative uncertainty feature good exception population suggest bipartite old agent attempt simultaneous effort create complex often agent idea recent advance high simulation social become research strong interest security biological well counterpart interface artificial intelligence social science analyze behind origin start example dataset interested reader may wish often interpretation social arise something mean characterize orient dedicated mechanism testing tend interested parsimonious formation common explain dynamic several find thought well biological similarity subgraph among understand find subgraph also help genetic interaction heterogeneity lot network purpose de degree structure prior graph biological analyze latent cell relate discover business organization machine know science statistic relate question member dynamic exist connectivity machine network often information likely movie box office application crucial business pattern state predict preference preference research name medium attention competition movie company netflix company million team able customer rating high house information propagation many domain infection work finding assume focus disease spread quick dataset ground break construction result length complete chain phrase separation study interaction relationship elementary student lot focus study gene mail web citation citation history collection subsequently title average author website raw along author connectivity static analyze two dynamic reproduce illustration ht network static nature activity statistic summarize os enyi generalization statistical physics generalization law free exchangeable graph edge ultimately structure connectivity strategy science community response social enyi model modeling model generative evolutionary set dynamic interpretation generating process generative node edge node often usually instance concern absence edge relation adjacency henceforth graph mostly set node contain direct correspond interaction os enyi nod science actor tie largely science terminology random equal edge model network simple usually back examine os enyi undirecte involve interpretation graph likely equivalently induce specify every edge expect edge modern simplify analysis binomial os enyi change remain mostly component contain node contain node use branching chapter concept useful os enyi mathematical study actual essence approximately network poor fit provide kind lead focused random second identify select physics several describe exchangeable simple extension introduce weak exchangeability attribute help connectivity source follow generate observation bit string edge node pair generation weakly conditionally string perspective exchangeable simple step generation string probable edge graph explore mathematic arguably interesting specification node dependent family hypercube dependent probability node bit binary string node direct node variability exchangeable direct argument string bit exchangeable main probability edge edge corner hypercube fit definition homogeneous leverage branching process intersect high form soon string match graphical illustration matrix correspond connect panel graph component bridge three bit string bit unlikely infer string set fitting assess observe leverage connectivity quantify retain bit plot correspond histogram exchangeable algorithmic model summarize illustration graph alternative independently likelihood need give fit sample profile histogram sample model support graph bit string bit integer distribution bridge member match practice specify correspond instance protein absence
jeffreys numerically chain case estimator error copula increase environmental study also currently finance precisely copula copula risk measurement compute simulate asset model copula provide book methodology copula within bivariate function uniform margin paper denote copulas lipschitz constant bound fr hoeffding copulas cumulative representation unique couple cumulative consider sample treat present function setup parametric margin rescale show estimator comparison likelihood rank depend describe latter exploit equation ci k describe v use copula call henceforth margin identical since inherent copula moreover estimator review optimal small think practitioner aware study rescale call bayes sample purely bayesian case invariant monotone transformation construct sup copula exist copula parametrize doubly mean numerical evaluation topic estimator use metropolis much problem copula write mixture doubly specify prior specify weight information reason candidate jeffreys main contribution derivation jeffreys generally difficult mixture literature face column well nothing kernel estimator every construction use basis unity partition unity nonnegative indicator function bernstein li partitions unity doubly lemma straightforward doubly absolutely continuous unity uniformly spaced restriction indicator evaluation copula constraint first know cx v give upper old mu copula appear extensively function equation copula generate doubly stochastic doubly matrix indicator multinomial experiment cell count define estimator empirical copula consider matrix copula q concentrated doubly matrix adopt objective jeffreys discuss doubly specification polytope polytope computing mathematics follow let function pt copula w ij h equality fact model vc w mm result matrix reduction greatly enable paper w w jeffreys proper specify consider b v dimensional hilbert subset positive prior uniform sampler distribution polytope another decomposition von doubly decompose combination matrix extreme furthermore doubly combination permutation see permutation exist weight lie simplex dirichlet realization couple structure know transform follow consider pseudo describe numerically bayesian estimator jeffreys call metropolis within approximated chain repeat select direction every take draw uniform accept eq expression prior previous draw stochastic prior eq jeffreys probability contain radius doubly meanwhile cc bm approximation jeffreys probability radius doubly doubly polytope show pt figure jeffreys estimator bivariate evidence important jeffreys prior six copula u u bivariate correlation univariate normal cumulative function u frank family popular family describe cross margin copula copula highlight cross dependence illustration extensive carry part family space determine value first four family respectively corresponding part experiment margin situation equally space range seven square margin sample size estimator jeffreys priors uniform g maximize expression estimator numerically estimator order doubly value latter commonly figure cc family family family pt c family cc outperform kernel estimator near frank family increase fr hoeffding bind call copula almost dependence bayes one remarkable result margin result margin latter estimator mention result margin margin case unknown margin worth case especially go stay boundary select life probable phenomenon marginal empirical procedure plug outlier consequently invariant increase transformation margin
previous especially prediction close principled balance seem automate hoc could compete base methodology use home familiar experience adapt interest rate walk tool predict outcome simultaneously several outcome single etc observation year outcome year player year outcome rate position past could extended experience prediction home consider multiple argue possibility player population refine home rate sophisticated model player population suggest dominate player mixture non limit shrinkage towards population home run prediction bayesian investigate status directly primary home prediction also estimate secondary trajectory position addition evaluate position home run trajectory note represent major represent conditional conditional prediction however interested unconditional trajectory sophisticated censor focus home make assumption throughout predict know maintain fair current modeling like discussion cm major bayesian paradigm principled balance player share player improve method held limitation performance award top assumption past course balanced age project observe player truly par consistent answer prediction within current include sophisticated player set player similar player historical past player effect effect recent major recent heavily methodology overall method prediction multiple publicly database day fit model player within follow datum home run age home position home run exclude leave nine position third ss lf center cf rf home use major vast majority player give player home home run walk effort home binomial justify coefficient age trajectory spline position nine position team share player home coefficient since play particular team home run home separate team effect would examine aspect performance capture outline firstly age treat home identically create home position share home run represent place mixture term force status model age extra odd home year player move status maintain year year address past performance trajectory would small year player favor prevent player build past status indicator player specifically status indicator player status player year markovian induce dependence home player share player position differ position initialize start status desire consequence player must must need model age share coefficient age intercept normal identity bivariate indicator non variance hyperparameter specific use prior parameter entire gibbs standard transition conditional posterior imply represent year observe involve sampling status use sum hide k hasting step player position variance variance procedure predict home player external validation include home performance several internal choice via gibbs distribution home home wide possibility home evaluate prediction focus external home run ab ab home run home fair external advantage full produce comparable term mae compare prediction overall percentage home home total among table give measure separate age year versus player ccc players mae mae competitive examine player absolute superior scale examine player rmse mae superior performance sophisticated position well past method root old player player position beyond shrink position fit validation encouraging method player principle past performance advantageous among examine year decide home run question indicator examine attention subset home datum figure player year player home fact player year home ij player player examine balance age question old player age examine versus entirely year contribution prediction home find good suggest information evenly balanced prediction age imply home run age ball status year position differ position status consequence age ss year vs arbitrary contain curve graph imply value gibbs output examine curve sample see ss shape home fact home position vs status substantial magnitude home overall restrict player equation difference part range surprisingly grow home across position examine non position distribution intercept imply home run age dataset
reference satisfy diagram look set observe standard gaussian white space assumption interested confidence quantity play value value integral weak strong separability follow auxiliary sensitivity prior arbitrary pp b w ii schwarz pp eq fix complete inequality nothing choose b easily apply estimate situation use w hand choice follow consider next assume utilize finally choose continuity pp pp b pp hold v q r r k r index convenient lipschitz observe observation r obvious accord immediately pp r contain norm give couple depend obtain use weight normalizing equal jensen cauchy direct apply place constant independently r pp proof low lemma recall thank lemma section satisfie pp pp pp h special inequality scalar assumption previous measurement possible choose condition yield special force limit present section indice pp define familiar truncation appendix proper variable follow norm formulate quantitative assumption pp r pp pp pp example difficulty state work circle identify measurement measurement finite function limit independent variable white indicator jx hoc random weakly take converge notice measurement behave proper condition violate proper white motivated white noise actually hold see smooth weakly space h j measurement white function dual verify complicated construct scope wavelet detail comment elsewhere discretization smoothness jx jt projection subspace duality value n jx q ni green differential imply lipschitz smooth almost surely old fix e every strongly identity infer define variation let consider e end vanishing orthonormal random jt variation gaussian stay linear example let dimensional disjoint triangle union function triangle b weakly variable zero smoothness dt white measurement well define verify proper measurement formally speak green study field formula dimensional meet difficulty therefore measurable measurement almost surely quantity could subspace surely value change subspace unfortunately intersection realization cm lemma proposition inversion space prior department mathematics statistics university measurement realization linear device inversion discretized formula kn nu kn nu choose achieve infinite case discretization priori wavelet invariant inversion norm wavelet coefficient inversion inversion wavelet smooth white object apply physics interested find measurement domain euclidean device realization variable operator relate device realization specific space brain imaging equation describe surface produce external model inner linearly device measure surface determine unit orthonormal model bayesian inversion quantity know value choose projection dimensional range finite source brain finite virtual sense appear particular relate random bayes formula exponential white statistic form information problem mean mean differ involve confidence approximately chain carlo software ray electrical imaging general reference inversion require practical implementation imaging correspond field discrete equation effect possibility lead gaussian fall discussion model measurement noisy measurement fully priori posterior work construct bayesian inversion discretization consequently sequence increase motivated phenomenon computational resource necessarily reconstruction perform lead reconstruction discretization converge mistake different discretization finite represent use surely state come realization prove strategy involve measurement dimensional white eq coincide km general analyze confidence infinite process achieve discretization invariance inversion total preserve reconstruction posteriori total inversion discretize particular estimate regularization band filter derivative processing prior discretization discretization proof quantitative convergence inversion use wavelet two know estimate norm wavelet us review literature inversion dimensional space discretization study kind statistical inversion hilbert space specialized work entirely within computational point inversion early invariance however appropriate realization emphasize inversion discretization organize invariant bayesian invariance value convergence generalized banach denote inner hilbert stand banach specific space denote realization acknowledgement discussion anonymous improve paper ml ss support national contract group project section prior inverse give result later discuss deconvolution smoothness prior regularization derivative ignore boundary periodic generality periodic cover compactly case dimensional space eq space inner generalized variable gaussian value assume hilbert space smoothness also stands take covariance parameter operator prior prior almost surely let differentiable index white gaussian operator function value hilbert space zero space negative inversion place measurement mean location thus omit formal rigorously practical range pixel truncate enable device us turn need series expansion dimensional converge estimate correspond smoothness discretization subsection emphasize practical use compare close discuss connection minimize functional eq define form convergence smoothness represent vary however sometimes priori prior gaussian geometric prior analysis discretization invariant inversion replace discretization dependent space prior band use edge inversion compactly measurement convolution wavelet normalization arise naturally scale function fine refer wavelet range choice efficient term probability mean approximately use chain carlo sampler hasting e modify involve fast transform converge either involve prior distribution define converge distribution phenomenon rigorous space immediate recall banach banach dual algebra topology separability borel algebra measurable separable banach space diagram separable measurement gaussian separable operator adjoint unique adjoint power henceforth space well call white domain consider triple tw z hold complete discuss tend weak value let nested variable variable u example mainly conditional pd pd vector fact value expectation ready variable deterministic measurement deterministic coincide thus formula realization measurement datum measurement essential model representation generalize value existence conditional expectation establish specific situation follow usual conditional present proof completeness measurable stand formula define equality integration thus integrable integrable sure pointwise since motivate would write derivative define particular formula see gm continue fm dm especially hold distribution dm respect well dm p sure formula assertion look everywhere correspondingly satisfy formula see appendix discussion respect point view practical inversion quantitative
min pls ridge hour adaptive lasso major rely cross scheme regression pls scale lasso adaptive adaptive nest partial parallel alternatively might consider construct network without cross method refine run propose partial application simulation assess reverse engineering investigate estimation perform fdr significance test fdr graph even sample dense network performance five world sparse except pls close mean replication yield structure suit compare stability regression stable difference non seem unstable constitute severe decrease topology cross pls slow fairly allow representation I partial correlation pls package repository assign nk study initial version manuscript package analysis concept manuscript acknowledgment support grant european network financial product constructive mr axiom axiom axiom axiom lemma axiom axiom axiom cm technology clinical popular undirected association issue greatly exceed sample partial correlation inverse poor scenario adequate need biased estimate scheme least investigate regularize regression respectively extensively repository investigate decide difference confirm know tendency towards select lasso provide reconstruct network six clearly obtain method automatically assumption uncorrelated replication lasso yield complex structure indicate suited approach auto model microarray article undirecte graph edge correlate correlation indirect variable correlation strength primarily article genetic microarray numerous study nonetheless reconstruct microarray remain covariance estimation observation array gene alternative regularize high dimensional focus comparative use various high dimensional extensively real truth true focus difference investigate examine result graph moreover dimensional represent whose dependency relation node assume quantify variable finite respect involve exist network plug partial formulate outline notation rest array unbiased q inversion estimate invertible partial denote least stand include intercept least column gene moreover sense microarray firstly estimate eqs ill condition even large criterion pose hence regularize regularize review secondly approach set hypothesis base sparse correlation separate testing discovery multiple review estimate estimation regularize ridge propose shrinkage construct control shrinkage assume covariance correlation whereas shrinkage diagonal entry variance shrinkage target intensity analytically shrinkage favorable subsequently gene association multiple bootstrap aggregating bootstrap help bag inferior shrinkage expensive augment closely shrinkage shrinkage finally recent base graphical corresponding sparsity conduct parameter challenge least square formula estimator define partial ensure estimate always interval method pls estimation two attractive least al pls pls method seed scientific field bioinformatic idea pls pls assumption mutually pls hence pls orthogonal component rescale vector lead formulation observation pls regression criterion freedom gene network empirically experiment well validation pls correlation testing use alternatively pls correlation mention estimate use pls speak root et recommend ridge popular regularize perform ridge term penalty avoid fold scale pls apply coefficient minimize penalize form coefficient mind lasso advantage yield assign gene regression successively scheme control connect e tailor however graph prediction determine stage adaptive require necessary among show procedure rather consider dependent manner specifically follow estimator pick estimator penalization lasso adaptive lasso use gene connect correlation selection determine validation split fit computational cost reconstruct network propose technique pls lasso covariance follow overview five characteristic package performance assess simulation set sample range investigate consider vary topology network topology scenario partial replication vary density draw rescaling ensure partial package correlation depend absolute file histogram partial correlation network high degree select correct effect eliminate simulation partial trivial lasso shrinkage estimator successively fold component penalty two follow parametrization ratio lasso estimate avoid phenomenon ridge range parameter range pls component estimation partial correlation ii recovery topology difference square panel figure mse display estimate base lasso mse sparse likely non significant ultimately vanish degree notable exception network conjecture pls number four reconstruction underlie panel number network regularization pls contrast conservative promising difference density difficult explain high degree panel rate panel come low decrease density argue report fewer less change relative method independently particular fdr detect non partial specificity fdr display roc supplementary pls slightly covariance lasso threshold depict roc finally runtime figure display lasso compute load lasso high pls ridge fast consume validation computation different regression discrepancy become apparent study topology consider topology simulate topology display show different simulation cluster shape cluster star star mse discovery scenario probably insufficient towards perform reconstruct network different diverse real set availability consider shrinkage ridge pls selection procedure approach simulation use cross datum ever performance power possible compare estimate percentage proportion six adaptive seem behave differently partly different select set except datum non datum likely explicitly standard regression independent implicit use fdr assessment estimate conjecture pls validation reliable method network strategy incorporation pls among fdr assessment pls conservative pls identify refinement present simulation
uncorrelated individual load correspond component uncorrelated explain achieve solve control however look plausible modify modification consideration component connection formulation pca technical integer define solution satisfie orthonormal r address problem orthonormal first whose consist orthonormal feasible hold also part show define matrix whose column consist orthonormal diag tv tv rp r therefore p tu u tu iy tu moreover together right hence diagonal let first respectively obtain identity v consist orthonormal correspond em orthonormal eigenvalue loading provide pca suitable nonsmooth constrain nonlinear programming problem pca subsection subsection minimize nonsmooth close suitably subproblem global nonsmooth constrain programming nonlinear necessarily presentation denote establish active jacobian expression demonstrate sake completeness cone cone tangent imply gradient condition satisfied identity definition hold order optimality let lagrange denote eq arbitrarily sequence equality second fx fx hold simplicity next contradiction nonempty contradict closed know convex unbounded px subsequence necessary xx closed side limit subsequence necessary contradict convex mild accumulation lagrangian method program feasible drawback novel lagrangian subsection make follow feasible moreover denote augment penalty roughly speak subproblem update multiplier penalty know assumption framework novel lagrangian sequence choose find lagrange multipliers method lagrangian augment step ii penalty lagrangian multiplier augment lagrangian address approximate subproblem lagrangian converge feasible subsequence accumulation point lagrange second gx k hx side inequality follow show statement q contradiction subsequence necessary x convergent identity fact imply cone contradict identity subsequence bound see accumulation see every accumulation fact first optimality discuss satisfy lagrangian method particular apply subsection able approximate second subproblem approximate solution th subproblem satisfy additionally propose subsection possess value iterate q minimize nonsmooth suitably subproblem augment lagrangian mention method relate known project method study propose subsection smooth lemma stationarity observe convex immediately change fast theorem conclusion immediately lemma indicator statement q eq definition px fx td fx fx td l together immediately statement set choose ij go differ distinct indeed lipschitz continuously method namely local convergence theorem two technical lemma first show hence exist whenever claim limit side contradict imply use relation provide conclusion objective converge observe integer eq bound limit hold definition together lk lk q second hold argument lead together use argument hence definition lk lemma ready show convergent uniformly accumulation method accumulation subsequence exist without simplicity since exist particular specify pass subsequence upon k relation boundedness k k sufficiently inequality contradict finally limit side h x begin proof convergence make establish inspire similar local nonsmooth bound gradient provide sufficiently constant kx use result lemma lie lk lk lk inequality lemma h ik limit side rearrange term q next follow choose integer h hx accord integer define relate proof global hold view smooth generally global proceeding exist satisfies lemma together lemma scale direction zero also converge suppose uniformly continuous sequence sequence together see limit hold replace immediately easily show use lk assumption lemma follow ready globally convergent level accumulation gradient stationary suppose contradiction accumulation subsequence subsequence assume limit side lead lemma stepsize monotonicity sufficiently fx px px k fx px px fx fx fx pt fx contradict hold establish rate suppose bound continuous set ii satisfy constant ix argument index ik lying follow lk lk lk lemma constant proof detail augment lagrangian pca reformulate subsection sufficient augment lagrangian include explicitly hold accumulation point trivially satisfied accumulation condition feasible due condition proceeding subsequently feasible v I ji fact orthogonal remain lemma reduce diagonal unique let assume condition b change thus suffice p conclusion ready show condition hold feasible feasible hold v ji j active inactive inequality inactive first diagonal addition equality one directly hold solution accumulation point augment lagrangian fail augment equivalently discuss outer inner lagrangian inequality constraint multiplier equality constraint convenience presentation matrix whose entry resp th resp stacking rewrite clearly complexity evaluating assume sample covariance efficient store compute initialization termination pcs eigenvectors lagrangian multiplier terminate lagrangian sufficiently prescribe equality augment second one method first choose subsection subproblem become choose scheme study initially termination criterion solution prescribe shall sake numerical lagrangian pp adjust lagrangian multiplier update speak decrease minimization lagrangian multiplier constraint decrease recommend conduct augment lagrangian method detailed subsection equivalently synthetic real term explain pc et specifically via penalty block via penalty thresholding pcs pca uncorrelated total pc variance pc pc pc correlate much explain overlap total total variance pc dramatically pc explain variance drawback account introduce pc pc uncorrelated usual pc adjust explain explain pc shall stress pc three pca nevertheless shall solve datum al test effectiveness sparse synthetic actual find pc independent pc table fact pc explain second sparse pc pc uncorrelated termination loading pc column pc interestingly one quite subsection rr pca pc pc pc first center subsection set choose termination criterion pc loading method properly method randomly instance column one loading average two three average orthogonality loading vector angle form loading column orthogonality pc present loading three pc almost uncorrelated loading method pc c c method pc average loading around sparsity present substantially outperform pc loading pc increase accordingly pc non sparse pc surprising orthogonality orthogonality pc data result introduce classic illustrate pc several method six pc ease pc pc shall pc sparsity one table pc large sparsity one sparse orthogonality pc obtain standard five measure third report measure absolute angle load fourth present maximum correlation pc pca nice table except give pc apply method test termination r r pc knot r variable pc knot pc knot length clear experiment uncorrelated pc sparse pca pc present sparsity orthogonality seven yet explain r knot l r pc knot pc orthogonality experiment choose pc pc pc orthogonality dramatically combine deduce seem possible six g nearly uncorrelated pc exist knot correlation control performance pc non orthogonality pc table pc explain variance correlate surprising drawback observe sparse pc still experiment six nearly orthogonal uncorrelated pc acceptable pc one r r pc knot c correlation formulation pca principal pc globally convergent lagrangian class nonsmooth suited gradient method lagrangian subproblem local finally sparse pca exist pc produce substantially method total pc orthogonality loading effective finding desire sparse set sparse uncorrelated pc loading word actual associate orthonormal since practice approximated sample question exist pc show also accumulation however open hold recently augment method nonlinear nlp program sdp sdp relaxation hard combinatorial augment generally converge due theoretically approach sdp augment mild nlp code available www extensive exploring application acknowledge comment west
correspond associate player instead balance game resource singleton resource machine assume profile strategy give scheduling policy player example schedule order clearly balance allocation game load balance load balance whose unitary introduce exist strategy pure I strategy basically intended initially select player lead random player player f consider validity probability say preserve function formally size support problematic please game stay allocation decision decentralize instant interest nash equilibria general consider weakly sense problem differential ode informally replace bp behave ordinary ode eq dynamic unity dynamic stay also mean field dynamic convergence limit nash equilibria convergence equilibria theorem ordinal game game admit lyapunov lyapunov take call lyapunov game balance game lyapunov martingale lyapunov lyapunov equilibria happen exact game potential load balance particular actually iff prove allocation game potential game value decrease player response correspond player good response move game ordinal potential game terminology ordinal pure nash take strict response must pure equilibrium turn player low load idea response investigate play result investigate require move choose version good response dynamic terminate uniform task machine expect time game pure nash equilibria game pls complete dynamic nash equilibria investigate nash equilibria occur extend different asymmetric game elementary partially game potential game prove equation follow incorrect happen towards nash unstable stationary super martingale rely theory evolutionary game set construction time discussion game consider lyapunov particular game nash equilibrium modify cost lyapunov player team define assume hypothesis convergence stochastic define limit approximation formalize piecewise interpolation bt k interested limit family variable converge weakly unique value presentation presentation mean law instantaneous variance covariance differential particular diffusion continuous consider markov clearly stay keep mean sup theorem correspond differential turn ordinary solution unique classical restrict like follow dynamic like rewrite equation lead ordinary rescale q limit point equilibria theorem evolutionary corner ever irreducible ordinary whose nash equilibria ordinary equation equilibria game interior mixed equilibria method problematic nash equilibrium set belong interior dynamic game dynamic generally provably convergent lyapunov argument terminology lyapunov say lyapunov game hence potential lyapunov degree game lyapunov game lyapunov function ordinal linearity clearly linearity I dynamic side game ordinal q ie ie lyapunov function ordinal take lyapunov function dynamic class continuous potential lyapunov definition lyapunov clear derivative ie define continuous potential conversely strategy equal cost case characterization continuous differentiable connect path pure part proposition part q proof lyapunov lyapunov dynamic ordinal game lyapunov differ ordinal lyapunov function accumulation trajectory lyapunov game dynamic entirely dynamic lemma slight lyapunov theorem ordinary stationary ordinary near prof stationary limit hence lyapunov game respect full lyapunov dynamic condition field nash equilibria field nc property nash equilibria unstable stationary fortunately previous continuous balancing game lyapunov possible avoid differential equation double proof second term lyapunov q ft definition hence variable ti observe indeed one move consider elementary lyapunov dynamic hand super martingale reach corner super eq martingale get stability subset enough underlying chain ergodic visit say closure positive neighborhood proposition one nash equilibria probability default require neighborhood lyapunov game point compact correspond nash equilibria fortunately nash iff ie q nash improve time current equilibrium u ie e perturb dynamic form b q u iteration near I opposite lyapunov lyapunov ordinal hence potential take surely nash equilibrium furthermore ie ie side equation already nothing otherwise sigma algebra generate whenever n proposition ordinal gain gain variation preserve course game game game potential give rt following say satisfy satisfy whenever nash player cost adopt would gain least believe perturb polynomially many game lemma fr france fr algorithms learn equilibria sense weak limit ordinary ode case stochastic dynamic ode turn dynamic fact convergence discuss dynamic game convergence ode equilibrium lyapunov lyapunov game stochastic dynamic prove ordinal game potential lyapunov lyapunov lyapunov game lyapunov function lyapunov super martingale way time martingale game consider include load balance choose portfolio interested converge rational equilibria game theory dynamic fully game description several game mainly deterministic good description market variation avoid consider
source negligible second calculate simulation nan complex assumption commonly numerically also simulate happen apply commonly remove contamination simulate filter filter proportion tool pp achieve comparable outline approach replace pp necessarily section describe see image pixel vary spatially clearly violate like background problem derivative use region stand behind multi examine derivative filter work toy location remain stay roughly reach big source point smoothed bandwidth occur tell alternatively source value smooth increase bandwidth get nan smoothed tp compare smooth source stand large smoothing quickly perform smooth filter origin bandwidth image scale image create pixel derivative high scale derivative cause source background smoothly vary enhance image flat background figure show look datum ground detect galaxy cluster detect leave strong source wiener combine three see middle detect galaxy wiener filter vary filter right highlight filter alternatively view incorporate peak create expect create design enhance make stand describe previously run derivative image confidence procedure source detect follow make detect object tp without image make underlying explicit pure design patch detection wide verify initially ability keep detection without benefit relax proportion grow publish follow optical observation detection contour ray background spurious ray count visually false follow object confident previous rigorous error find recover outside optical unable whether spurious still accept expect proportion false source determine wide set collect follow scenario control reach aim analysis beyond imaging place ask carefully arrange behavioral dimensional brain acquire regular perform subtle flow response brain location compute statistic change image area see detail process ask visually visual image acquire fmri resolution surface david use primary responsible processing call fmri roughly visual first phase location goal region appear series visual band problem region prefer inferential location fisher stimulus phase stimulus type location want location response phase location stimulus want test phase false cluster adapt definition deal multiple oppose pixel grid classify cluster belong define test two location less large location confidence determine percentile uniform total number information previous connect location great analogous limit band surface false proportion situation control region detect show control object regular variety source guarantee false check science behind multi type enhance false cluster follow detection whereas run false verify control comparable detect control without critical generation provide manually generalize type plane grid object proportion false future apply proportion technique wide clustering try false token token token token token token token interesting product important input detect error control several rigorous control detect aggregated pixel technique rigorous statistical ray source source detect detect previous paper extend blind detection david student department statistic pa mail edu department pa mail work support national nsf grant dms national health ns space grant center like thank ray david fmri helpful discussion record light section various detector count detector exposure record solely interest reduce seek coordinate source survey input scientific early star object produce direct visual improve much require year collect comprise change digital imaging design computer available power storage relative could wider deeply fast ever search automatically collect previously digital hundred million survey collect comprise past decade poor lie challenge lie answer challenge lie rule often come generation operation entire prohibitive yes several new cut massive controlling source incorrectly reduce object comprehensive human scientific likely automatic misclassifie method control advantage especially large survey although context wide similar give later array value pixel arise include object anomaly essentially poisson base denote source respectively disjoint poisson random apply good report base observation image reasonable approximation gaussian utilize generalize problem pixel contain pixel hypothesis coarse want characterize pixel accurate construct criterion coherent localize wise operate pixel false proportion control effective control false image show procedure power technique wide source take procedure powerful technique provide excellent ray even source maintain although setting concept band activity multiple give rate underlie center problem functional response field region contain fmri dimension general detection processing operation effect well detection typically operational planning typically simulate real detect source depend simulate qualitatively effort construct simulation run fail detection use fall peak consist cutoff wise statistic classify fast computed popular software apply raw signal match filter simulate create ray image simple thresholding simulate image pixel false south small million second observe exposure mean able resolve distant interference galaxy approximately angular size also location complementary figure scientific source think galaxy center ray make rate publish combine source modify version background source region pixel look real select include several detect refined observation effort replicate design patch follow back reject spurious exchange fail large make impractical follow automatically reliably control one pass conduct pixel appropriate pixel apply pixel unit inference pixel compose collection therefore alternative pixel wise proportion introduce false random treat field derive location unknown confidence pixel decompose component pixel pixel measure tolerance come proportion detect sufficiently cluster envelope false proportion wish search proportion detect equation calculate look q pixel intensity p pass value quadratic homogeneous gaussian field locally field satisfy equation incorporate appropriate detect cutoff less henceforth see plug software software way active galaxy detect make spurious classifying really ray detector model background pp assume deal smooth apply pixel smoothing filter count structure poisson make close zero square background root normalize root low rate normal different rate normalize transformation pp rate source conservative pixel select power nothing know nature assume check pp calculate publish since many nine tp boost
combination classifier interpret hyper plane space select decision weight find maximize separability criterion criterion technique asymmetric address cascade unlike discriminant analysis criterion wu optimal lda technique neural feed forward class lda consider neural increase superior adaboost key contribution introduce detector algorithm combine detection performance objective word two separated detector flexibility well beneficial consider highly detector criterion incorporate hence well adaboost exponential loss detector detection incorporate classifier propose experimental result show section technique adaboost concept lda handle asymmetric also sample select analyze training discriminant lda cast eigenvalue decomposition pair matrix covariance separability maximal lda q count nonzero integer become np nevertheless infeasible extend efficient add number select predefine meet eigenvalue case column vector determine inverse greedy adopt suboptimal simple rank computing reduce computation mainly forward index calculate q drawback control number control tune decide predefine sparse explain classical explanation cascade reader refer detector operate initialize train weak classifier focus haar later haar examine continue predefine algorithm set haar like rectangle feature minimum acceptable rate cascade cascade false train parameterize threshold error add weak classifier add misclassifie cascade cascade cascade would prefer high rate false positive bernoulli easily sense linear classifier equal covariance linear matrix normal input image feature minimum example central limit close asymmetric cascade trade rejection eq lda minimize total covariance select asymmetric easily adaboost detection little positive think negative prior number lose contrast sample hence sample artificial consist show weak adaboost select small weak introduce since classify adaboost weak classifier try introduce positive adaboost weak classifier yield classifier less false positive adaboost distribution adaboost experimentally significant boost round term cause gradually pay attention boost separability boost boost popular originally design problem combine accuracy least minimize hinge property majority vote boost adaboost additive classifier example final decide label receive determine significance weak boost computed adaboost select update final rule weight coefficient learner weak classifier dimensionality linearly project direction introduce concept domain learn save small error remain however decision achieve decision boost training receive weight decision u ix margin minimize subject weight select weight update base detection replace line normalize weak classifier optimal add classifier output yield weight manner remove correctly classifier misclassifie boost cascade layer need classifier classifier calculate complexity total dominate spend weak classifier organize experiment efficiency haar rectangle know haar adaboost face fast adaboost haar feature technique classification consist consist face rescale image pixel face face patch non obtain image split train experiment classifier selecting set remain measure test alarm set mean dual forward fig haar like comparable adaboost fig slightly haar adaboost slightly haar classifier confidence error indicate adaboost curve result figure adaboost experiment layer positive previous cascade bootstrapping cascade terminate negative bootstrap fair train technique cascade backward dual cascade detector face detector low resolution face mit test set contain face merge overlap ground truth overlap bound ground box exceed face fig receiver curve produce classifier weak cascade classifier meet roc curve adjust threshold adaboost instead weight adaboost weak perform unlike adaboost train algorithm sequentially whose process continue meet classifier experiment svm lda detector believe another provide finding open object rectangle feature cascade select haar like cover compare classifier classifier average number haar evaluate detection window adaboost number classifier classifier adaboost decision nevertheless classifier indicate classifier class suitable domain skewed separation compare different decision scheme correspond skewed asymmetric boost apply figure classifier adaboost surprising adaboost adaboost low haar rectangle feature alarm positive evaluation curve adaboost train cascade roc curve roc classifiers cascade predefine meet evaluate performance asymmetric asymmetric multiplier every show classifier roc curve train false rate might suitable domain example gain small adaboost lda problem reduce table indicate perform comparable slightly cascade intel cpu ram total hyperplane class weighting conduct scheme remain previous roc configuration outperform adaboost classifier high positive perform positive bt stage total haar adaboost adaboost scheme apply difficult face dataset scale set contains analyze two six roc conduct three haar scheme experimental fig roc curve face conduct manually time sample preserve contour extract haar poorly haar feature however applicable dimensional overcome dimensional brief stack covariance project onto space decision discriminant drawback slow assign classifier store project technique project space replace lda feature train classifier threshold rectangular filter subsample approximately stage objective meet detection rate cascade
tree hypergraph density arbitrary measure distribution implicitly take depend disjoint union vertex determine density factor way factor set denote respectively factor even decomposable eq may specify give determine law parameter hyper inverse wishart useful factor far hyper previously little typically take decomposable perhaps os r enyi encourage express arise model posterior graph factor independence entail selection graphical either little represent discrete space obvious geometry case challenge flexible metropolis mcmc parametrization intersection region proximity form less ball radius intersect diameter range intersection convex consider compute concept finite collection distinct nonempty hypergraph three paper complex denote set alpha denote construct family example complex diagram vertex plus illustrate alpha display table alpha complex illustrate index intersect produce isolated vertex index intersect triangle vertex index intersect face include known abstract complex subset collection set hypergraph arise skeleton skeleton complex cardinality skeleton uniquely skeleton obtain nonempty intersection obtain skeleton induce alpha exceed although still disjoint index dr parametrization space set determine implicit whenever obvious generic element alpha induce parametrization two advantage need hypergraph vertex hypergraph mcmc induce distribution integer radius intersection cube ball widely clear choice write pt height pt set os edge model exhibit class construct describe asymptotic count alpha complex regard distribution area recent survey discuss count joint vertex suggest result sometimes multivariate sometimes require approach infer specify law inverse wishart normal principal law strong build construct random dr construction transformation angle simultaneous scale change restrict distribution reduce feasible represent ball r triangle inequality must great fit ball center segment separate diameter translation rescaling may diameter complete proof fix simplify writing understand induce configuration uniform edge lead take enough empty feasible rich graph converse prop dr r slightly strong attain dd tuple distribution wish draw distribution vector fx metropolis hasting begin diffusion walk parametrize spherical radius euler angle informally radial walk brownian radius angular brownian radius fix walk begin reflect expression dd random walk generate local simplify ergodicity option global move uniform one hybrid walk ht draw graph begin hasting distribution invariant accepted multivariate law close efficient variable parameter depend configuration reversible mcmc use auxiliary variable keeping need nest auxiliary auxiliary instrumental move metropolis last metropolis ratio non move graph proposal ratio denote p feasible statement recurrence markov section strictly global move replace draw p markovian nevertheless prop encourage specify factorization point vertex display htb complex use list skeleton complex alpha factorization factorization factor similarly figure fx particular graph set contrast os enyi edge clique plane mat ern core radius asymptotic ern iii comparison make os enyi os r enyi inclusion joint ern draw os enyi joint radius enyi inclusion width depth clique mat ern er mat ern mat ern mat ern mat ern er er c complex sample draw core construct change hypergraph maximal represent need normalize monte store I give point radius triangle hypergraph encoding point unlikely lie radius hasting proposal vertex mixture walk pick probability random pick prior proposal burn posterior bold prior vertex lebesgue simulation result concentrate section set investigate methodology class use model consider class conditional marginal class metropolis hasting random enforce uniform draw summarize table mode alpha alpha obtain alpha feasible alpha complex marginal observation class summarize observe alpha mode section sec mode match l alpha alpha propose describe subgraph graph triangle cycle cycle star estimate network compute count subgraph network os regime assume vertex prior vertex radius move move subgraph convention induce procedure lasso decomposable graph display decomposable c incur exception star also outperform regime exception fit highlight green another simulation random package formula triangle specification encourage triangle produce triangle os enyi experiment triangle surprising triangle os model tend decomposable proportion decomposable table graph decomposable compete incur produce error table runtime discussion term clique distinguish estimating regard hypergraph method estimate scale multiplication lasso require scale regard method complexity compute skeleton scale lead compute alpha large scale run employ normalize decomposable ghz gb ram h variable width length datum geometric graph assume walk proposal markov specification law hyper adopt prior center deal normalize constant decomposable graph run burn display posterior concentrate coincide model attribute leave missing report appear conduct assess gaussian via miss model incomplete set summarize simulation average average predict vector surprising since contrast decomposable daily exchange rate consist model interesting trivial miss yahoo finance r data assume vertex unit intersection different law keep simulate burn assume miss posterior decomposable parametrization perspective support prior also useful characterize intersection set particularly suited strategy generalize subtle proposal lead move graph hypergraph proposal perturbation consist resample produce model law hyper law encourage feature think lot possibility specification worth couple representation restrictive gaussian graphical model general convenience graphical develop relation hypergraph state decomposable hyper markov put mass conjecture characterize feasible consider graph embed propose ii segment straight every complex geometrically realize dimension iii ball different idea deep subspace span design prior specific graph tree width acyclic insight core incorporate tool prior future development involve process parametrization structure dag obtain strong topology convex subset abstract theory perspective construction modify communication accord grid may improve behaviour infer among easily direct hypergraph think encode individual picture people picture people arguably properly complex topology efficient computing alpha r decomposable decomposable adaptation construction example central idea decomposable complex decomposable decomposable formalize I c produce decomposable initialize decomposable empty graph test finitely decomposable compute use decomposable simple show figure table induce complex decomposable present clique edge reject ht edge accord note reject quite point unit algorithm skeleton restriction radius appear complex preserve htb complex display b decomposable complex ht pt section parametrization geometry idea induce place prior configuration retrieve alone hasting monte move great comparative evaluation abstract complex copula model inference dependence random observation dominant formalism graphical formalism specify carry detail distribution undirected problem among distribution random stage distribution summarize form index edge say markov pair simultaneous decomposable proposal promise extension take parallel computing batch incorporate local move
signal classical cholesky lar lasso single gram small first iteration step lead large prevent phenomenon implementation use dictionary never happen practice solve practice problem case difficult highly choose strategy gradually method algorithm reference improve inverse hessian matrix efficiently yield remove optimization hessian reason share illustrate major modification formulation unconstraine course original formulation nonetheless recursive formula algorithm sequence fast implementation reference therein tool process prove reasonable support impose convex semi definite great consequence invertible strictly verified experimentally consist dictionary dct wavelet enforce replace definite omit penalization sufficient uniqueness coding present briefly optimality denote necessary unique eq correspond eigenvalue equal eq build invertible linear lar aim elastic net replace improve homotopy objective paper optimization neither one good practical proposition almost surely meaning prove variation prop assumption surrogate short calculation growth f prove smooth support feasible continuously x continuously f therefore one restrict unique appendix statement since continuously second immediately simplicity slightly condition optimize unique imply matrix convex show moreover e x smooth show almost surrogate surrogate surely process quasi theorem variation obtain fact tf past obtain bind u f f tt strong corollary easy hypothesis verify differentiable exists bound applie exist u prove surely u sure tt appendix almost converge note addition prove prove final result asymptotically stationary assumption almost matrix compact possible extract converge moment converge case converge u limit converge taylor follow optimality condition cone close accumulation close normal address optimization matrix code prior coefficient regularizer verify positivity regularization homotopy handle regularization elastic regularizer case note sparse induce regularizer subsection claim unit step solve column keep solve project u j j classical amount ball convex long union negative elastic j induce regularizer analogy constraint elastic constraint control negativity constraint problem factorization slightly look piecewise consecutive prove array replace projection ball onto constraint converge optimization onto additional thresholding onto set compute onto net constraint extend ball homotopy solve lasso piecewise part solution fuse problem present efficiently numerous application scope allow fast solving fuse could use complex constraint test address regularizer within formulate dictionary regularizers nmf follow x matrix force negative lead solution face localize one address project research objective control sparsity vector negativity vector work tool datum interpret direction maximize variance matrix propose different formulation sparse analysis extend pca maximize formulation enforce orthogonality sparse formulate matrix factorization regularization net low operation well optimize matlab lars matlab measure performance test method objective act surrogate column setting full version version subset set systematically outperform batch counterpart desire similar speed choose try try power objective datum plot optimal though plot curve contrary set curve set large datum one bring tune hand observe big slow cycle obtain mini give sampling among give strategy except mini compare sgd improve use new much learning form instead tune high value lead bad asymptotic try power couple refined trying select show curve curve see curve still asymptotic set different initialization initialization lead similar variance computation classical negative code face size pixel mit database face image extend f compose patch pixel database even though implementation advantage speed matlab spend multiplication well optimize ghz minibatch original experiment choose face objective matrix run run sparse vector report initialization great curve average initialization term test even nmf bit implementation cm nmf method report computation logarithmic present address various type face qualitatively nmf learn principal component analysis dictionary vector unit figure e dictionary set result different level scalar indicate compose represent value white systematically face database neither nmf localize feature patch hand sparsity among demonstrate technique analyze genomic expression measurement dna copy comparative genomic try analyze determine gain therein measurement order analyze correlation set suggest correlation obtain recursively orthogonality v p u center good r furthermore regularizer gene experiment gene non coefficient select divide repeat split factor tr te te experiment result curve report purpose genomic compare carefully substantial conclusion need lasso average denote demonstrate remove use implementation patch minute ghz eight core text remove thorough comparison art instead wish indeed trivial dictionary use mode cm stochastic online dictionary adapt significantly batch alternative million training stochastic gradient moreover extended matrix factorization negative solve already course need bioinformatic plan propose computationally demand video extend framework address sensitive paper f pf uv differentiable x stochastic measurable realization stochastic f converge index x suppose element borel measurable e f additional address project extend formulation project efficiently lemma solve elastic solve define lagrangian minimize calculation b condition l denote solution complementary imply q close necessary short optimality b j k modification similar convergence solution exactly u u b handle replace scalar problem term homotopy solve constrain u u solution natural lasso propose use lar exploit formula require operation admits close c c p allow homotopy cholesky factorization operation adapt require path stop whenever still note lasso solve eq improve modelling linear focus factorization basis adapt variation signal negative propose base stochastic approximation scale set million sample naturally formulation wide image genomic demonstrate art optimization large online principal analysis atom predefine wavelet recently lead state art task image denoise texture synthesis classification learn natural signal decomposition base principal basis allow datum machine slightly factorization paper algorithm address prove art computational challenge include million address design generic capable various topic approximation atom show modelling code many processing natural predefine dictionary wavelet dictionary instead although look tuned algorithm batch iteration minimize training change time video address mini batch particularly common dictionary several million patch per frame set online stochastic attractive descent sometimes low consumption low batch experiment scale large million training problem convergence problem generalize dictionary learning devote demonstrate suit make smooth desire solve quadratic cost constraint stationary experimentally fast approach dictionary small task dictionary show matrix procedure project onto define p sake n tu space norm denote product kronecker dictionary represent good signal sparse usually process say overcomplete dictionary basis pursuit norm q admit direct analytic sparsity prevent constrain k minimize cost respect rewrite k decomposition rewrite x f alternate variable minimize also prove behave general pseudo dictionary computation coefficient dominate accurately suppose unknown effort inaccurate optimize algorithm whose fact certain theoretically empirically reach solution cost batch set overfitte may become speed memory project iteration orthogonal obtain train speed way fall process one mini batch structure allow without rate section kalman compose distribution loop iteration dictionary minimize
measurement polynomially bound information polynomially observation conjecture infeasible presence classification generator generator specialize binary quantum define quantum formally quantum generator generator tuple state unitary transformation finite set quantum nx symbol basis degenerate generality hardness hold highly link measurement quantum system certainly formal identify concept exist stress model relevant generator distribution assume take arguably hardness practice turn symbol quantum copy hardness appendix improper learning notion divergence learnable kl output satisfy appropriate polynomial improper learn efficient definition boolean efficiently learnable input output representation polynomial learnable quantum generator pac learnable kl gate cc k gate unitary gate particular quantum representation net distance generator size distribution converge calculate former give divergence infinite string support prevent minimum symbol accordingly kl divergence symbol string perturb divergence net generator n hardness distribution generator formally noisy polynomial circle state belong come output quantum generator divergence acknowledgement thank question relevance thank quantum convenience quantum quantum generator unitary starting gate kk k perturb similarly whenever put np px claim hoeffding range perturb therefore take circuit evaluate gate merely involve follow rate construction generator think represent explicitly generator describe transformation entrie nonzero entry appear together zero unitary unitary index preserve application decompose entry real suppose two entry column latter contribution q sum similarly entry weight sum unitary generator basis outcome satisfy previous quantum generator verify induction inspection measurement generator basis claim induction state case inspection support sx b symbol output generator precisely accord suppose could quantum particular learn circuit function easy verify calculus contribute
similarly exploration search embedding e non operation focus split interested split base split specificity sensitivity trade think bayes b estimator tree estimate ht nj neighbor tree sample estimator ht assign left figure b b h conjecture thm thm thm problem thm remark bayes bayes estimator reconstruction p w li r biology institute building pa department work department phone measure address sample maximize expect closeness distance unify focus especially distance euclidean notable hill likelihood consensus metric reconstruct reconstruct wrong slightly refer reconstruct issue help cope bootstrappe bootstrapping occur almost highly support regard similarly common close way closeness rf know reflect common likely reconstruction least tree likelihood accurate representative yet ml goal closeness true optimize likelihood approach bayesian view accord tree many distance easily express vector euclidean statistical understand square minimize distance rule estimator closeness closely distance hard compute hill popular heuristics hill compute hill comparable hill difference hill bayes hill tree compare ml encourage pilot study bayes conclude discuss improvement direction develop sequence specie many evolutionary exist express give observe could topology method create bootstrappe tree tree nj entirely obtain bootstrap notation whether bootstrap tree expectation distribute expect regard bayes close common decision theory give dt estimator simply say tree call recall vector popular embed distance tree branch length branch denote split half size realize squared euclidean maps vector correspond partition induce size symmetric realize euclidean figure square map distance define study branch length topological distance topology length dissimilarity distance analog count leave figure leaf combinatorial distance interpret mean near depend project split onto v consensus consensus tree bayes project nearby analog tree reconstruction onto base see cm bayes minimize frequency since weight set weight find compatible maximal weight traditional split frequency tree sample frequency apply though hence hard considerable toward see special collection tree easier least square ols evolution square dissimilarity bayes estimator reconstruction ol I ol I first length topology length comprise ol I however difference I square map ols I dissimilarity sharp contrast summarize govern underlying sequence bayes treat perturb tree tree exponentially hill ml work hill minima hill move topology combinatorial move subtree move composition move tree move detail quickly ml choose hill move hill must definition situation much express additive constant tree begin hill need expense calculate bayes choice distance consensus extensively lie foundation dissimilarity choose length prevent short topological estimator believe property suggest importance conclusion state path evolutionary tree study tree choose square think believe bayes connection outline deferred tree vector compute simulate briefly review tree process k seq program datum available website sample burn run total hill software compute nj pairwise house software euclidean hill hill along various choice trees nj tree five ml tree sample call briefly hill list hill input compute pt neighbor end practice allow hill might several statement loop study always code write available compare reconstruct ideally would obviously unless particularly set frequency v pt attention topology probable tree topology fairly probable topology compute three record tie topology proxy set hill nj start start bayes five plot nj ml figure report interpret difference typical plot pairwise pairwise also analogous plot empirical estimator true might global true plot table hill nj ml hill
local super propose xt xt minimize likelihood connect source due chose candidate function lag structure sequence cause reliably maximum adopt regularization suggest shrink zero functional brain connectivity fmri property shrink point coefficient appeal connection might also appropriate area eeg datum penalty hoc adopt lasso correlate assumption instantaneous hence order apply split original since correspond source propose put e sum group p call connected extend causal discovery eeg connectivity correlate source observation mix volume effect compare fit sensor space instantaneous e source effect ideally turn ica employ usually g explain arise model nevertheless regard maximum perform instantaneous source possible connectivity ica connectivity model possible filter even connectivity invert generally equivalent source sparse carry sensor able comparable ica allow cross latter see traditional ica measure mutual htb step compute ica ar fitting second statistic cause drop compare obtain vector l bfgs optimizer converge sign difficulty compare retain df become l bfgs special care gradient bfgs df uniquely subdifferential straightforward subdifferential sign inversion care must exactly corresponding norm minimum subgradient attain minimum norm care practice outline procedure sparse shorter obtain heuristic pruning connection might composite alternative alternate justified expectation call unconstrained importantly parameter convexity concavity convex great unique guarantee bfgs eq regularizer method bfgs unlikely solution however joint problem lagrangian introduce minimize additional loss conjugate hessian source gradient conjugate evaluate eq conjugate give turn nonconvex exactly convergence part autocorrelation however increase especially use augment function minimize assess simulate seven accord seven model draw mixed source eeg channel spread seven place spread realistic forward build image head see generation never noise none explicitly evaluate robustness additional variant pseudo eeg variant differ degree temporal correlation variant variant source share mix xt variant simulate cover brain distinguish noise n distribution instant variant temporal ar note since delay model causal snr snr norm eeg source dataset construct htb six structure create correlation introduce distinguish noise source coincide contribute sensor sensor n ica reconstruct seven source instantaneous ica fundamentally source fulfil dependent connectivity analyze variant lag temporal disadvantage temporal filter reconstruct source computation lag provide author information selecting lag constant either jointly bfgs variant since relevant source unfortunately generally reason perform similarity fit achieve another goodness scale possible coefficient goodness fit whole matrix quality decomposition match pattern location typical example estimate reconstruct finally source ridge coefficient area auc correctly discover significance true connectivity order equally necessary connectivity however could examine coefficient ridge testing non actual reason preferred regression different noiseless six variant plot outlier red remove result simulation follow difference box correct affect localization good achieve fig occur estimate fig sometimes superior amount criterion another might instability role current performance solely regard noise say relative degradation source seem problematic uncorrelated sensor spatial mix temporal affect however error de quite connectivity processing finish short em medium still room htb make source inconsistent studying require minimum brain hence make sense innovation ar pay price effectively stable stability connection observable eeg presence background emphasize assume innovation non innovation brain complicated question correct decomposition scope address channel significance decrease channel brain connectivity hard problem volume measurement rise spurious novel overcome elegant numerically detail eeg model source jointly drive super gaussian spatially use group achieve interpolation two ica cross extract connectivity usefulness excellent work study stationary analysis novel assessment addition aim connectivity enhance interpretability e european fp technique brain connectivity eeg signal volume follow eeg autoregressive source parameter overfitte avoid extract appropriate manner functional usefulness compare exist excellent two decade become possible thank progress make field mathematical imaging modality allow spatial temporal simultaneously record series brain functional relate connection sometimes region significant correspond quantify spectrum coherence g causality place outside head arise rather measure site sensor superposition brain instantaneous detect spurious recently eeg analysis account volume cross instantaneous effect couple robust volume conclude
tendency class grow vice versa notation element alternatively member tendency member around see alternative parametrize one generality dx fp segregation pattern association segregation alternative family association alternative segregation restricted lie remain denote vertex b j segregation occur association occur around alternative family parametrization triangle triangle segregation association alternative triangle segregation parallel segregation standard arbitrary away vertex segment segregation available association alternative segregation depend explicit finite asymptotic association alternative asymptotic much segregation uniform furthermore number follow mh om nh nr nr r mm segregation alternative proof r hold limit hold association notice segregation degenerate cr cr triangle segregation define one triangle number segregation association furthermore order extend manner translate segregation alternative extreme segregation association expect segregation association triangle segregation alternative extreme segregation small standardized test statistic critical sided segregation normal distribution segregation association segregation yield binomial base carlo power investigation realization segregation leave middle association number segregation test segregation test association give mf p n mf r mf normality consistency prove similarly hold segregation multiple triangle case triangle j gr gr mh h mn gr gr gr gr segregation sense theorem efficiency investigation involve detailed discussion segregation alternative applicable association direction hand segregation alternative association sensitive segregation st nr nr mp around choice pattern seem uniformly distribute support intersect intersect clear violate detect test statistic tend segregation support intersection point intersection support intersection calculate record per triangle replicate repeat monte carlo times critical base empirical conservative determine deviation significance level size conservative upper figure level value side tend small conservative association power restrictive sense realistic crucial lose substantial proportion might outside extensive simulation outside affect empirical monte carlo respectively set respectively form deviation generation record x expect area realization independence support segregation point away square restrict segregation overlap segregation segregation furthermore large segregation outside segregation finally associate level association e generate combination combination proportion point outside denote mean value ht fit value various proportion point outside relationship base solid fit carlo propose adjust proportion namely outside suggest per triangle suggest eq hull affect segregation consider large adjustment seem correct increase equation hold might large guide sample correction extensive suggest statistic adjust mn mn ht j mb extension proximity let polytope vertex faces form opposite vertex fall vertex face opposite hyperplane rx polytope rx rx xx r j x r nr pe nr pe ir nr pe conjecture iid simplex intensive calculation article bivariate segregation knowledge theoretic cover unlike mathematically tractable computable numerical dominating proportional minimum set nice triangle base region obvious particular sort dissimilarity find empirically work perhaps complicated edge geometry triangle class vertice construction imbalance abundance imbalance interaction use provide tend remove hull nan pattern since point circular result alternative parametrization alternative invariant likely might parametrization segregation point pattern segregation parametrize small parametrization construction convex hull hence correction pattern might pattern inference ii triangle recommend binomial approximation simulation randomization finite sample section correction segregation recommend power edge building manner acknowledgment air force scientific contract grant project theorem remark identity mail edu tr parametrize family multivariate spatial relative position extend proportional edge segregation spatial randomness binomial size class whose point constitute infinity normality size infinity evaluate carlo prove consistency restriction small find optimal segregation association discuss article keyword map interaction implication specie relative allocation method graph approach test spatial association respect spatial interaction give together tend occur convenience call characteristic observation example segregation investigate specie spatial neighbor contingency table nn pattern rl nn test design spatial mostly pattern rl realization arbitrary bivariate interaction graph gain popularity analysis move metric landscape suit concerned connectivity conventional explicitly reference reduce utility geometric system path location preserve spatial datum usually lose spatial spatial require adjacency allow meaningful edge interaction reflect patch relationship quantify patch propose association interaction class nonlinear correlation theoretic spatial arcs relation pair order arc vertex place arc vertex proximity near set arc iff near first triangle trivial proof omit short proof give article graph vertex correspond arc define bivariate datum introduce arc utilize radius dx ix r ir involve multiple dimension one prove uniform rather appeal dimension find minimum dominate np tractable maps alternative introduce type call parametrized family parameter number one sufficiently arc design distributional mathematical new family applicable classification proximity parametrize apply spatial testing spatial segregation relative purpose proportional extensive treatment article investigate proportional spatial segregation furthermore extend range expansion arcs small probabilistic behavior segregation association applicability pattern new choose section present one triangle section segregation association present suggest adjustment outside hull sample extension proportional dimension proximity associate point think close name x proximity map building survey map vertex arc define iff call arcs author vertex dominate minimum dominating dominate set proximity set proximity refer iid square triangles iid hull ht circle point distribution unit square set arc relationship symmetric rather iid proximity dominate point random depend explicitly implicitly n example briefly define high exact set size interval proximity map associate rx arc iff open pure contain element interior sphere natural proximity rx proximity extension higher spherical introduce whose view dt iid illustrative purpose point triangle preserve transform scale vertex vx xx r line interior triangle lie e mass figure region line edge possibility vertex region assign arbitrarily edge pass distance let triangle orientation opposite hence ht proportional vertex cm I line vertex n pe pe xx pe rx proportional bottom vertex arc iff arcs n pe likely small imply probabilistic spatial segregation association transformation also similarity rx rx r z rx rx additional degenerate f special falls occur probability star shape necessarily ht ne I f rx n nr pe ir ir rr basic triangle disjoint figure triangle region orthogonal region orthogonal important role proportional uniform triangle complete spatial randomness desire variable nr pe pe r vertex geometry segment edge line invariance us special iid variable distribution geometry note geometry theorem pe vertex notice geometry projection hence geometry triangle projection map orthogonal
test see interesting degenerate alternative realize gaussian noise alternative alternative give statistic limit hypothesis test regularity function continuous derivative converge process type use equation wiener let fulfil write explicitly follow calculus done write q wiener put wiener q eq mention slightly simplify solve equation introduce limit random come observed wiener function usual allow write kalman theory value innovation elementary hence correspond alternative equation eq q function condition write hypothesis show consistency function uniformly kolmogorov test hypothesis test hx correspond hypothesis start behavior fix write direct yield equation put inverse yield limit test statistic put wu wiener signal white case power condition fulfil alternative condition degenerate power already different put continuous hence put n test lead think relation eq replace forget wiener formally let formally modify fourier coefficient write integral mathematical meaning starting introduce statistic hypothesis introduce quantile lead valid important observe condition fulfil q fix contiguous sx put alternative course minimax alternative weight special choice white gaussian poisson process representation time start play ergodic asymptotically proposition moreover convergence weakly let c statistic eq constant asymptotic limit construction ordinary differential weakly local wiener equality q integrating put hence outside suggest q free case local time limit close white suppose hypothesis impossible property coefficient solution test take smooth consistent normal fisher regular smooth limit coincide integral another consistent structure see et calculus eq empirical integral test way integral formula contain integral I converge uniformly process statistic statistic limit kullback
pick edge theorem obtain formula correction miss tree totally backtrack error bind length shortest derive chain inequality k intuition accurate interaction omit equivalence product compute aid z correction express whereas correction still include serves totally backtrack non basic idea underlie depicted cycle base cycle copy tree root illustrate tree undirecte understand allow computation follow omit embed equivalence cyclic walk trace cyclic computation reduce infinitely tree walk totally backtrack segment tree walk edge direction step break walk embed cyclic walk calculation let weight single loop graph product intersect subgraph determinant corresponding matrix equivalent tree loop diagonal consequence impractical however walk correction block k lb w maximal block negative submatrix approximation elsewhere however insight gaussian w definition weight error result scheme converge correct decay exponentially estimate cover estimate cover cd z z exact hard distinguish bad correction need need cover graph grid four may shift increment see block loop short intersection block add weight complexity compute determinant block correspond numerically periodic quality approximation rapidly correction determinant interpreted totally backtrack show estimate way involve cycle particular product grid method address system leave plan model explore factorization r k kk direction model walk corollary propagation determinant determinant backtracking group equivalence furthermore multiplicative correction efficient correction length belief propagation numerous application sufficient also also insight sum within graph estimation determinant inspire recently study new perspective tie determinant cyclic walk determinant backtracking computation universal cover group interpret determinant graph edge solution truncate specify grid estimate differ heavily multiplicative expansion additive expansion loop calculus walk truncate let treat undirected edge direct adjacent direct end visit cross precede vertex begin end multiple shorter primitive close primitive walk walks shift cyclic walk follow classification walk play role said backtrack unique walk denote walk totally totally walk irreducible elsewhere set disjoint equivalence class denote backtrack irreducible totally backtrack neither backtrack sparse symmetric partition j h dx normalize vector px x dense use gaussian graph complexity growing distribute parameterized dx reduce follow iteratively message convergence marginal I method terminate elimination cover computation converge correct variance covariance jj concerned link partition motivation graph unstable solution interpret walk describe covariance determinant independent message correctly mainly concern correspond covariance approach walk diagonal diagonal walk sum eigenvalue interpret walk count occur walk regardless add walk require absolutely equivalent spectral wise sufficient variance walk sum reweighte walk interpret recursively sum imply walk model walk message interpretation compute correct mean walk variance totally backtrack totally backtrack walk estimate walk call w w primitive primitive cyclic totally backtrack totally walk play role walk interpretation estimate analogous z totally backtracking seem intuitive prior proof previously prove theorem summarize useful use identity
fix verify decrease iii original theorem proposition specify choose nontrivial fortunately necessary achieve substantial improvement already considerably set constraint upper leave remark iii equivalent ii surprising enter picture study algorithm introduce reduce result without ii limited availability expect little iii simply channel nevertheless convergence proposition v proposition follow entry also obtain symmetric eigenvalue proposition radius eigenvector unchanged particular restrict precede eigenvalue stochastic frobenius theorem prove involved place proof prove reduce special omit q final output eigenvalue right calculation eq deduce proposition capacity preserve provably alternate capacity calculation channel appeal geometric possesse monotonic extension study variant aim speed focus channel capacity calculation convergent typically valuable insight slow construction usual measure global substantial approach differ acceleration proximal iii broadly term overlap monotonic iv theoretical discrete specifie letter mathematically e nonnegative logarithm loss identically zero algorithm eq convergence original generalization constraint guarantee iv reflect intuitive explain end iii illustration fig iteration appear start give comparison algorithm henceforth define entry nonnegative write scalar satisfy doubly upon stop convenient restriction discussion reduce ii slightly implement determine time simple minor substantial consider inspection reveal regardless easier verify ii iii lead fast possible restriction recommendation conduct illustration example channel accord satisfy algorithm ii record use evident display bivariate sometimes hundred throughout reduction summarize acceleration ratio acceleration around ii support preference still implement satisfy guarantee intuitive setting maintain write study property calculate break restriction equivalent indeed algebra specify reduce useful proposition iv reduces conversely satisfy deduce equivalent iii converge monotonically bad slow channel matrix try overlap iii simply call proportional iii call work iii long I inspection regardless mention vector obvious relation key iii slightly straightforward follow maximize maximize jointly maximize maximize tucker condition iii never decrease alternate monotonic sequence generate iii exist solve throughout general convergence theorem comparison theorems convergence iteration mapping emphasize assume lie interior enough hence eventually fast notion global version see formula
I complement spectra ideal spectrum perform turn strict f orthonormal accord fix th matter eventually fast da per iteration lebesgue count k mf obvious minor symmetric component let q density datum version highly intractable moreover explain every interesting equal furthermore vary exact mode separate area despite year careful description facilitate fs algorithm density integrate pt pair joint give integer density datum density xy section conditional first follow eq formula reveal two fact independent conditional draw sequential example define know available concern section deal suggest chain converge move mode section describe alternative chain move iteration encourage transition symmetric mode move proceed choose permutation get choose permutation call fs explore effectively chain establish fs chain fs respect develop formula permutation represent switch fs choose simply observation cluster choose suppose satisfy give depend hence argument show indeed establish couple demonstrate balance little thought thing happen either r uniformly obviously step uniformly I fourth fs indeed theorem applicable operator compact spectrum pt eigenvalue fs fs actually chain I move exact recall graphical latter marginally fs target density fs spectrum compare spectra fs chain mixture quite flip fair coin take tail fs r four eigenvalue along early easy see nontrivial eigen switch replace theorem order case add replace replace affect fs evidence fs substantially small base appear surprising minor huge function form conditionally q proportional bernoulli complicate complete via routine eq analogous form imply chain consist probability entire idea use approximate eigenvalue fs section express da respect write operator fs dp carlo idea spectra must furthermore heavily generate random mixture result observation contain third set ten observation use classical carlo row carlo da fs row calculate eigenvalue record large dominant close dominant eigenvalue fs eigenvalue increase clear f may eventually would begin fs estimate monte carlo random seed eigenvalue correct place element must estimate thus simulated mixture process purpose result analogue nearly irreducible section show reversible respect eigen equal row could imply determined eigenvector eigenvalue element eigenvalue element yield two root correspond eigenvector eigenvalue row acknowledgment visit author wants acknowledge first support nsf grant third author la paris thank universit paris paris author thank project visit anonymous suggestion theorem paris paris paris reversible chain augmentation da self adjoint encode convergence generally quite handle spectra augmentation finite operator da compact dominate spectrum operator former less study density associate bayesian particular compare da fr random label switch intractable monte resort monte augmentation da algorithm liu build da must density say p algorithm free satisfie obviously densitie da goal joint da good formalize unfortunately ideal da fast function da draw mind inherent tool reason da possess two variable explore da simulate reversible adjoint whose encode property chain zero define p gx dx da chain formal value operator definition da operator imply eq invertible fast unfortunately I even get currently yy yy fx consist element directly call reversible chain yx small prove associate alternative chain close ideal draw call draw surprising van decade great deal modify da liu wu liu van yu alternative auxiliary reversible new chain move routine show chain reversible alternative name yu base despite perturbation negligible relative drawing concrete liu wu van operator condition close spectrum consist small addition eigenvalue dominate uniformly da gold standard monte hope strong quantify auxiliary degenerate start illustrate huge gain new involve sample take j negative course mixture thus make dirichlet proper let denote observe result posterior highly modal matrix note maximum else da introduce augment level prior eq consist use conditional call da chain simplex move mode lee switch movement applicable spectrum operator fs chain dominate illustrate extent switch speed study fs chain toy get frequently eigenvalue monte carlo conclusions converge affected sample organize brief operator use analyze reversible chain da appear review proof fs fs chain example eigen special equip irreducible hilbert x xx dx dx act follow show adjoint exist ig il eigenvalue eigen eigen define application jensen negative univariate define p extend quantity call define liu particular geometrically ergodic driven geometrically central theorem asymptotically standard estimate ergodicity reversible operator rather call whose probably satisfy eigen solution less da suppose integral intractable direct simulation indirect liu important spectrum da dx dx dx yy dy kf quite compact operator function ce particularly compact iii accumulation eigenvalue compact along eigenvalue define conjugate da chain geometrically ergodic finite indeed eigen chain start stationary liu conjugate calculate integrable function abuse double inner norm exploit
factorization joint probability depend q therefore presence arcs variability whole indicator goodness bayesian criterion develop call summarize interest difficult value marginal success distribution collection simultaneous subset vector useful depend approximation multivariate bernoulli vector consider link correlation univariate bernoulli bernoulli eq complete variable independent pair conversely equal pair eq independence sigma sigma set correspondence analogous variable apply disjoint subset bernoulli bernoulli l bernoulli variable bernoulli dependency uniquely identify probability bernoulli covariance matrix two zero imply several moment attain minimum eigenvalue show multivariate let negative prof hold eq simplex preserve whose variate moment underlie unique direct arcs state naturally bernoulli include bernoulli via dag undirected empirical use obtain variability network structure undirecte simplify bootstrap variability structure learn sample outcome frequency moment bernoulli structure bad outcome independently half univariate spread normality multivariate associate unstable network structure due easy equation hadamard determinant non respective maxima generalize maximum reach equal rank instead norm structure rewrite multiplier method coincide associate network high distance case second moment generalized variance frobenius matrix three matrix limit statistic interested relate total achieve hypothesis significance correction inequality distribution respective correction one compute correction bold associate frobenius spectral decomposition see significance value unlike display sample raw asymptotic statistic compute bernoulli specify diagonal matrix completely marginal normalize compute monte eq evaluate remove distortion cause lack problematic dimension low bootstrap carlo small network pt multivariate bernoulli significance various size bootstrap use derive property along arcs ph school article give useful suggestion thank pure university help respectively frobenius property maxima depend multiplier multivariate bernoulli prevent direct interpretation quantity matrix statistic k line global correspond
result actor ii likely connect actor mixed membership case actor mix role within affinity actor obvious discrepancy truth percent actor percent still retain employ different scheme mixed possess membership could accuracy mixed vertex difference display ground estimate metric result ten time error bar perform slightly significant difference model compute experiment question simple log derive via table goodness fit model dominate dirichlet normal assess fitness consist time point remain role furthermore network adjacent point certain similarity compatibility display figure performance average membership vector value percent suggest indeed integrate test confirm membership grey learn role compatibility entry arcs value outside example record social ranking toward study major member interesting look separation undirected vice static point researcher study static fit role select estimation label correspond mark member place place demonstrate estimate role compatibility appear intra pure role boundary leave bic score suggest role compatibility network vary role infer illustrate big change mixed membership time occur overall dominant early mixed membership grouping static role dynamic roughly role later besides support member change time membership trend member isolate finally lead email communication email process email network email pattern property gene engineering fit coarse quality dl life among high dynamic role evolve plot trajectory wide simplex dl diverse pattern formation axis specification system compound development relate heart role invariant diverse gene cluster cluster combination across evolve biological role consist functional role group different gene functional group heart cell development fourth role development study level ensemble path node mixed blockmodel propose relation dynamic social biological role independent structure actor network role third membership actor vary provide extra expressive network rich temporal practice proper development fulfil move cycle evolve change drastically stage process specification posterior axis may dominant many gene interact various aspect hence lead understanding ingredient prior membership context diagonal dependency structure role clearly readily couple tracking role drawback therefore learn develop necessity role indicator possible role focus develop efficient thousand rather million appropriate web offer want economic extend explicitly clique state enforce mixed membership interesting derivation simplicity drop subscript eq q derivative ks kx exponent x tx tx ts x second derivative k jensen apply approximation specifically analytical set derivative mle approximation likelihood tractable bind log grant nsf cs ii award fellowship dynamic social biological environment actor systematic evolve offer infer actor underlie topology build mixed membership blockmodel many actor account interaction actor allow actor behave differently carry interact reality approximate communication gene interaction network full case pattern dynamic role actor fundamental form science entity also actor communication study reveal position pattern biological evolve investigate statistical inference evolve vocabulary imaging network internal network gene network relationship vertice event evolve terminate stochastically semantic static stand role biological entity affinity compatibility algorithm infer dynamically concern evolve episode major therefore behind change email communication network company record perhaps behavioral trend business time span development capture inference function system social company capture role individual interaction among community behavioral process biology translate latent genetic interact protein topology molecular advance understanding mechanism biological broadly lead consequence diffusion hierarchy organization formation appropriately simulate mechanism discover change actor network network various trend network include free formal characterization characterize detect characterize additionally progress traditionally toward semantic review major limitation modeling actor biological label etc interact actor realistic role multiple influence play actor role actor response exist relationship stochastic molecular systematic distribution understand biological process phenomenon capture actor role evolve modify enable link specific connection separately fractional role capture thereby actor statistically infer embed membership characteristic membership intuitive community model latent infer actor biological entity function observe actor position dimensional simplex role actor role functional actor among actor reflect distance actor dynamic process drive evolution furthermore tracking position actor blockmodel emission shall short allow infer role result membership wise microarray actor remain briefly study simulation model algebraic derivation vast body network traditionally focus exponential generative cause actor position latent base latent explore membership blockmodel idea model node belong multiple block e fractional population model analysis membership project aspect space normalize reflect weight aspect role etc serve surrogate develop early role network use aforementioned actor multinomial actor role sample interact actor interaction may contexts interaction protein context space tracking trajectory infer entity termination underlie adopt extract text author network network invariant j possibly link vertex class relationship point vertex role realize predefine latent single membership paper different membership stochastically role role role compatibility realize role interaction mechanism link pair actor actor interact interact specifically different role pair role actor unique actor role combination basic mixed membership blockmodel static mixed interact vertex draw indicator role j j ji j unit draw specifically generative define among vertex reflect latent e identity link actor actor existence link package send person vertex label undirecte ignore semantic capture membership unique interaction strength interaction compatibility pair membership actor actor expect role role actor actor different interact neighbor role affinity role dominate actor role connect differential preference role rich pattern role compatibility complex role represent capture probability actor actor actor membership membership employ multinomial nontrivial correlation among within role role cell employ logistic simplex result normal logistic membership break draw simplex follow constant constrain parameter freedom need draw leave description model role every vertex compatibility evolve conditioning sequence trajectory function state static logistic random mixed compatibility relationship behavior generative transformation mix adjacent membership represent transition shape trajectory model emission define dynamic propose dynamical change network entity sense termination function topology generative graphical membership k k compatibility coefficient compatibility subsequent compatibility probability via point dimensional tb link assume actor mean evolve accord topic unlike outline mixed directly kalman smoother intermediate principle membership capture vertice dynamic simple walk membership compatibility semantic membership unlikely time expect membership difficulty base prior vector intractable additional logistic normal direct infeasible mean approximate b factor show expectation variational approximate marginal apparent description subsequent variational simple building simplicity inference role observation role indicator latent marginalization hide intractable approximation update continue formula multinomial multinomial therefore laplace base possible previous number repeat multiple pick simplest via em style current maximize log posterior intractable use update formula em derivation
plain see graphical refer problem restriction generality correspond independence graph follow edge require dependent detail copy analyze infer regression require global put later penalty likelihood obtain consistent edge glasso mutual neighborhood fisher matrix counterpart generalizing via infer graphical general condition closely relate pursuit regression conjecture use glasso study section also glasso limited restriction via consider many regression recognize hand behave problem glasso study adaptive strong every pair irrelevant recover correct selection assume mutual incoherence pairwise eigenvalue examine property procedure relaxed design design parameter know framework multi study probability rather analyze ensemble satisfies incoherence open lower sample assume adaptive general property present restrict section estimator design design consequence distinguish fix design proposal use lasso variable make assumption relate denote eigenvalue throughout consequence bound constant describe assume allow use notation fix row index active behave throughout hold imply also relevant property initial true hold optimal argue set hence multiple unique emphasize require random require upper use nice estimator section selector could restrict design moreover hence proof analyze later section adaptive standard lasso specify sensible view build work assumption formalize therein selector condition introduce integer vector index value outside integer hold often fix condition adaptive lasso possible eigenvalue result argument know size positive definite cover sparsity corollary condition follow restriction sparsity weak assumption derive discussion arguably upper thresholding follow hold set large restrict every regression pursuit assumption model assumption among derive adaptive correctly relevant ordinary stability sufficient easy stability eigenvalue high relation former require assumption restrictive stability appear trivial condition certainly derive additional thorough relation direction frequent appearance dimensional reasoning check analyze glasso algorithm clarity denote parameter weight solve distinction assume former pre specify latter slightly strong support sign pattern weight design matrix ordinary lasso coincide impose lasso incoherence let nonempty j general recover sign furthermore statement define q variable n ny c bound throughout rest follow plug fact kx large union bind unique c p crucially loss show bound lasso triangle fix satisfy strong hold hence ks ks x conclude finish check invoke finish regard dominate k design shorthand hand coordinate assumption ks ks design subset perturbation design set define hold subset ks ks ks sparsity hold proposition invoke finish sufficient event let submatrix j positive invertible weighted hold weight I solution j subgradient simultaneously subgradient p cs w cs c direction reverse guarantee q simultaneously hence similar detail illustrative adaptive program w eq sparse standard uniqueness definite elsewhere define relevant j x x x condition event prove least straight hence finish check incoherence condition invoke exclude event incoherence satisfied need concerned sn shorthand event design term section thm thm thm pt van f two procedure dimensional model accomplish gain stream feasibility provable penalization scenario easy convex oracle require condition design refer coherence compatibility restrictive restriction severe penalization shrink also correspond become infeasible alternative multi procedure example adaptive loose penalization analyze gaussian framework
diameter horizon ball arbitrary x play bx total tb plus term reflect location metric uncertainty insufficient active distance follow confidence reward pre ucb payoff center ucb obtain observation refine use observation ball well ucb expect reward ball ensure property ball activate allow accumulate activate ball dominate active relevant context ball round oracle call current ball let running time oracle implementation run similarity upper ball prove third must played ball put tr rr jj start ensure several maintain confidence time active domain ball cover similarity space ball radius immediate ball cover ball radius activate center notation active ball center set x increment hoeffding n tn tb n ucb clean run heart clean active b activation put piece together tb clean activate activation lemma claim simply ball ready ball activate throughout parent ball activate round active ball activate select moreover corollary activate center lie distance another fix round select regret contextual provide difficulty correspond definition carry without add fold matter crucial ball accomplish center form packing make efficient radius active ball full child give bound constant similarity active radius potentially much radius ball full center pack additional assign arrival call packing packing fall region tailor application informally context optimal suboptimal radius ball center select round contribute contextual therefore consistent r rr consider payoff contextual contextual absolute dimension cover modify activate ball radius become ball full become ball child factor give upper pick upper uniformly must possible namely pack mab stochastic payoff contextual bandit theorem extension context free payoff payoff parameterize pack collection pack number packing note r n metric point context similarly arm sequence permutation repeat problem instance payoff define completes summarize regret choose lower contribute contextual regret proceed lemma derive arbitrary horizon increase function write tn mab condition brevity write loss exist p rp simplify define whenever obtain contextual random contextual consider separately form free bandit round payoff instance pick informally know full reveal contextual context independently expectation respective remain handle separately draw use divergence arm exactly instance stand alone theorem fix ensemble describe mab slow change stochastically evolve bandit discuss recent learning publish free mab payoff specifically payoff round priori known algorithm adapt change converge arm mab problem maximize payoff context expect payoff benchmark see dynamic regret sublinear time term performance quantify dynamic goal limit bind payoff mab payoff arrival payoff specialized arm constraint q mab long performance contextual suitably horizon every round version prevent strong provable provable provide theorem tractable mab contextual period whenever period analyze period take obtain ok theorem packing number time modify section omit corollary contextual stochastic payoff combine temporal time across arm mab literature payoff variation analog regret depend cover arm mab problem volatility arm average ok corollary periodic contextual first bind covering cover arm work former take bind easily arm generalize mab walk payoff evolve uniform analysis particular name mab defer version call mab marginal assume payoff evolve stochastic mutually marginal arm dynamic q strong uniform period ok interestingly whereas seem mab walk walk stay boundary assume require accord walk contextual space q corollary dynamic arm parameterize round arm bandit payoff result mab arrival every arm pair lipschitz set high index payoff easy mab payoff arm payoff e arm context arm follow sum contextual mab problem extend bandit incorporate contextual similarity preliminary publication apply bandit interestingly contexts motivated web round user rank document click document namely relevant specify vector document minimize click extension user style connect expect click separate contextual document sequentially slot round document relevant treat algorithm condition expect click suffice guarantee lipschitz corollary contextual stochastically evolve payoff corollary dynamic periodic non periodic mab show benchmark payoff lipschitz likewise let let result provide completeness conditional hold let obtain corollary implicit rt namely sake r see definition fix pack rt rt six fix time marginal least claim fix exist claim follow show winner winner proof eq q take possible simple namely ok tt suffice periodic payoff satisfy constraint consider failure implicit corollary plug section maintain partition thus advantage algorithm bandit call reveal pick observe payoff section arm payoff round specific fix adversary advance generalize regret free mab mab arm tie break define metric arm loss metric reduce stochastic setting set context mab randomize statement achieve cover arm space flexible leverage distribution take bandit line bandit payoff payoff achieve notion quantify guarantee refine outlier cover cover slack mab instance fix bandit multipli slack remark version raw number page contextual parameterize bandit use subroutine finite collection ball ball radius stay instance create parameterized radius proceed call report time round ball break activate specifically see space diameter set data ball counter loop b b else x bb bx specifically activate activation active contradict continue satisfy basic ball active radius separation immediate note specification round activate child ball hand else activate contradiction child active ball within fix horizon contextual ball ball select regret claim definition collection full ball radius radius say full heavy exist heavy ball round specification claim q ball q plug derive payoff contribution maintain adaptively partition time refer resolve arm pair either slowly nearly invariant payoff special change adversarial payoff maintain partition context adaptively advantage payoff non choice tailor concern requirement quality provable would weak version obtain several result make meaningful would suffice setting upper difference payoff arm pair want provable contextual publish publication question payoff desirable payoff context knowledge author armed manuscript comment anonymous improve presentation analyze construction kl divergence technique implicit stand alone contain along relevant definition section minor feasible payoff function function rely subset child feasible payoff exist coincide payoff lie play arm round incur subset child ball payoff end subtree root feasible mab payoff bandit algorithm regret payoff assign payoff arm preliminary manuscript early address issue point journal mab round receive associate case strategy focus exponentially recent literature similarity extension round payoff round contextual motivated place crucial particularly similarity information bandit arm payoff prior bandit similarity approximate payoff context advantage central fine partition several mab bandit descriptor abstract bandit henceforth armed bandit mab present round past payoff operation economic clean exploitation trade crucial sequential uncertainty regret optimal mab arm understand arm infinitely unless assumption bandit need find investigation discussion work assume certain assumption efficient work start arm mab give contextual bandit directly place crucial cast ad payoff click page perhaps follow bandit simple bind contextual arm round follow choose payoff fix expectation subsequent definition set whereas payoff mab set demand arm free space call lipschitz word lipschitz without generality generality context metric mab similarity context suggest choose create bandit adjust horizon box mab pick run point horizon regret idea adversarial payoff cover space arm potentially payoff adaptive partition adjust frequently occur context instance payoff two main payoff payoff payoff region correspond frequently occur region prior space maintain one context arm develop provable payoff match uniform obtain matching bound divergence fix upper bind mab contextual ad scenario per se incorporate spatial constraint contextual meaningful recover one context contextual recover publication version contextual bandit constraint arm context study mab payoff random payoff case problem choose mapping context case arm applicable contextual mab essentially algorithm match use partition expert bind distinct translate context reduce mab define mab payoff notation sequence context arm break way ensure attain assume similarity hold space cover related paper diameter less minimal covering similarly relate packing point maximal packing packing cover namely small follow know introduce know dd euclidean radius
display likelihood simulation technique public explain illustrate need analysis tool practical side grow complexity available approach handle identifiable infer graphical estimation dense time markov bayesian perspective support presentation comment acknowledgement relevance unity neutral agnostic manner bayesian handling model increase proportion past irrelevant keyword choice informative computational statement believe valuable bayesian toolbox elaborate argument science paradigm perspective I toolbox set theoretical ensure coherence procedure proper encourage historical people jeffreys answer greatly towards bayesian procedure popularity even trick bring keep mostly usual already text selective run support advance practical past anti rational asymptotic normal conjugate first statistical much book bayes incorporate seem class parametric mostly prior parametric much acceptable work common parametric nonetheless approximation choice obviously state handle model really give reasonable answer minimal type go beyond bayesian question draw run obviously opponent inference impossible immediately discussion paradigm operate truth true production parameter since even inferential optimality alternative model agreement demonstrate coherent statistical relative support drawback paradigm contrary strength thing true reference besides belief test analysis check little reason besides name state theorem toy position wide jeffreys stein optimality bayes surprisingly bayes express formalism economic constant ignore inference validate measure conditioning approach give mean properly datum report engine raise experience inference view decision automatically derive automate derivation optimality explicitly search square minimax sometimes also economic utility bayesian whole update recurrent bayesian perspective whole inferential upon straightforward come decision answer take impact would plus bayesian allow infinite range item advantage incorporate focus prior e surprisingly completely parameter thing informative expect moderately remain probability make particular generalised manner space mathematically mathematical fail criterion family like haar mention prior rely classical difficulty natural factor rich bic bayesian law test time specific contrary strongly bayes factor contrary maintain specify theoretic rarely improper prior mostly set lack proper constant solution domain ad hypothesis whether inferential question justify assign type hypothesis define factor direct pearson fisher namely coverage percent interval live coverage thing question frequentist jeffreys occur theoretic relate classical consequence nan must subsequent towards decision theoretic perspective posterior bayesian bayesian se acceptable cover seem I fundamentally sound since statistical limitation fisher paradigm impose perspective measurable hypothesis sum measurable mathematically probability co classical belief dimensionality nest contain vice versa bayes properly applicable interpret point apparent absence standard output illustrate default bayes bayes standard lm obviously answer else frequentist strength region bf intercept strong substantial ten bayesian provide carlo computer recently extreme elaborate less mind reason simulation past decade detail computational seem infinite inferential assessment simulation discussion allow inferential question nature namely generic principle simulation abc confusion reliability development allow mcmc essence method output convergence apply mcmc sample help stationarity therefore happen similarly software fail important numerical little often detect together
infect individual ultimately infect probability avoid infection simplicity community infection dependency inherent avoid infection argument hold z infection new final describe model natural parameter original new size size hasting algorithm three give center size period epidemic outside quite result final outcome set example median l simulate giving decrease community dataset c true mean l table result point table size alone sufficient effective precise sense former low latter reduce even dataset threshold similar kind finding sir affect extent individually dataset conversely four experience hard infection drive epidemic reflect complete conversely correlation distinguish community infection estimation interesting note likelihood poorly consider final estimate nothing either true average seem likely occur surface need really level model mix ignore e group zero final hope considerable second consider divide belong two kind model population secondary contact go school go individual child allocation allocate school child school school allocate go respectively go obviously numerous possible allocation seem impact model approximate branching epidemic epidemic group community transmission ignore recursive formulae e g p initially become infected course epidemic far mean initially ever infect early epidemic receive external contact average total individual infect infected therefore type individual matrix denote number individual community contact child half period infect equal school child infect equal child plus child infect school infect school probability number school school similarly large work place school ever infect ever infect individual ever school child respective infection child infection transmission within exactly infected explicitly number infect school place community child ultimately infect number child infect infection event second epidemic child behave individual infect initially individual adjust type child infect eventually infect amount final treat complete final size perform analysis key finding use follow simulate epidemic typical parameter infect comprise child difference infect purely random identical dataset except result epidemic far infect comprise child c mean mle complete final size pt median median mle remark parameter analyse table broadly deviation especially suggest comes associate less form infection picture transmission occur infection within contact school community datum great data utility datum affect former attack infect information deviation datum large explanation arise detailed consideration dataset particular child individual whose infect contact infect individual know conversely possible infection argument apply analysis already far source infection would precision heterogeneity estimation heterogeneity compare size purely question study try question one consider intermediate final kind enable considerable precision distinguish infection temporal epidemic distinguish community occur secondary group structure homogeneous factor generally threshold change ignore turn level would affected level mix unity could natural two individual similar seem broad qualitative finding unlikely basic acknowledgement partly uk engineering sciences ep pt minus pt pt epidemic secondary typically school place accord infection parameter understanding infer precision thing considerable inferential heterogeneity considerable keyword number inference epidemic disease classical mathematical epidemic mix individual disease e rarely reflect disease propagation therein broadly speak g contact structure child heterogeneity chapter cause assume mix within g considerable essential population mathematical address issue various disease propagation example age structure school location national characteristic period possible kind study intensive simulation identify disease realistic assign plausible inform study mix population large precisely enough relative transmission transmission measure school restriction aim precisely reduce transmission al recent transmission longitudinal report school transmission child intermediate main aim establish procedure longitudinal use assess simulate epidemic incorporate assume two kind epidemic time kind maximal actual mix child former go belong belong paper epidemic derive epidemic devote community finish community group school sequel terminology belong consist applie label individual thought type potential difference similar equally apart behaviour individual individual remove contract individual capable disease remove individual matrix sized group transmission contain ever infect ball final infect initial recall threshold section complicated turn statistical kind I individual throughout disease start epidemic knowledge initially motivation extreme scenario gain extreme evaluate collection social know secondary grouping method extend take account likelihood leave limit th infection infection infect period contact straightforward potentially censor last carry period transmission product contact respect parameter ji jt jt j j ds ji jt I cm ds extend equally plus number number define hasting obtain
although formulate operation global optimum still need initialization multiple neighboring class state e c gradually increase well mechanism explain organized notation motivate section explain derive present latent lda conclude assign latent th denote element available assignment assignment datum indicator vector product product kronecker matrix b lb available class state quantum use extension conventional call classical statistic example well indicator indicate occurrence indicate diagonal probability introduce quantum quantum real however trace finite distribution specifie unit quantum correspondence generalize distribution mixture third employ phenomenon involve vector diagonal vector therefore quantum uncertainty expect useful vb variant proportion maintain term mixture via uncertainty introduce describe enhance generalization explain apply define indicator density log posterior conditional introducing posteriori variational free maximize temperature follow identity classical explain approximated posteriori distribution maximize assignment state approximately see j boundary accurately take completely sum free energy approach index interaction initialization label update indicate class explain b quantum denote inner update temperature decrease field estimate projection implicit precise extract solve whose latent allocation probabilistic corpus document vocabulary schedule temperature process schedule temperature simulation obtain mathematically rigorous consider schedule paper schedule schedule low inverse temperature markov varied fig run five average amount five repetition batch fair term execution time average execution outer iteration iteration try lda novel increase interaction limit work think well eq accurately quantum variational bayes vb simulate annealing vb density generalize add look demonstrate allocation actually typical poor complexity projection class work construction quantum suitable schedule mixture gaussian aid scientific work research area physics material grant generation super project program thank institute physics university google institute physics university technology university present anneal variational easy implement find free latent allocation relate machine quantum mechanic conduct density adjoint one connect mechanic learning maximization concept quantum use quantum mechanic generalize density bayes
fan chen li sir popular regression chen j model ann quasi unknown j regression graphic idea study regression graphic new york li ann comment dimension smooth noisy spline functional measure explanatory fan local regression pursuit ann p index partially york demand air environmental economics management spline coefficient ann ann li projective resample reduction li data visualization stein li reduction response li asymptotics ann york york convergence partially p york university ann adapt link ann h li l estimation reduction r distribution yu spline linear l asymptotics ng check confidence partially index x discretization manuscript university simulation angles eps bandwidth bandwidth dot htbp scale pt plus true linear model wang california china china associate model link function index parameter model establish estimate convergence error variance result facilitate region study application illustrate extension index primary g bandwidth attract combine nonparametric recent comprehensive book curse dimensionality accommodate covariate reduction assume collapse single nonparametric partial index specifically response index component metric index inverse sir li li discrete fan assume yu confirm also yu difficulty employ link fall dimensional flexible smooth h mild suffice often reduction justification suffice summarize predict reality select analyze census area build eight describe neighborhood describe center air covariate specify house binary demonstrate chen li variable interpretable drop consider datum part reduction structure note component detailed procedure et estimate sequentially simple optimally approach plug lead difficulty come remove impose identifiability positive approach sir resample li estimate smooth residual get estimation accomplish sir proceed via residual since plug employ square result employ obtain estimate conclude role estimate specifically regression indicate efficient importantly equation normality equation perform well procedure directly target iteration convergence asymptotic normality attractive method limit variance compare h reduce function h small limit provide asymptotic elaborate present asymptotic report proof independent identically error general explore stage elaborate dimension versus estimator smooth perform versus initial dimension find use profile new residual update step complete theoretically practical iterate simulation report benefit iterate elaborate case index sir response need recommend reduction li already choose smooth fan link bandwidth sequence minimize estimating calculation smooth q estimate replace smooth estimator correspond x smooth bandwidth minimize respect linear smooth need later achieve rate relate specify asymptotic normality estimator fix estimator nonparametric reality possibility term partial spline correlate profile smoother short obtain desirable smooth express estimate advantage current estimator g regression constraint hence include et et model worth variance estimator asymptotic estimating asymptotically efficiency attribute make least possible solution equation true tp remove yu obtain motivate estimate least estimation theorem al final n final theorem follow additional need avoid curse high smooth index moreover asymptotic straightforward beyond scope assume extend great smooth remain unchanged except change sir sir save employ sir perhaps sir major intensive sir difficulty covariate dimensional dimension may hold show li sir save li correction contrast sir direction sir employ conceptually however per wang sir obtain asymptotic regard two theorem estimator normality estimator condition hold r convergence rate corollary give optimal convergence estimator obtain result quantile plug need follow x ig theorem q theorem let quantile far great smooth multivariate estimator sir sample sir use slice slice scalar uniform probability case one choose check scenario orthogonal throughout bivariate save computing pilot reveal residual estimate frequently select generalize utilize estimator bandwidth bandwidth minimize bandwidth calculate bandwidth correct magnitude ng note satisfy ii sir notation sir sir slice estimate iterate sd mse last serve gold column iterate iterate column table table pursuit sir direction sir might iterated typically estimation angle lead far sir fix replicate plot seem parametric chen partially model investigation try et procedure use unable meaningful seven difficulty analyze mention determine variable describe census median census variable covariate final age coefficient hypothesis versus coefficient determination estimate chen li examine drop fit sir inverse meet single choice well thus transformation describe sir met either try suggest satisfy sir sir well new handle slice sir lead slice sample much sir slice lead slice chen li point sir sensitive slice try slice estimate small bandwidth choose sir approach chen li high sir statistic reduction estimate yield significant inclusion minor show estimate axis effective variate variate make transformation use smoothing theorem proof divide appearance denote show central easy divide step existence prove normality condition c expression z obtain minimize much separately existence enough omit ii eq follow equation r r definition recall j decomposition page find complement far cc cc cc cc r cc cc r asymptotic give definite inequality easy proof vector theorem imply complete corollary
product q identity counterpart motivate penalty propose drive far call orthonormal coincide component form jx programming desirable theoretical property problematic thorough overview combinatorial estimation estimation machine estimate penalty let index indicator inequality user hold least first inequality coherence number satisfy oracle coherence require bind correlation cf almost suggest maximal local satisfy f deal gram entry clearly obtain immediate gram sparsity assertion condition valid gram simple frame consider necessarily vice minimal eigenvalue choice without modification useful wavelet drive corollary choice remain valid replace deterministic inequality replace event n j triangle hoeffding sum independent view lemma leave f f f apply inequality easily proof theorem analogue j j theorem q side replace modification rest identical cauchy schwarz j yield xy proof sum jx f ef j xy conclude exponential inequality independent j l n f r x r x nr represent mixture density identification convenient normalize write form q unknown clarity exposition simplify weight cf clarity exposition upper recall true paper general sparsity dimension strong mind dimension model valid equal require correct selection typically mixture density specifically use estimate mixture substantially regard deal identify close let component investigate need ensure quantify mixture quantify follow I restrict nonnegative inspection result section indeed introduce burden section replace suppose belong probability lemma f kx j k hoeffding recall f consider event index elsewhere pt recall position reasoning sum apply rl lk collect bound conclude density mean q identify density square euclidean large eq apply density grid approximation unknown target identify correctly index constant compactly support wavelet basis compactly assume orthonormal n nm nj k converge follow probability constant allow attain minimax logarithmic simultaneously c various old appropriately basis reference condition inequality find probability expression rate logarithmic factor satisfy construction logarithmic pointed logarithmic sometimes asymptotic constant cite therein class density uniformly fix transformation drawback rate class densitie q wavelet assumption depend minimax obtain minimization study differentiable adopt descent instead descent technique start step choose order find direction optimum observation norm optimum derivative th coordinate become iterative coordinate minimization direction minimization procedure apply quantity detail drive choose candidate validation describe principle construction avoid preliminary give parameter tuning proceed queue general queue queue back queue queue method experimentally discussion times accuracy dimension whole partition candidate determine index criterion jj ji pl final estimator take need slightly approximation density good practical validation goal validate bic bic array numerical context beyond scope research simulation investigate ability tuning parameter choose ii identify component conduct density random gaussian choice true weight consider bar begin evaluate accuracy estimate investigate relative mixture panel panel consider three size vary increase significantly increase large size investigate ability mixture figure percentage time mixture find versus consider correct accordance recall indeed simulation need identification l finally percentage small result requirement small happen another present sufficient see error crucial suggest percentage time correctly decrease function dimensional left center least density choose obtain candidate density circle exactly finite component natural exactly approximate isotropic many application reflect sufficient constitute crucial step analysis offer approximation application demand constraint determine weight display right figure successfully approximate number minimizer minimizer nf kx rv k v k minimizer nf kx f f lr nf kx k lr coordinate fact therefore event prove assertion component minimizer minimizer recall distinct point minima constant jx derivative mx jx mx continuity j lemma newton institute
delay error sf denote time sf case analysis bias ci range mse avg ci ci avg ci ci range mse avg sf pair test sign test sign test sf sf sf sf sf sf sf minimum apply version sf sf contrary scale noise short delay day significance ci sf statistic therefore accurate method sf measurement hence practice wiener conclusion outperform method bad stress optical novel automatic delay successful drive result promise approach involve one way procedure see fitness technique may fitness costly moreover approach presence uncertainty test huge next scale monitoring project dedicate http www http edu produce multiply available effort currently one would automate cope gap develop estimate represent delay context distant mix within evolutionary artificial detailed delay day readily optical monitoring involve regression statistical evolutionary algorithm delay delay arrival path observer time optical distant light nearby fact come get pass massive galaxy observer receive various direction phenomenon massive object source sequence path delay depend direct method universe often dark scenario underlie pattern time intensity gets delay corrupt observational sample possibly gap availability weather system currently long period delay claim multiply discover claim generate currently delay common employ optical multiply observation inherent modification largely optical well novel evolutionary delay fitness mse novel procedure decomposition integer direction evolutionary introduce form novel principle automatic propose drive iv study evolutionary optimisation devoted type come optimisation series instrumental fail etc sample availability influence collect way assess dispersion spectra b refer delay generate employ artificial wiener outline justify also significant method observation observe several describe outline conclusion future optical monitoring usa measure filter table sample day bc observe image b measure filter measurement deviation std bar weather scheduling monitoring monitoring peak light curve day correspond c time error day day day delay real attempt generate synthetic performance method kind large wiener delay offset optical et five set gap l h without gap bar correspond ds five bar set function variance represent sample periodic gap monitor eight series delay delay shift day shift model noise represent true day fig noise shift bar previous introduce approach paper detail derivation repeat detail section come monitor either multiplication offset image optical option time model zero underlie light curve whereas eq delay image generalise regression superposition kernel width imply width delay light curve image specific goodness square involved eqs variable may system time involve inversion decomposition svd tell may artificial optical singular pattern find fall change estimate fall range furthermore fit inside outside none aim fall review c axis versus singular good evaluate trial range noise gap increment concern pattern true actual optical unitary increment table formulation optimisation optimisation discrete variable force validation apart deal consume manner section estimate delay landscape unitary increment optical band explain landscape set hill search also local show combination shift follow three ii width value fitness loss follow avoid minima refer square validation might apply artificial genetic mutation population generation well accord fitness block include validation compute mse artificial global evolution evaluate generation link individual correspond use employ represent population fitness population sub population index individual population individually mutation link fitness repeat procedure mutation double mutation mutation population individual unless evolutionary good refer hereafter unless mixed type kind population fitness real artificial dispersion spectra width observational large spectra involve nearby correlation real observational synthetic wiener observational optical outline evolve integer fitness ten run table good solution individual accord column generation day approach es gray neighbourhood mean parent select produce evolutionary es choose es fitness function fitness superior benchmark problem world medical variable allowing reach fitness fitness variable es mse c run day fall pattern crucial estimation omit yield estimate regardless es iteration evaluation fitness tend iteration across every correspond fitness es artificial since fitness similar objective small measure day day table however day q report predict therefore delay increment ratio true value light curve besides validate range highlight delay quantity delay mean estimator delay estimate square delay absolute eq ci l r ci ci range also hypothesis estimate group underlie gap student zero significant show significance level value dot group highlight test significance nonparametric ht k table within ci group illustrate ci e c detail locate delay result group previous give grouping table observational explore various noise group level regardless well highlight bold table ht mse l mse performance pair delay pair bar represent delay
control subsequently identify approach outline note nan properly cat score centroid pool mean reduce cat cat extensive group comparison ranking refer cat centroid group gene respective score table discriminant function form pool centroid reduce definition weight splitting product page however selection simplify interpretation selection level center predictor interpretation involve feature typically substantial nan overall variable group cat feature greatly grouping contain cat lose usefulness lead instead exclude discriminant prediction rule fast inverse root stein estimator detail normalize nan summary employ fdr software apply cube transformation normalize shrinkage selection cat reference compare feature et investigate set cancer patient analyze summarize correspond plot gene fdr plot generalization box median prediction error balanced fold repetition control local small gene nan gene need include fdr cutoff gene figure note recommend feature differentially prediction balanced repetition split feature ranking estimate avoid select feature lda hc fdr based hc hc statistic maximum hc top feature hc limitation fdr shrinkage rule hc hc obtain fitting mixture nan cat employ remarkably hc imply framework hc choose large fdr hc threshold buffer dimensional discriminant analysis prediction contain stein predictor rank employ stein shrinkage estimator well perspective shrinkage improve cat rank among predictor cat lda dimensional score predictor note induced score differ procedure interesting propose efficient term high variable fdr lead inferior dimensional difficult discriminant analysis procedure hence shrinkage lda cat score computationally shrinkage accuracy typically intensive large lda cat rely stein shrink shrink empirical toward estimate propose shrinkage link present detail gene correlation soft selection bayes false discovery thus provide correction error modify lda stein shrinkage pool matrix employ greedy algorithm optimal criterion regularize ridge expense problem find unique discriminant selection public later critical comment helpful grateful von ki ki signal problem lda first pool centroid predictor give adjusted cat thresholde cat control third stein analytically overall effective shrinkage procedure implement package profile genomic study specific challenge see large prediction sophisticated proposal forest conceptually effective prediction applicable recognize essential specifically datum particular care need take covariance yet highly dimension zero employ discriminant classification high dimensional advantage straightforward selection relevant group multiclass mean centroid commonly software regularize multiclass gene procedure see essential ignore include imaging correlation spatial dependency pathway suggestion include approach version offer automatic correlation lack elegant feature optima paper framework high analysis base employ stein shrinkage train analytic fashion expensive resample second correlation score cat lda equation presence correlation third thresholding detail discriminant subsequently genomic conclude lda start multivariate centroid weight application bayes k lda evaluating choose maximize variable way prediction pool mean eq centroid group observation discriminant center interpret ratio completely equivalent careful simplify eq mahalanobis transform predictor make variance remarkable discriminant test much variable contribute group lda function centroid rely different stein shrinkage rule ridge estimator variance proportion frequency stein shrink analytically precise advantage stein rule large stein lda stein shrinkage compete approach vector lda adjust cat scale version pool adjust minus
graph remove associated uv uv argument follow kullback distinct st uv leibler st uv uv exactly uv uv st uv use base bound st cliques separation conclude uv claim apply require uv least construction conclude prove third ensemble second bind binary stay claim ensemble markov contain previously valid family large note certainly remove set z distinct uv pair distribution uv uv uv uv x uv expand recall st st x st establishes claim kullback divergence st uv st uv uv x uv x symmetry term decomposition x decomposition expectation st uv uv x x uv uv inequality apply bind shorthand denominator uv combine b correctness claim theorem maximum likelihood ml graph model describe provide deviation performance remainder term edge structural collection rescale likelihood give set uniquely choose attain conservative failure consequently decoder vanish deviation previously notation apply chernoff thereby obtain associate exploit deviation claim derive low divergence divergence graph make recall theoretic terminology match distinct matching pair comment great auxiliary elementary hoeffding vertex st sample hoeffding yield st st edge vertex u distribution unnormalized obtain sum fix ising define distribution manner lemma hence inequality yield analogous exploit bind quantity quantity involve bound obtain tv tv definition neighborhood uv combine bind theoretic prove four main necessary demonstrate succeed succeed characterization constant remain minor gap open immediate summarize gap binary appealing computational theoretically binary technique construct variant deviation apply would interesting extension direction acknowledgement nsf grant recall divergence uv definition uv via proof contradiction contradiction unnormalize little shorthand notation xx st appendix characterize change configuration agree definition observe original assumption sum yield root denote achieve quadratic formula since use uv equation u contradiction root contradict remain exploit lemma appendix distinct sx st ss set definition namely notation proceed proof contrary stay fix note add together yield equality equation note ix ij c inequalities st st thereby complete cm proposition corollary example ex ex em p decoder succeed family similarly exhibit fail succeed nsf dms preliminary result international ccc j department berkeley berkeley structure social associated graph independence recover domain attract search vertex reduce problem hand theoretic constraint base relaxation analyze selection penalize form particular increase address problem allow vertex sample line consistency method variant pc graphical practically appeal interest graphical selection identically sample size index triplet goal address triplet correct conversely triplet although applicable issue analysis paper field physics phenomenon model analysis ise recover approximate class perspective view observation channel parametric accordingly develop relate kullback leibler divergence statistical capacity practically way computationally theoretic optimal constant reveal date motivate indeed type sufficient edge vertex model theorem indirect precise consequence devoted proof whereas devoted condition discussion reader throughout standard constant notation mean background field formulation degree denote edge variable vertex specify graph ise form factor nature tx little calculation conditional x physics physical phenomenon modeling network represent vote negative edge impose collection structural depend property obviously mask three ccccc field parameter arbitrarily separate ise model identifiable nonetheless finite identical distinguish require exponentially markov field parameterize low maximum neighborhood weight minimum satisfy analogous belong class markov observe definition precisely consider refer loss risk probability incorrect scaling sample specifically size maximum decoder variant edge decoder know decoder graph know unknown variant low discussion follow sufficient begin state upper weight let comment regard interpretation consequence weight compactly observation scaling make although dependence refined increase grow case grow wish impose family method recover correct roughly speak fisher distribution recover graph factor theoretic analogous upper decoder comment consequence sequence compactly weight subtle number construct form subgraph require least correct subtle compare graph total edge development sufficient class true small degree necessary case adversarial choice completely condition match discuss bound size complementary condition discuss exist give edge weight decoder worst graph statement sample compare theoretic scale require guarantee careful grow degree like scale exponentially wish growth follow family succeed probability use theorem provide bound within condition et guarantee restrictive incoherence sufficient case satisfy decoder unknown weight exist decoder condition condition imply exponentially parameter stay control discussion case scale minimum scale know decoder succeed estimator half section introduce sufficient state definition model quantify kullback leibler special ise kullback leibler kullback leibler two divergence symmetric natural secondly via note symmetric calculation divergence family st give straightforward leibl denote divergence summarize bound b turning ensures claim group vertex discard property component edge permutation use permutation vertex permutation
transition system paper one lr decaying law std annealing temperature complete dynamic way compare result organize description brief subsequently conclusion state ise hamiltonian spin site assume summation perform length take periodic lr hamiltonian lr interaction I order apply briefly discuss update metropolis carry choose possible dependence temperature critical evaluate fluctuation l spin autocorrelation critical uncorrelated ground state average start analyse expect short start uncorrelated al static exponent account large critical setting become q hold short length lattice law initial follow law critical exponent depend exponent equation hand randomly generate configuration initial exponent avoid correlation determination ct r std measurement started fully order one validity mean std temperature validity g ground configuration pointing anneal q scale law valid take logarithmic reduce evaluate critical get exponent slightly furthermore worth std free size effect finite truncate short regime investigate worth know influence critical time size find search equation bar assess close deviation fit number configuration detail finite size carry several see range distinguish source cause finite often obtain interaction physical depend effective temperature cause range simulation system interaction spin hamiltonian periodic condition carry relaxation effective law behaviour short figure sum overlap suitable temperature meanwhile analogy behind range scaling eq aid get value fit report obtain calculation transfer method already size enough critical within time statement complete show second fit power list exponent significantly principle expect depend short ise nevertheless discrepancy attribute ising perform adjacent effective critical derivative temperature obtain observable exhibit behaviour table exponent correspond one equilibrium solid fit aid correspond effective indicate htbp worth evaluate source insufficient statistic interval power estimation former observable fit measurement bar account account major error report bar value estimate std std exhibit temperature simulation use obtain evolution suitable power behaviour function figure bar estimate way htbp c contrast measurement perform measure size get exponent std value obtain std calculation asymptotic expansion yield relationship excellent std estimation ht average indicate configuration indicate configuration autocorrelation increase exponent figure fluctuation calculation function consequently bar time close obtain agreement estimation average indicate obtain exponent scale spin range panel hold result bar collapse form show space excellent agreement self result different cr tx rr panel configuration discuss extensive lr ise interaction decay regime affect effective temperature law contrast study range finite analysis order temperature agreement std estimation exponent measurement agreement two may monte carlo dynamics present report dynamic relevant critical long range evaluation dynamic critical work
algorithm design condition rip sparse good application requirement super overcomplete fine thus highly correlate relaxation inconsistent see necessity objective practically parsimonious model representation stable parsimonious account datum never encourage penalty convex norm maintain penalty statistical setup group group desire family penalty allow regularization scad addition penalty family application briefly summarize important absolutely setup soft thresholded solve coordinate view recently linear approximate least penalty guarantee may another popular dc solve nonconvex represent difference similarly neither apply penalty group address grouping concern assumption group penalty group penalize algorithm design attain trick penalty include scad predictor group less impose penalty contain conclusion apply mild organize introduce thresholde rigorous present concrete discuss high study ridge section propose selective super resolution reconstruction example methodology leave setup go assume observation fy il fy canonical link fisher group r want keep predictor whole super size singleton reduce associated function nonconvex zero great exist nuisance feature directly use build parsimonious nonconvex class thresholding rule solve somewhat interestingly convenient tackle viewpoint tool define rigorously real value odd unbounded shrinkage version monotonically increase introduce thresholding define group satisfy canonical avoid influence ambiguity thresholded correspond assumption rarely application easy norm sparsity refer turn solve problem dimensional k kt corresponding limit theorem threshold cover essentially penalty practical indicate matter predictor group arbitrarily simple guarantee appropriately glm glm rule glm classification except multiplication predictor identical approximate original penalize logistic guarantee hand experience small computation concrete bind seem implementation give example estimation figure illustration attain discrete penalty group glm scale comparison predictor net elastic net q solve penalty justify group predictor solution attain scad shrinkage mcp scad focus minimum pg g numerically give property function odd equal offer coefficient fy normal introduce outli intercept intercept vanish center involve inversion improve dimensional computation incorporate accelerate asynchronous recently update penalty convex original reduce dramatically must take account run correlate avoid greedy preliminary b perform iterative feature step threshold nonzero similar long reasonably maintain set safe much fast quantile screening independence base correlation apply penalize estimation penalty lasso scad penalty multi similar calibrate restrict predictor weight construct ml behave apply neither introduce hard penalty thresholding grid search look good validation seek performance understand ideal situation allow parameter setup norm penalization lie group prominent predictor predictor vary generate well tune combination measure performance simulate sde n fy run successful joint probability probability simulation much serious dimension intercept transform suffer power bp resolve cosine atom ignore resolution high similar high corrupted frequency performance additional effective error report goodness fit frequency joint bp hard ridge regression compute mm tune c large large ideal experiment validation observation tune hard penalty spectrum reconstruction ridge validation bic show minimum necessary start good regularization maximum see solve time acceptable tradeoff resolution cancer real cell patient normal probe label iterative quantile ridge small ridge penalize tune aic bic correction test near centroid penalize refer tune get optimistic cross outer evaluation classifier hard ridge behave give parsimonious htp c error mean median median aic identify parameter tune plot replication give select gene frequently appear visit triple bootstrapping annotation show associate solve arbitrarily group sparsity require group treatment condition
q measure tight boundedness w notion apply system irrespective boundedness trade uniformly boundedness b critical behavior critical mark transition boundedness operate infimum eqn obtain unstable operator computable bound critical relate probability unstable invertible clear upper proposition system operating boundedness unstable boundedness necessary operate whereas need unstable important boundedness operate strictly next main paper operate unique initial sequence convergence stochastically theorem system converge invariant irrespective operate distribution operating may guarantee invariant discuss implication system boundedness unstable note existence uniqueness boundedness verify operate invariant case proposition boundedness invariant measure denote closure dirac concentrated eqn natural dense unbounded study example show positive exhibit next devoted rely dynamical establishe complete proof prove subsection sided space dynamical st mapping xt xt iterate guarantee iterate non negative sided transformation purely see later iterate sense suitably algebra define sample sided sequence projection random family path side equip shift assumption dynamical iterate eqn follow transformation jointly virtue construction pair initial distribution iterate construct indeed iterate map construction probability investigate distributional distributional carry paper sequel generic one generic empty real banach solid banach order hold sequel space preserve sublinear eq strongly sublinear solution eqn almost equilibrium transformation preserve eq eqn iterate object technical convenience asymptotic analyzing lead understand asymptotic sequence possess u boundedness sequel boundedness back eq compact back family compact topological property conditionally result sublinear preserve banach cone strongly sublinear order preserve ergodic one equilibrium follow space preserve sublinear establish order eqn consider eqn fix preserve point obtain eqn definition topological clearly tt sx sp continuity permit sp tt tf f sp xx q eqn reverse closure eqn establish inclusion inductive unstable hold follow preserve property eqn eqn thus since interval detail structure scalar characterize support study scalar show dense interval highly subset self explain proposition prove interpret reflect proper scale restriction restriction alternatively restriction placing show dense interval yy yy ni start every use eqn q show inclusion reverse scale rigorous self explain describe nature large iterate system countable explain contain thorough studying pattern consecutive proposition visualization component separate one look see top version consecutive study ac assumption plot cumulative eigenvalue approach dirac entire concentrated cdf vary assertion ergodic measure attractive transition process example consequence moment probability obtain complementary say moment condition instance process long belong consequence moment need costly simulation generate empirically generation suffice positive iterate possess support open set empty measurable existence uniqueness situation possible unbounded functional sequence important operate invoke mean present study kalman lose analog channel sensor novel analysis steady filter critical steady arrival steady show characteristic latter combine ergodicity provide numerically steady amenable address general provide theoretical evolution paper control channel interaction via random proposition consequence also thus verify map eqn property weak every verify eqn linearity chebyshev part obvious ii trivial unconditional reach suboptimal estimate point ii unstable since invertible substitute eqn eq tx large eigenvalue zero indeed imply inequality possibly loose purpose follow mm mm theorem axiom problem summary htb paper equation arise arrival sublinear boundedness limit preserve strongly sublinear asymptotic random matrix converge exhibit boundedness arrival weak convergence operate arrival rate possess moment weak distribution closure countable general characterization sure ergodicity distribution non self name first great kalman filter linear kalman powerful steady consequence steady implementation time complicate naturally identify arise study consider sufficient filter conditionally finite instant need identify recursively recently wang interest system concern control namely sensor purpose area network characteristic channel additional source analog channel drop limitation limit quantization address fundamental common quantization kalman suffer delay sensor arrival observation model bernoulli process receive kalman optimal differently asymptotic depend critical arrival time provide close special characterize critical relationship spectral widely adopt extended author although present chain observation establish stability I boundedness information grow characterize behavior distribution asymptotic bernoulli boundedness see subsection provide question existence invariant weak irrespective consider boundedness operate arrival boundedness operating arrival lead whereas critical boundedness instability mean ensure finite preserving explain later limit preserve strongly sublinear invariant distribution take transform compute bernoulli process numerically sound assume infinity asymptotically first weak theory iterate e invertible overlap satisfied satisfie contraction existence invariant use unique stability show operating stability enables characterize countable functional algebraic general dense highly self finally explicit identification ergodicity enable easily moment context complete analytic result example address follow characterize invariant arrival organized subsection preliminary present formulation establish proof present scalar subsection conclude euclidean natural subset indicator otherwise partially order banach banach space field banach space cone induce namely solid non order cone monotone various ensure banach compatible ordering induce supremum focus banach equip norm close solid interior matrix partial induced notation denote denote semidefinite theoretic metric
fig interaction initial static interaction radius apart scale avoid boundary robot use infeasible apply user use possible every hence box user interestingly robot minimize perform hamming rest centre marginally bad hamming e conclusion might sensitive location stroke graph show summarize centre robot user strategy feasible since centre nearly ham always centre call robot misclassifie hamming result hamming variation weight boundary differently investigate hamming evaluation giving ham particular fair quality user want form fact setting inspection image visually error visually incorrectly visually discriminate large due runtime robot colour want small confirm considerably fix interaction different free predefine minimize loo estimator sensible important thing big suffer fit train addition value rough perform define perform static interactive static start reweighte error system iterate optimisation dynamically learn strength course interaction summary illustrate plot differently lead study also interactive static look close learn system dynamic change introduction participant infeasible robot user run user good smoothing comparison segmentation system start contain number max exposition reference adjust segmentation image simplicity involve operation respect vector balance trade symmetric degree fit segmentation give margin rescale read rewrite quadratic exponentially segmentation fit cut kn graph cut connectivity approximately train empirically interaction binary one partial fed unlabeled stay optimisation choose maximal reweighte pixel formulation interaction prove less convenient compatible cutting difficult basically geometrically possible remove closely regard selection kernel special kind selection explore discrete try variable short guarantee conceptually optimisation user removal k k user human relax strategy k concatenation properly proxy forward optimisation unary gmm potential ise contrast robot begin look little final include unary potential also potential unary unary behavior without either learn optimisation evolution user interactive system demonstrate approach show grid parameter segmentation approach infeasible max framework show solve optimisation art segmentation include enforce unary potential handle future tackle challenge enable interactive learning grid parameter learn interactive segmentation microsoft microsoft microsoft cb microsoft com successful vision parameter traditionally interactive manner interaction propose bring user loop human art interactive segmentation popular propose computer hard automatic show challenging input condition past sure laboratory decade research primarily interactive system help interactive interactive popular area interest interactive lead vision graphic interface interactive crucial automatic counterpart come surprisingly little devoted learn interactive system try bridge gap interactive segmentation efficacy interactive interactive segmentation aim separate rest treat assign label foreground interaction come pixel mark user help belong question interactive segmentation system interactive system imagine generate possible change segmentation evaluate one efficacy system interactive go margin automate summarize study evaluate interactive thorough segmentation algorithm margin user discuss give segmentation explain na I structure interaction conclusion problem development world choice set test measure traditional vision learn misclassified interactive choice hard presence behave differently prefer interaction learn interactive intuitive want achieve quickly literature interaction mostly choose researcher encode assignment give evaluation consider user current interaction refer static user prefer user good compete system prefer segmentation cut ground result performance scheme static tool newly propose involve group use use correct segmentation example full advanced job static evaluation participant significance participant car fast normal car experience car evaluate independently participant infeasible try segmentation thousand million crowdsourcing attract lot community primarily scheme collect datum crowdsource excellent platform interactive vision could imagine user system require suffer study light new fix human current ground output code middle label alternatively interaction interactive market similarity agent reinforcement one learn task summarize effort price user model yes yes yes crowd yes slow bit user yes yes infeasible static far low paper ground truth code consider kind input blue ignore database image ground comparison keep ratio create computed ground truth segmentation indicate foreground user interactive system gmm make short ratio get system cut undirected node correspond segmentation color background foreground unary learn color foreground computed channel pixel concept practice term ise
recent rely ultimately density algorithm ergodic use argument mild additional bound away imply evolve matrix position covariance x k nx dp density stand characteristic determine portion proposal template appear decay verify setting correspond apply instead original essentially fit differently covariance symmetric metropolis increment proposal deal constant improper target distribution recursion template weight result suppose adaptive walk hold speed original setting value scale smooth proposal behave almost grow reach decay slowly however covariance figure therefore use significance successful burn may ensure space borel algebra lebesgue n k n n unbounded follow walk vector almost surely dimensional uniformly adaptive small enough n fx converge probably target compact support extension however handle convention compute independent mean determined recursion analysis first show increase estimate imply also substitute algebraic equivalent strictly suppose conversely consequently geometrically imply contradiction strictly ultimately additional sequence growth respectively assumption index lemma another q show sequence clear z hold z g similarly decrease sufficiently small sequence suppose decrease imply establish obtain contraction consequently triangle converge constant latter satisfy lemma first us eq imply combine point sufficiently n hold k x section define stand behaviour recursion express simplify first follow symmetric degenerate identically distribute real measurable nr n n invariance notice particularly unity behave quantify behaviour random non degenerate kolmogorov set thus n conclude technical mention require zero assume degenerate variable measurable adaptation estimate j j nz sufficiently hold whenever use write choose sufficiently I kn sufficiently right first estimate conclude adaptation adaptive walk satisfy process surely proof estimate apply martingale convention assumption martingale converge limit satisfie due implie converge hold simple result similarly process adaptive small walk increment walk sx sx sx n select n appendix b j construct apply fix also jt ib surely sufficiently assumption therefore whenever hand consequently establish surely converge martingale difference trivial infinitely indice one stay index infinite inequality sufficiently must whenever infinitely index infinitely exist whenever thereby hold trivially conclude proof strong number run ingredient check simultaneous ergodicity next suppose everywhere non increase x condition measurable vx sx lc adaptation target satisfy event eq us truncation construct truncate start truncation function n coincide law select law number deal component proposal mix fix adapt result ergodicity result key speaking regardless adaptation compactly measure absolutely measure constant measurable dx ds da interested ball fulfil eigenvalue relatively use adaptation stand sphere independent auxiliary n use variable measurable p nx w write lemma hereafter denote define n martingale ny w write w w bn b cover suppose measurable satisfy w w almost absolutely measure du measurable bound stay tail contour eq process weight moreover adaptation satisfy sufficient fact show cone ax bx r hold fulfil density fairly easy verify practice hold exclude unbounded contour theorem corollary hold proposal require use record ergodicity large differentiable stay super tail contour use mixture stand weight neighbourhood origin adaptation surely imply compact set hold vx dd put auxiliary truncated process ensure law large letting
education internal web page site external user visit period consideration record list page website visit internal part context attribute external user site target incidence pair visit site analogously page target pair visit preprocessing receive visit visit total visit page site describe term site tackle dimensionality case reduce size datum follow observation give information interest target site group site page merge domain bank page page page take month reduction size large context size give concept interesting group employ index concept construct stability extent context factor show much stable motivation index formal obviously stability concept indicate particular extent index close several several stop site part concept stability lattice large threshold look correlate set group fig part lattice site www external internet visit user www month concept many concept correspond political read fig present order several survey stability web site site paper group decrease amount arise employ group formal proceeding terminology attribute
choose translate mild shape bind question establish principled either dependent dependency mix process specific ec fp give let binomial upon classification km obviously jensen directly markov suffice see q p f version block measurable bayes compound need differentiable concentration respect second stand eq thank eq give q lemma step inequality q let increase function take nonnegative end mp corollary replace thank equation correspond appropriate effective identically independently make follow concentration random range fractional make distribute inequality rise generalize bind applie allow function draw z bound corollary theorem range identically allow retrieve tight given establish pac bayesian mixing follow stationary denote order bound mix suffice follow concentration countable z suppose function bayes mix mix sample corollary align ne iid particularly classifier even guide classifier situation data dependency iid state framework generalization call dependency encodes dependencie datum graph fractional fractional sufficient result train way note bayes stationary mixing rank much progress bound bound bayes bound bayes refine appeal advance concern bound classifier directly bayes show bayes risk also variety outcome explain able learn classifier real e bipartite first constitute iid principled establish bound setting establish convexity exploit contribution cope dependency subset bayes bound non distribute mention dealing variable establish concentration fractional cover derive inequality extend function provide bound complexity subset dependent beyond inequality call bayes illustrate ranking ranking function performance area exhibit interesting strong skew imbalance negative besides rest vc dimension rank algorithmic stability qualitative somewhat bayes uniform bound already observation carry bound consider result deal use new process recently bayes price dependency graph straightforward shall calculation lead sake provide bind process together tool allow derive iid new fractional provide iid mixing give rise generalization stationary mix possible product space bind iid probability predict draw iid mh throughout generalization bound value appendix kullback distribution success kullback posterior absolutely prior straightforward tt apply argument deterministic margin focus generalize dependency identically independently result rely build accord dependency state bound role dependency graph dependency edge independent connected cover fractional fractional proper fractional cover cover cover colored color way problem fractional number graph come precisely graph order clique split independent dependency fractional bound draw assume dependency distribution equal proper exact fractional cover follow stand q n defer technique proper cover proper fractional reason cover weighted classifier distribution ne q z classifier n z j third iid prior enter scheme cover factorize look factorize minimize iid accord respect readily choose cover least mc side q z km comment say iid iid amount bayes valid prior posterior depend get give side decrease reader check indeed suffice nf vs induced subgraph draw amount subgraph induce situation empirical error I see fractional must color fractional subgraph remove twice big differ carry circle inner sep fill gray origin size gray might comment comment well subgraph compute subgraph bind subgraph let least random h simply subgraphs probability form small induce node fractional graph subgraph size replace keep still iid fractional obtain bipartite rank negative fractional dependency graph figure totally see plug give fact distribution look rank rule hx yy allow pair high rule predict conversely make measure natural question rank difficulty sum distribution henceforth clearly simply need upper bind xx bind rank claim bind fractional theory u rewrite permutation iid random index cover decomposition appear permutation permutation take fractional cover definition knowledge ranking argument analysis easily fractional even require practical situation value limited pair rank minimizer able appropriately moment bayes course base scoring state similarly important usual tight clique clique make equal odd bipartite respect e one interested sequel whenever consideration opposite sign situation straightforward iid q estimate fraction pair incorrectly incorrectly express scoring notation minus name consequence bayes provide evaluate score negative early share dependent depend figure read resp entail positive building bayes step part necessary deal depend sequence check cf say early identically exist determine p z q hz q propose call event directly unconditional p fractional part finish fractional clique define construct addition check cover mean minimal proper fractional cover note skew express therefore score classifier acting carry assume gaussian posterior parameterize bayes follow straightforward bind parametrization done choose minimization selection iid empirical turn proper fractional cover moment empirical error convexity linearity z q z side decomposition dependency tackle order datum another graph depict easy clique besides proper implicit reasonable easy bayes sake theorem datum mix stationarity stationary identically denote algebra integer coefficient process say note detail mix process dependence difference datum rademacher bounds mix integer p z block even odd drop block variable within stationarity block dependency variable connection block theorem qp
pls algorithm svd extract factor pls selection stream pls pls assume discrete streaming datum introduce challenge offer update bridge pls simplify bridge pls latent inversion loading reduce division pls stream efficient svd weight unable least square least current datum essentially adaptive individual covariance eq exponentially contribution past current bridge pls construct sum pls point eigenvector perform iteration sim column orthogonal power column eigenvector schmidt follow modify iterative bridge pls algorithm estimate simplify pls latent vector since effectively loading sparse regression detail full initialize c ty tc tm tu tw ts initial mutually orthogonal also initialize choose place penalization make penalization design line govern ar autoregressive coefficient factor means give time stream hide create stream bridge pls accurately select line algorithm lead convergence take place point create response coefficient center variable inactive ease visualization interpretation define univariate pls response multivariate result sparse pls simulated coefficient line correctly factor pls pls factor area pls quickly line suggest period pls algorithm correct furthermore pls use stream univariate strongly regression select center analogously second mean assign period third pick automatically rapid switch keep switch gain show pls able accurately mostly component neither inactive suggest change fast control report solid percentage second carlo data pls little percentage decrease quickly select time result place adapt change change quickly factor cause become see large monte carlo case pls select capital return track algorithm portfolio tracking propose lasso latent tracking suggest component reduction latent regularize application publish index motivate present enhanced perform index tracking target asset return plus pls algorithm figure enhance track static stock static portfolio period poor financial suggest portfolio test pls p benchmark stock stock assess track randomly sure fair portfolio weight vary square make ability portfolio composition really advantageous tracking see pls consistently portfolio achieve exactly target portfolio index result suggest importance stock detect certainly advantageous drive advantageous market heavily portfolio entire period stock whereas pick drop period suggest adapt track every transaction cost change place return trade tracking algorithm streaming aware show able factor stationary pls able accurately pls specify pls per towards development tune several method literature automatically update adapt achieve evaluate use minimize aic incorporate pls selecting retain reconstruct ensures retain lower retain retain new likewise retain high important incremental pls pls method number pls initially projection recursively keep function component pls pls increase add pls per choose change achieve covariance maximize explain threshold quantify however remain pls stream fashion technique literature concern neural network plan relate application trading application mining efficient multivariate problem stream recursively latent select factor singular line fashion simulation artificial dynamic observe stream time report track financial data stream consist line asset allocation benchmark streaming arise web monitor asset management contexts quantity collected analyze often incoming stream stream refer may multivariate common fundamental regression penalize coefficient problem arise firstly select truly important component correlate arise ill pose special care difficulty take approach relationship change quite development adaptive method dependency datum generate development methodology unify vary stream numerous analysis amongst track recursive yield regression propose lar know penalize solve algorithm domain finally aim propose incremental sparse square pls track stream pls linear assume existence latent property deal format review pls emphasis development pls propose pls allow bridge thresholding singular although second contribution incremental pls incremental pls pls real time simultaneous iteration method bridge pls ensure full pls component eigenvalue possibility pls ridge notice covariance covariance regular pls ridge far notice small prevent becoming follow pls eq eigenvectors loading loading final pls pls direction may extract computing relate necessity computation pls single svd review bridge pls perform pls find pls computation pls svd efficient bridge pls svd recently device loading briefly achieve pca low approximation
replace simulation smc sensitivity smc bayesian molecular dataset smc apply type deterministic stochastic computationally efficient allow discrimination candidate selection give sensitivity review abc application dynamical formal bayesian model aim computationally intractable costly exploit computational modern simulation vector abc generic distribution tolerance tolerance algorithm approximation replace summary dynamical without statistic simple simulate accept reject disadvantage sampler acceptance rate introduce proceed simulate go go abc correlate couple acceptance may get low probability period avoid abc monte carlo smc method smc sequential sis sample represent sample gradually evolve target principle get area proceed population indicator indicator else perturbation return candidate return particle go normalize go particle denote perturbation kernel walk smc abc selection employ concept selection find let model define prior uniform simplify bayes factor weak strong compare hypothesis testing firstly need secondly equally traditional insufficient explanatory reject translate direct evidence true additional specific particle specific estimation particle proceed ms initialize ms indicator ms candidate calculate q every normalize output happen factor directly use however explanatory extent inference typically great particle thereby ensure estimate poorly posterior belong wish independently abc smc selection parameter perturb particle ode solver code scientific library implement ode solver code delay part deterministic dataset typically world molecular highlight abc smc dynamical demonstrate describe interaction specie add simulated sum conventional noisy low reach tolerance accordingly rejection sampler approach order particle approximately need infer distribution apply outline comparable abc rejection gaussian distribution next apply distribution summarize range median obtain substantial need reach result need time abc abc mcmc ccccc step inner histogram result try describe equation population fix resource inference master exact simulate point smc population laboratory setting replicate also experimental three run purpose system dynamic one repetition rate generation prior parameter magnitude large accept result summarize gene gene loop protein gene loop gene six parameter display cycle behaviour available point function inter infer histogram four infer posterior large credible interval recover change change change hence sensitivity analysis intermediate construct intermediate visualize distribution project intermediate accepted pca scaling pc explain variance parameter sensitivity problem increasingly difficult visualize behaviour secondly determine varied individual pca output abc go sensitivity particle rank parameter eigenvector describe orthogonal pc define ellipsoid accepted particle eigenvalue specify pc variance proportion sensitive pca account posterior summarize pc first interest lie pc extend across region distribution pc large pc last mainly combination change two third composition pc outcome agree obtain range smc stochastic transform correspondingly process condition correspond include protein choose particle average summarize get hard infer infer deterministic figure analyze compare deterministic stochastic parameter sensitivity link system insensitive impossible vary stochastic scenario variation one case infer stochastic disease sir describe spread infect recover r individual simple delay get infected ability individual individual death irrespective infection rate recovery realistic adding gets infected time incorporate delay include infection state infect equation rate latent allow individual become rate obviously basic four impossible inspection alone sophisticated need select experimental group gaussian deviation similarity intermediate inference simultaneously bayes calculate basic time population bayes factor weak compare true rt da isolate approximately observe break arrival use day stochastic abc infect recover individual extra epidemic previous expect period particle use prior perturbation kernel uniform interesting population probable local model pass selection describe marginally draw infer wish posterior estimate compare whole general epidemic broadly establish abc select plausible realistic suggest smc tool reliable dynamical generality abc unlike deterministic time obtain merely intuitive meaning frequentist advantage abc shape cost information different parameter simulation sensitivity infer narrow one infer population localize change population resemble distribution correspond straightforward number care selection domain care prevent reject population e observe bayes change tolerance perturbation variance care need stress way posterior range test goodness successive intermediate crucial signature proposal impose limit procedure different abc systems particle develop sequential carlo abc compete approach apply chemical science reaction sensitivity abc smc also critical identify quickly relatively insensitive financial division molecular college sciences trust helpful discussion david manuscript especially grateful helpful manuscript abc sis improve rejection impossible sample weight dirac delta proposal reach series importance sis sis draw set stop sis intermediate take sis base define abc smc sis proposal intermediate distribution q dataset indicator tolerance particle allow dd bx equal proposal perturb accepted kernel walk calculate define approximation particle intermediate weight calculate b dd bx bb tb tb abc write indicator weight particle simulate particle go weight go simulate base smc sis disadvantage smc sampler optimal good backward suggest simplify prior result approximation weight affect credible toy example distribution three population abc poorly figure approximated population particle run schedule show variance green weighted particle blue line result smc variance abc rejection use population
case matrix overcome follow rectangular define shift k entry loop construction confirm lemma add make sign gaussian notation enough model rescale hermitian use loop parameter input k f kf channel decision rule matrix approximate complete adjust accomplished linear channel appear extensively storage characterize noise ts definite often channel medium bank match physical assume ambient noise error mmse detection suboptimal mechanism linear equation scheme maximum introduce algorithm mmse detector work apply sequence implementation typically convergence spread since indeed axis plot color depict well next loading loading axis show residual option loading first force dominant radius dd radius iteration tradeoff amount show outer loop weight dominant iteration loop increase average loop iteration per tend number inner global h present iterative propagation essentially involve add diagonal become walk add feedback cause loading believe numerous detection give art linear iterative fail convergence compute mmse direction importantly overall double trade beyond converge make outer slowly balance compete choosing loop develop lastly class beyond loading computer science support foundation thm thm assumption condition thm remark inference converge vector compute converge compute constrain linear construction link engineering discuss force convergence originally fail consequence able pass gaussian link fundamental science processing rank social machine recently exist instance densitie condition graph numerous novel convergence definite application discuss practical original section introduction describe extend system positive information potential solve optimization rescale variable zero diagonal correspond pattern reflect markov distribution edge radius condition imply definite key idea loading walk perturb convergent solve walk equation propagation obtain walk normalize diagonal walk dominant hence satisfying condition explore solve iterative effect iterative follow objective basically version newton minimize size typically ensure hessian optimize
inactive fixing fuse inactive set instead path choose nature algorithm define set determine fuse becoming fuse everything go fuse set specify define base active correspond correspond inactive fuse correct also fuse finite short optimality locally w f get active inactive coefficient inactive remain subgradient unconstraine get determine setup find set maximal flow let associate define assume source fuse lasso k kl define fused status time define activate activation activation active putting length remain variable split inactive maximal come source capacity two iterate step nothing inactive active fuse result proposition assume set give denote calculate use rule fuse valid accord necessary outline extension inactive set particularly fuse additional q function regular subgradient start see kk np ii ff ii time activate ij h f ia gain fuse update fuse hand active composition include inactive treat calculation necessary possible old fused computationally lot apart moment delay program lasso later date subgradient equation value eq show node see come capacity flow come flow go every maximum imply eq proposition first continuous actual proof j unless condition f jt kl lk fail restriction existence satisfy construction graph node merge immediately otherwise occur loss generality therefore continuity fuse fuse necessary flow come use grouping optimal solution space linear flow flow derivative force break thing remain sure set piecewise easy continuity linearity inactive active l g active thus inactive definition stanford university supplement article path fuse give complexity alternative article increase fuse become proof result fuse lasso predictor article fuse order however calculate derivative identify fused small require operation use save fuse stay derivative per decrease therefore entire possibility store retrieve store whole coefficient fuse linearly impossible efficiently memory requirement store node fuse coefficient parent store end example retrieve start next interpolation store node look vector value therefore get complexity speed article want result assume flow fuse sufficient edge node node essentially situation f rsc rl ks inner flow whereas maximum flow go residual also graph node know definition merge penalty turn fuse immediately split lead fuse split want expand result fused predictor want difference get calculate whole fix additional lar see inactive fuse restrict development jump
dimension h compose proportional f f corrupt additive snr observation burn number iteration chain precisely output example depict middle converge reconstruction iteration sufficient ensure ghz course challenge mcmc gibbs blue reconstruction source result active detect amplitude implicitly indicate absence paragraph indicator regard active estimate consider histogram fig component line mmse active mmse estimating depict range paragraph sparsity identify follow atom mix patch source illustration dictionary depict complete blind separation problem orthogonal prior source hyperparameter prior unsupervise collapse sampler study generate noise approximate generate sample estimation simulation conduct svd favor investigate unsupervise jump extension currently investigation domain would thank university suggestion relate grateful fr address identify formulate mixed unknown orthogonal formulate model process prior inference gibbs sampler joint matrix synthetic illustrate representation chain carlo year representation consist identify decomposition signal among main alternative pose recently compressive sense reconstruct projection reconstruction sparse formulate penalize numerical unfortunately np problem reconstruction solution well pursuit mp orthogonal omp norm exploit property devote constrain generally estimation mp design course completeness redundancy atom recover activity signal machine community procedure problem assume recently formulate strategy negativity orthogonality demonstrate gene datum analysis blind sparsity signal image recently molecular source unknown source assume therefore source mixture zero problem among hyperparameter stein instability noise manner couple monte strategy consist level assign informative hyperparameter parameter bayesian paper hyperspectral image standard noticed standard process lead demonstrate deconvolution result partially collapse sampler van strategy efficiently dictionary ensure source orthogonality main impose recover property preliminary address source base mix knowledge level precisely unobserve noisy bayesian description idea decompose free orthogonal column conduct assign couple von fisher strategy assign mix generate sample develop formulate blind source separation problem derive quantity algorithm accord posterior illustrate application natural processing future consider unobserved stand notation q accord center unknown source consequently index sparse propose variance unknown mix section likelihood observation independent vector stand full numerous work gamma choose noise main simplify column introduction regard mix gamma eq generating highlight require orthogonal accord column firstly nan sample sphere set unit set unit sampling detailed achieve strategy matrix accord distribution element therefore prior recommend coupling classical locate deconvolution frequency sparse approximation molecular take dirac center hyperparameter degree assume strategy adopt would source sparsity level times explain introduce subset active jeffreys prior hyperparameter distribution hyperparameter variance eq assume hyperparameter hyperparameter q define graphical acyclic joint nuisance lead infer matrix generate bayesian estimator mcmc method allow one generate collection asymptotically reader consist highlight detailed lead mix weak alternative deconvolution recently study rely presence collapse van drawback inherent cite sampler consist replace conditional summarize step subsection h von hyperparameter independence vector source consequently achieve successively f derivation needs explore mainly gibbs sampler local introduce overcome introduce active conditionally indicator rewrite active govern hyperparameter sampling successively notice efficiently initially introduce adapt account orthogonality rely cholesky decomposition inversion avoid calculate compute intensive determinant sampler u
gps include mat ern entirely herein generally describe classification case generative variable via generative draw trial require class without generality prior sample hybrid monte carlo hierarchical almost suggest sampling prefer hasting mh draw carefully consider variable notable drawback popular typically stationary stationarity categorical operation necessary modeling computationally mention natural suggest sampling model partition categorical separate input gp predictor treat cart fashion cart cart branch predictor split assign branch assign recursive arbitrary partition cart standard real value output cart theoretic cross may cart partitioning hereafter model otherwise identical model proceed open detail computing default begin start queue root recursively queue accord child define explanatory uniform distinct informative leave sensible requirement take automatically overlap region partition region define extension intercept wishart hyperparameter aim tree set hyperparameter treat monte carlo gps gibbs exception integration gp contrast difficulty change leaf simple mh instead reversible jump markov transition model parameter leaf node proposal mh ignore leaf maintain collection allow jacobian leaf gp set jacobian factor ratio unity leave describe proposal make move call move acceptance largely exception move solve predictive distribution conditional normally eq mean r boundary aggregate near partition boundary grow swap integrate tree gps posterior uncertainty gp flexibility datum gp fit almost indistinguishable cope quite ordinal categorical split make marginally input code binary treat handle real input perhaps standard wu wu input scale apart parameterized scale parameter output tend order function binary even variable cause unique consequently tree handle input present indicator ignore come leave exclude ever boolean indicator upon lm zero beyond produce speak boolean gp parsimonious accounting relationship value include model mix separable presence indicator necessarily setting allow whether particular categorical input visit tree split binary indicator technique require tree relationship remain value chain propose operation benefit context possible drawback allow tree relationship predictor gp relate real input separately region mechanism directly handle input predictor imagine correlation input part set partitioning categorical ability learn real value account correlation different categorical predictor argue possibility influence take back speed interpretability benefit section benefit classification single softmax case imagine possibility recall partition fit model consist output gps variable region entire might tree could tree call note latter tree imagine perfect copy partition differently partition region across expect may model imagine datum set occur tree capture split partition greater likely split immediately relevant class gps enough move directly proposal grow change grow take advantage structure essentially hierarchical classification formulation code modification package henceforth tree region set hyperparameter model identical prefer mean incorporate freedom less generative incorporate softmax summarize diagram extent sequentially region three modify extension latent along would along newly newly propose f grow move play direct indeed define try determine variable grow benchmark input real value set handle categorical tree inference identification categorical last would difficult consider modification covariate categorical normal noise q original reverse irrespective response linear irrelevant response note encoding split indicator note previously context cart categorical predictor split categorical encoding generalize arbitrary predictor split predictor range achievable cm median training random compare summary root square rmse sample behavior method I gp ignore categorical account rmse test clearly indicator include cart appropriate input constant leave indeed fit unfortunately improper indicator partition proposal leave respectively parsimonious smoothly vary cart improvement real balance cost categorical translate bayesian cart proper drawback boolean come leave boolean partition lm full implement scheme consider linear map value input lower rmse treat value fitting leave direct linear model boolean process smoothly indicator represent categorical boolean partition predictor matrix cart lm partition indicator indicator leave indicator become function well parsimonious monte carlo value input analysis trivial consider generative assign great distribute evenly depict order expect cart one partition another perfectly class set divide likewise capture mechanism represent htp latent follow vary smoothly enter determine class correspond latent mark distinction amongst latent class likewise negative region remain modulus reduce almost smooth limiting exhibit divide divide remain latent constant similarly latent variable divide separate latent variable care separate difference arise separate two would parsimonious class tree place true capable nonetheless much likely partition partitioning fundamentally predictive interpretation tree model meaning finally see run benefit take second roughly difficult trivial candidate amongst easy model hessian synthetic partition illustrate amount sd add convert assign base indicate direction exponential concavity change quickly similarly require fit hessian right multiple maximum posteriori encounter separate regression able upon relative grid misclassification location rate relative range well represent softmax partitioning suffice even take hour execute whereas fold thing context strength compare gp mind shall categorical uci database group class aspect six input distinct category set predictor treat continuous form attribute neural network implement library test library hide ten training evaluate via validation select parameter setting keep partition rp select base default avg rp fold slight misclassification standard misclassification fold slightly cpu use hour per fold hour appropriate worth map tree principal split binary therefore predict success extract devise single figure show chain log figure tree illustrate many benefit separately success process latent softmax powerful simple drawback implicit assumption stationarity handling categorical evaluation due repeat decomposition divide gp suggest partition input advantage divide handle categorical gps leaf drawback behave inefficient involve basically integrate direct preferred however categorical value appropriate categorical gp real one dimension result interpretable highly acknowledgment sciences tb research support thank anonymous comment statistics california berkeley uk laboratory uk interpretable regression gaussian process gps application output utilize classification formulate gp sampling expand helpful develop handling categorical collection synthetic inference produces interpret keyword gp function ease typical prior stationarity
shall underlie currently requirement adopt practical experimental verification algorithm usually observe metric consider reasonable verification main contribution establish assess experimental global instability drop instability multi verification reinforcement agent reinforcement algorithm learn receive theoretical prove action arguably simple system domain consequence verification agent researcher stability evolution global surprising since optimize global allocation eventually usually reasonable verification argue end global challenge widely establish framework assess global instability system drop alternative local entropy experimentally propose traditional metric instability multi system organize throughout review use experimental initial global lead conclusion algorithm use instability conclude simplify task allocation assign agent service minimize scenario figure execute option forward reach although experience task well without know appear world yet capture analyze delay effect action appear message time delay consequence delay message link unit agent queue simulator make agent state feedback get agent agent make good unobserved agent gradient ascent neighbor practically load learn incoming request neighbor receive later indicate load ascent adjust successfully domain load balance concern main stochastic expect benchmark game equation intuition similarity neither analysis nevertheless mention elsewhere update ensure policy update adjacent arrive x sub center time unit execute task unit arrival service illustrate htbp algorithm look plot safe actually spurious ideally would evolution learn parameter include small aggregated summarize agent neighbor case ascent policy learn learner effective policy count neighbor deviation agent agent policy
penalty example long hence fashion guarantee detail argument statistical paper concrete norm regularization give decomposition thresholding create grid initialize initialize compute assign norm shrink retain validation shrinkage choose solution svd iv iv meet positivity negative sign singular vector correspond minima warm reasonable process version appear speed demand calculate matrix assumption explicit p rewrite factorization svd computational heavily upon matrix indirect iterative calculate lot algebra sophisticated requirement effective count compute absolute maximum error algorithm rank bi manifold approximation svd use suitable output algorithm uv normal iid completion meaningful standardized error nuclear panel exclude snr percentage indicate unique correspondence nuclear plot rank nuclear true minimizing summarize perform simulation size iteration convergence time version acknowledgement support dms national health miss entry miss entry true noise middle get type b post process poorly predict apart type get however snr time lemma relaxation matrix regularizer simple nuclear norm algorithm replace element svd warm efficiently solution measure represent entry netflix competition primary datum recommender correspond rating movie potential rate movie movie yet rate lie much represent recommender structure suggest movie small score movie affinity recently theoretically assumption unobserved recover within observed contaminate entry constraint make singular svd modification make convex nuclear norm norm explore programming efficiently modern software algorithm second method propose large sized criterion equivalence rank problem al singular scalable comment suggest prohibitive problem force minimization lie path grid value warm inspire svd different step prohibitive exploit structure require large iterative partial adopt notation onto observe complementary form basic ingredient svd refer proof follow look minimize lagrangian control norm minimizer solution warm z initialize grid iterate assign repeatedly replace guess figure curve see test smooth objective fix satisfie nuclear norm sake brevity expand singular product arbitrary lemma obtain svd minimize tuple pass subsequence k w pz pass
formula quadratic give minimization amount unconstraine aside interesting note thing low correspond norm one big little say nevertheless might useful construction finite library construct potential among recognition tracking physical extract image want library dimensional contour transformation projective object might collection estimator x selection sequel recognize actually priori solution respect basis profile obtain example know class discretization vector class bound lower typically one proceed construct r fm linear suggest devise select among collection estimator rank inferior identifiable identifiability statistical possible identifiability hypothesis body know priori noiseless uniquely noiseless formula shall impose type detail please p recover subspace though generally little norm write obey derive expression recovery least norm say state start derive expression one one equality power equality derive say estimator quadratic equivalently eq ideally minimize hence refer oracle serve benchmark assessing add challenging indeed inferior quite derive satisfactory drive identifiability one write new formula hence formula remainder possible useful procedure propose estimator penalize criterion q arbitrary penalty minimize risk shall penalize help estimator assume fix eigen positive eq defer consider inverse problem argue give might help motivation like emphasize begin identifiability might might enhance risk performance linear estimator select risk introduce denote large positive eigen value matrix large value number number function select simple fact quadratic risk amount consider result issue suggest reasonable bind penalize accordingly assume remainder obeys identifiability choose penalty put penalize formula performance compare predictive oracle course hence constant positive take satisfie shall choice assume positive eq obtain upper put find equality universal say model model start obtain collection performance bind situation q sense measure write start intuition subsequent assume moment discrete integer l sense since among difference I achieve low term risk take make actual instead hope expression reasonable way achieve quantity diffusion gaussian bandwidth expression proper bring enough constant formula penalize risk reasonable decrease versa nevertheless exist intuitive fact take one p hence amount energy concentrate interval regard role prevent upper risk model find instance trade constant reasonable remain parameter optimal speaking suggest reasonably knowledge examine penalize firstly far unless happen increase decrease dominant hence great adequate default reasonable penalty constitute course turn enforce exclude optimize penalty study attempt involve penalty application dedicated reader optimize whenever estimation concern inferior identifiability address selection put forward noiseless x real satisfy unless belong respect identifiable recover estimation highly hypothesis call identifiability know write equivalently happen practical engineering field one like one respective basis zero mild happen realistic numerous practical application basis capable extract vector construct enforce two cx hold one generally cx might priori know solution identity sense mention illustrative assume multiply smoothing kernel increase kernel correspond cx one meet identifiability eq actually idea semi definite symmetric formula formula might ellipsoid know semi know construct follow decay law appropriately like add investigate identifiability scheme scope please penalty directly concern concern statistical ill linear regularity signal mail request stand constant take filter signal summarize condition matrix check ill conditioning ratio find stand third discrete follow stand three stand differential achieve degree recover triplet filter repeat experience instance penalize risk experiment need first reason treat since large upper penalize strong oracle recover perfectly error recover almost perfectly oracle latter relative error order assessment please direct inversion practical noisy green drive smoothed signal visualization green drive red estimate shall framework address estimation penalization work non mainly show good difficult application penalty constant simulation well propose note paper present plan mainly process constitute attempt satisfactory inverse problem course challenge address along near final answer could though make lemma nevertheless penalty small penalty however bind behavior penalty mainly open question solve near spirit finding identifiability really rapid question would yes assumption go direction recover interest among infinite impose nevertheless exclude generic belong devise identifiability comparable result noise modify simultaneous believe future answer question currently amount try question practical point sometimes fact find fix subtract quantity latter equality find put derive find need control put q eigen value value put number derive simultaneously large derive sharp let put integration concentration real square vector eigen symmetric prove derive stand respective eigen eigen stand two follow inequality hold true lemma consider rewrite let compute put derive technical detail k finally technical proof derive f proceed inequality inequality inequality derive r concentration variable acknowledgement author france paris university discussion video scientific stay en remark grant propose address estimation statistical depend inverse problem drive mild important due e sharp paper estimator propose approach find code name recover interest trade fidelity term regularity selection achieve summary notation shall sequel problem lie hilbert endowed euclidean follow shall euclidean matrix identity mean diagonal element write one solution case countable framework section collection quadratic risk extend quadratic mahalanobis risk rank interest linearly identifiable know priori adopt via penalization estimator stand upon propose remainder great inequality q constant collection generally collection choice sharp universal stand correlate dimensional estimate resp vector adopt via penalization regard deterministic matrix general ill great therefore seek paper allow number viewpoint refine point consider example go shall latter coefficient orthonormal otherwise property indeed recover library thorough discussion estimation certainly continue researcher either work appear indeed early model concentration line consideration collection oracle inequalities propose extension aforementione work mild model selection quadratic risk extend simultaneous probably present subtle author subspace orthonormal basis estimator construct error admit constitute contribution author might suffer result especially family basis play relaxed estimator construct construction encounter computer highlight proceed unconstraine matrix generally compete h fidelity illustrative quantity regularization operator pass band filter enforce smoothness recovered solution generally regularization however want mahalanobi optimally application collection putting obtain finite linear estimator low find selection representation family orthonormal complexity g wavelet correspond proceed mp solution put collection estimator among
value ridge fold multidimensional computed roc construct varying significance range sparse illustrate correspond situation underlie box reconstruction base causality ridge regression white noise approach discovery subsequent testing var show outperform research problem give split subproblem solver regime estimation flow fmri accuracy support european discussion technology institute technology institute interaction autoregressive vanish belong respective parsimonious causality assume discovery consist absence causal lag zero achieved regularize recently outperform recover causality use causality widely accept assumption effect cause series cause knowledge improve series spurious causal relation common factor detailed form elegant handle causal vector var graph ol fitting var model coefficient enforce estimation lasso account absence ar belong pair furthermore detail give extensive briefly relate autoregressive causal strategy autoregressive tp gaussian noise cause causality vector autoregressive causal coefficient interaction conduct matrix set z p var latter transform partial correlation copy coefficient time copy estimate correlation propose case control high sparsity alternative white ar square straightforward way ridge however fully dependency graph ridge improper estimate bootstrappe explicitly estimation k z kt furthermore x hypothesis multivariate normal sparse causal discovery ridge suffer assumption direct sparse via desirable would causal lag suggest overcome coefficient vector correspond incorporate belief influence restrict estimation positive modeling lead eeg problem vector first univariate coefficient hence overfitte avoid scenario eqs solve order carry efficiently coefficient solver lead considerable reduction memory combination solver conduct experiment recover compare lasso ridge causality four case reconstruction cross multivariate time generate var accord choose distribution var randomly causal randomly test look e eigenvalue accept var datum noise strength across
decrease perspective open bring low radius achievable single datum rich geometric streaming provably store empirical classification theoretically many ball improve upper upper identical alternatively analyze stream set specific place around appear streaming mean probability toward stream could ellipsoid ellipsoid upon inclusion unnecessary ellipsoid axis scale variation allow ellipsoid expand addition approach along line gaussian space weight incoming maintain uncertainty ellipsoid covariance exist streaming however conservative stream pass svm streaming prove despite conservative experimentally competitive technique learn much believe careful classification alternative cs edu streaming svm leverage connection streaming impose allow formulation minimum ball idea learn require multiple streaming weight way use stochastic perform computation storage even efficiently accuracie comparable solver online give extension common traffic fashion streaming apply disk store propose allow pass data storage severe stream streaming successfully employ domain geometry adapt streaming setting formulation streaming naturally motivate efficient technique approximate stream context encourage high latter geometry admit stream adapt provide outline experiment synthetic machine kernel provably generalization formulate quadratic typically solver require requirement svms decomposition method divide small scaling approach rigorous guarantee due success stochastic stochastic base quite requirement recent considerably single pass easy train approach suffice require several converge reasonable define w n ny difference form nonlinear map far assume property fix isotropic dot kernel criterion label unsupervise one second account misclassification vector except entry instance metric product margin induce ball extensively geometry inner programming become dimensionality cardinality svm turn radius solution extract small subset originally core easy equation correspond center dimensional space kernel pass mutually never store vector compare perceptron radius store exposition kernel kernel store weight vector initialize kx kx kx weight update lagrange dy input example slack rw dy nx ht example slack support vector initialize nx x sm l sm use storage obtain conservative classification well end therefore weight ball choose ball ball merge ball end ball merge variant zero amount store radius income ball store buffer whenever buffer closed rather take size size whenever buffer take number practice less synthetic art evaluation pass accuracy compare online iterative batch pass accuracy solver table accuracy pass dataset suggest pass use perform single solver htbp cc cc dim c
snps discount option namely multiple snps even computation computationally c snps uniformly use snps show reasonable nominal one amongst true reduce snps considerably bring close improve nominal false maintain misclassification still sensitivity overlap distribution sample apply apply column snps another correct purely collect set nan see return bar quantile blue situation individual publicly available nan threshold incorrectly half line illustrate choose sample resemble choice sect similar european descent converse well match illustrated incorrectly classify majority outside population sample nan empirically sample poorly match separate threshold false negative power specificity practice remain unknown population produce false greatly error less nominal positive nominal false rate sample accurately importantly threshold specificity assess match specificity describe considerable quantifie post prior theorem posterior prior write plot specificity accurately need exceed post probability subject specificity assume sensitivity belief difficulty assess specificity practice great positive fact result specificity sensitivity versus nominal yield difficult match depend strongly underlying meet consider table severe resolve membership individual positive sect frequently effect family child parent another child snp test child value child large yield center amongst reflect see resolve member group become big homogeneous move close note know true positive know positive distinguish positive similarity ideal situation snps independent ideal hold carry sect population control sect simulation sample snps identical rate fraction varied show identity find half time identity note fractional identity percent simulating varie percentage fig rate exceed yield misclassification eqs together true member eqs member possible variation cause turn green show distribution close close genetic inferential significant responsible misclassification example far characterize genetic quantifie allele frequency classifier positive classification sensitivity quite classifying specificity distribution eqs utility test circumstance dominate see different amount nan assumption positive sect high false result nan I amount snps sect meet despite sample adjust deviation nan obtain sect additionally conclusion neither positive assumption meet classified despite assumption sect amongst even threshold adjust produce classification considerably expect nominal nominal report table practice result difficulty yield probability probability high specificity test sect correctly sample sect finding privacy become topic positively identify mixed pool unknown scenario evidence need population obtain sample composition discount value sensitivity assumption sensitivity difficulty threshold rate party identify relative concern one access subject careful need member well show classified value sample narrow sensitivity need probability truly status eqs discover despite limitation eqs quite sample likely neither also little worth investigate column fig lie sample fig nan control control shift snps statistically close control statistically control disease individual conversely find subset snps snps use snps comparison location successful positively presence dna finite genetic useful appendix nan underlie writing two trial eq per person invoke binomial value notation consequence write analogously absolute value consider separately treat expand polynomial simplify notation rewrite perform expand well indirect indirect become dependence uniform sample nan hypothesis sample range varied circle plot uniform obtain closely national health md fellowship national cancer institute national md individuals population control control control case child child specificity specificity nominal snps sensitivity specificity specificity specificity nominal rate apply metric quantify genetic suggest infer allele limitation characterize strength limitation reveal individual member several circumstance crucial yet explore method improve additionally specificity may ideal circumstance despite identify disease strength limitation situation implication finding individual amount dna highly complex mixture snp author may material minor allele idea genetic sample subtle systematic introduce dna comprise binomial question contribute variation contain intuition statistic measure distance underlying identically reference detect versus hypothesis major minor q assume central large modern normally denote average snps exploit individual neither article propose composition e appropriate reference positive individual intuitively author construct near rate presence specific genomic specific summary many publicly available study method concern participant false assess rather nominal quantile expand investigate robustness inherent I hence difference independent central g via enough assumption begin attempt address analytically empirically explore conclude finding well individual dna reveal sensitive small nan frequently finding false individual likely exactly although finding may reliably detect valuable find extended genetic explore analytically empirically attempt publicly available source retrieve genomic breast sample describe comprise breast cancer case comparable match participant participant american european sample array snps share snps additionally european individual project individual member comprise parent snps frequency control three pair group even reasonably resemble allele difference minor allele sometimes allele frequency magnitude lead non size small representative sect derivation fail separate case analogous center situation separation leave control misclassifie achieve use nominal accuracy control classification sect three frequently nominal simulated identical genetic method individual possess eqs quantify furthermore positive sect sensitivity appropriate pool comprise individual case snps use positive sample fairly correctly classify sample misclassifie group positive nominal sample test participant classify subset yield misclassifie rate amongst false positive rate amongst amongst specificity reason
test negligible therefore bp justify dependence mention situation causal nonlinear demonstrate noise verify lead et tu ab ir tu wu valid normalize w provide basically argument worth asymptotic case box test powerful box test power lag whereas detect lag goodness problem long memory commonly long long true value lie test goodness attract lot construct roughly distribution usually trivial power alternative type asymptotic parameter chen al smoothing alternative neighborhood model disadvantage kind asymptotic martingale chen utilize practically far chen et justify process conditionally martingale exclude interesting invertible observation follow similar nk present literature even pseudo likelihood memory series reformulate equivalence series chen test prove case allow process iid partially solve conjecture even limitation assume know turn modification main reason series population slowly rate become adjustment point chen statistic mean invariant mean adjustment possible extend directly beyond scope short affect hold adjust residual seem natural central might nan transformation use et al care unknown fourth innovation happen error feasible base show large sample white residual still aspect although chen fit series error find skewed adopt chen domain along correction device yet certainly university j goodness series dependence journal series autoregressive integrate series autoregressive integrate move average journal american j day em chen goodness chen induce methodology variance ratio em j bootstrap test frequency distribution fit statistic r test martingale martingale journal em daily flow science estimation plan recent model j diagnostic uncorrelated journal american introduction bilinear time w j model economic relationship york university series york drive specification p theory em bootstrap stationary journal em serial lee serial unknown bootstrappe box robust l test parametric mean alternative long dependence statistical statistic wu stationary j journal american li li autoregressive average li integrate autoregressive conditional journal american association forecasting fractional journal zero theory r ng testing time series journal new york p journal american statistical association extend theory em wu asymptotic theory wu national science wu principle wu dependent stochastic process em wu limit functions journal wu em wu approximations uncorrelated ergodic martingale f x f kk wu accord wu cr contribution achieve approximate double well study wu wu wu among directly major set martingale bound martingale presence separate along lemma respectively ii wu basically wu omit eq part covariance martingale u u u u u k k j u c j l jt jt jt tu j tu u tu tu u tu u tu u u kt v j k j schwarz tu tu c j apply f f l l proof since nu nu pn nk nj nu nk nj suffice show cauchy schwarz nz jk nz jk side view hereafter term nz n j cm nj nk nj j nk imply nj j u u nk nj absolute wu n nj nk nj k write martingale difference nj nd om regard k nj k nj n n r nk nj om martingale nj r martingale difference constant nk n nk nk nj nk sequence difference martingale central suffice verify nk n nk nj cm nk nj nj write argument follow uniformly td j conclusion nj nj td pm proof j td pm view notational n nk nj nj n nk nj nj nj nj since dependent get cauchy absolutely wu derivation z r nh j h r u cm regard j table cc th establish assumption nk nj nj td r pm write nj nj j n n nj n nk nj nj nj nj r j r n om nj nj nj j l nj r dependent cn nk nj rd rd om pm appendix ny kt n nk nj nk nj j nj pm let follow show tu n absolute since mean get k kn kn e k k k ny p ng ng ng n part nj pn cauchy proof theorem need nj pm schwarz end nk nj nu nu nu nu n proceed u u k u om wu bound om n show k u u u k u u cf wu u k fourth term n derivation conclusion assumption n kn kn n j k j e l suffices k n e k j h h pn k u u k h h h j g p partition k h h h contribute shall partition nonzero j similarly involve restriction fix typical fix schwarz fashion similarly argument assumption n om kn absolutely sequence u exist let q noise statistic box lag truncation nan distribute assumption popularity conditional imply autocorrelation understand asymptotic nan box dependence lag diagnostic checking address go early finding series white lack serial correlation process variance covariance function respectively alternative statistical common checking systematic fit statistic domain time probably admit form eq call truncation empirical nu u reduce frequency rigorous contribution derive nan work lack stress statistic presence dependence show bp lead inference uncorrelated demonstrate test uncorrelated modification dependence statistic asymptotic shall seminal distance density lag normalize density nonnegative function bandwidth sample quadratic distance equivalently nk worth regard case iid establish nk j stand normal lee establish martingale general white establish replace residual uncorrelated distributional theoretical asymptotic fix negligible change contrast fourth appear negligible result restrictive
equation corollary kl get problem constant ignore margin linearity discriminant pac bayes bind generalize multi base wrong independently draw definition pac bayes easy reading copy pac bayes continuous margin threshold respectively independently e discriminant random draw define distribution f hoeffding achieve mh mf result fact two last hoeffding substitute mf expectation side cm exactly statement get f h h hold finish remove mf since pac bayes true possible cm maximum prediction lack probabilistic dual probabilistic model bayesian allow hide combine extend generalize markov style entropy discrimination discrimination paradigm know mn model similar mn simultaneously type plug combine mn pac generalization bind combine learn generative model structure maximum discriminative outperform compete structure extraction maximum entropy discrimination network margin infer modal g web intelligence scientific genome discriminative graphical structured dependency field network specialized graphical model explore learn remarkable success structure output flexible capture impose desirable bias reduce structured maximum margin machine concentrate output due dual depend optimality arguably desirable paradigm prediction context lack unable domain sparse irrelevant extensively likelihood estimation learn linear model svm univariate output turn extremely later guarantee estimation discard generalization discrimination simply formalism offer formal paradigm generative discriminative principle bayesian regularization structured spirit discriminative structured prediction mn maximum discrimination margin offer advantage mn learn primal label formalism extremely contribution concentrated define solve generalize arbitrary insight general solution reduce prior achieve effect laplace primal mn convex thorough compete include mn mn synthetic mostly superior comparable formalism general discrimination follow laplace offer sparsity laplace discuss web extraction language parse annotation decode map sentence parsing scene consist denote combinatorial structured interpretation object input entity structure must arrive satisfactory represent pair function without class optimization maximize input structure finite discriminant predictive possible f column structure likelihood formalism classical label discriminative boltzmann mn feasible discriminative principle elegant generative regularization find advance structure crf method utilize discriminant optimum frequentist furthermore concentrate directly map elegant probabilistic cause obvious easy derive sparse paper bayesian obtain infinitely propose formalism objective mn special achieve resemble knowledge base principle substantial principle yet explore important exposition approach margin solve follow cm slack adopt iy j equal one otherwise constraint explore label reflect specific pair wise duality cutting pass optimum directly employ mn learn max margin employ question average devise constraint spirit scheme optimum sequel regularize formalism offer simultaneous primal generalization traditional bayesian style use entropy principle additional predictive basic discrimination learn binary general ratio px bb find solve follow optimization slack keep optimization input feasible mn feasible subspace expect entropy discrimination principle optimum correspond relative entropy measure minimize kl entropy theoretic favor among feasible appropriate accommodate separable prediction usual kl optimize entropy close put everything formalism discrimination discrimination nf relative cm optimization function expectation problem nice calculus variation p brief insight underlie optimum markov lagrangian multiplier solve slack program lagrange function saddle dual dual defer appendix program lagrange kkt kkt condition show enjoy lagrangian multiplier correspond active equality enjoy mn due constraint learn duality conjugate slack easy c equal hold infinity training perfectly ignore multiplier problem c uncertainty slack expectation side successful rely estimator optimum special generality offer important mn admit primal enjoy generalization third incorporate generative predictive train partially analyze partially combine possibly latent discriminative structured choice parameter subsection standard mn somewhat surprising reduction offer totally predictive counterpart respectively underlie mn make claim explicit assume p posterior ip mn state primal assumption detail corollary slack shall exist mn applicable solve trend markov extension implement sparse graphical context margin gap merely due discuss mn adopt strategy primal regularize mn directly dual problem constraint non trivial due complicated solve mn preliminary expensive real another possible method interpretation resemble regularization effect parameter reveal mn normal weight standard regularization model around dimension standard infeasible mn dimension employ learn apply regularizer mn variational express e heavy tail encode around nice concave negative convex exploit estimation laplace mn obtain f k intermediate note laplace hierarchical mean admit hyper straightforwardly new multivariate vector p k multivariate substitute hierarchical normalization column due derivation avoid infinity convex conjugate arrive function depend logarithm problem laplace qp mn variational laplace cm solve develop corollary posterior primal subgradient cut optimize respect take set since univariate gaussian dimension representation get calculate expectation convexity optimum apply posterior distribution shrinkage laplacian irrelevant zero qp qp hyper pac provide theoretical average classic pac rule prove follow mild boundedness mh definition pac pac bind bound margin cm event true spirit bind extension structured margin detail present stochastic structured average predictor pac dependence pac classifiers posterior generally evaluation laplace mn mn regularize regularize newton method mn use correspond good mn mn open thorough beyond scope appear elsewhere experiment implement equivalent mn slow apply ball mn evaluate structure e synthetic pre field dimensional feature vector correlate sample sequence draw define crf gibbs synthetic randomly transition capture pairwise independently sampler generate iteration draw rest mn validation sample setting outperform mn encourage outperform un case derive note convex try optimize mn perform mn algorithm consistently performance reality correlate evaluate model linear generate contain input relevant partition group get feature mean method solve qp variational fold cv datum like part k search get state generate compete average correspond average mn large mn exhibit shrinkage plot generate assign label predict fact learn generate variance since variance two variance relevant group variance whereas figure sparse structure partition fold cv sample fold put web cutting method slow warm simplex example stop iteration hour finish cv thousand performance approximate project sub combining sub cut plane synthetic figure instance small variance consistently outperform mn regularize bit reasonable examine accuracy obvious fitting contrast robust unlike usually small ability lead instability regularization put weight regularize fitting suggest later mn stable log likelihood right question magnitude regularization regularization fix sensitive regularization mn mn stability instead thresholde zero max margin regularize primal dual less outlier last conduct problem regard web extraction web page record web page characteristic type structural tag well flat construct construct vision accordingly product image hierarchical long dependency level level item web attribute image price generate template page evaluate record record accuracy extract attribute record page testing record criteria comprehensive measure four price identify percent datum model especially mn mn regularization outperform regularize max outperform third mn enjoy primal especially number training mn suggest margin mn mn scope defer later motivate discrimination method propose average maximum essentially structure version mn structure svm lead powerful new paradigm structure prediction enjoy primal raise algorithmic inference propose example along relevance machine unlike optimize margin function logistic sigmoid although sparsity due justify unlike style learn margin margin also explore interpretation hyper advantage free hierarchical laplace encourage two replace weight explicitly add applied maximum discrimination straightforward fundamentally state interesting online laplace conference input formalism offer formal paradigm integrate discriminative principle bayesian technique vector machine entropy margin
covariance satisfy approximately briefly let good desirable inverse covariance concentrate find pair positive approach inverse covariance give make complete choose suitable subsection update perform instead vary amount incur ij discuss convergence nonlinear nlp nlp associate problem follow penalty establish penalty nlp assume exist cf nlp bound finite optimal value easily fact first know due together ready generate optimal update inequality maximization value observation first see proposition respectively exist saddle immediately desire definition diag derive give immediately far rewrite concave conclude convex solution smooth let similar solve simultaneously solution next nesterov method equivalently subsection adaptive project gradient solve equivalently spectral al smooth close integrate al classical namely respectively optimal view moreover close set suitable ease subsequent describe detail also h problem integer g k k k establish generate suppose interpretation observe observe q q follow nearly optimal expect nesterov study show u iteration nesterov method unique nesterov slowly propose adaptive nesterov subsection continuous general replace corresponding constant ease detail method assume give definition subsequently inactive ready inactive kx u u f sd g u sd kk end subsection convenience sparse adaptive project method nesterov solve instance describe al generate generate contain value uniform positive randomly q instance experiment purpose code matlab method addition initial terminate find ghz table iteration evaluation cpu rr rr rr cn rr nf cn c rr rr rr rr nf cn table able within time spectral generally outperform nesterov carry entry nonzero sample pattern covariance approximate solution observe capable recover independence know formulate norm penalize nesterov demonstrate able half within amount outperform online www namely completely straightforwardly lemma author research grant inverse conditional formulate method nesterov smooth instance able solve least constraint nearly reasonable amount inverse nesterov undirecte capable describe among zero zero year matrix notation norm estimation control trade nesterov approximation scheme coordinate lin propose suitably et al capable discover graphical nesterov substantially outperform exist addition maximum estimation conditional formulate pair method practice graphical partially know sparse inverse covariance partially completely covariance constrain penalize pair control sparsity worth observe ii view easy reformulate convex homogeneous self barrier suitably interior al bad iteration ip ip solve dense arithmetic ip prohibitive et convert large apply slight square method dual ill surprising slowly square often fail huge consist found project adaptive nesterov smooth paper rest introduce first generate conclude remark assume write cone matrix resp operator unless explicitly norm one dimension clear context absolute determinant
material balance pseudo piece square preference single single preference opponent master piece multiply rank factor expansion relate four complex preference opponent c relate preference isolate isolate non isolate open file isolated feature view limitation move game avoid early move normally program search depth check account evaluation carry carry game train random validation record train natural temporal learning position game black loss generality game white black actually play white mae feature divide position without white game classify player I identify playing play actually still valid game black perspective replace define opponent play cardinality cf white summation black perspective e white perspective note discriminate player game examine examine perspective obviously undesirable know white black algorithm game subsection first mae player differ much bias mae selection evaluation may useful choose train move difference position detect example nevertheless reflect structure point consideration opponent situation break move varied method note mae counting game despite moderate success replicate low value enough weight feature evidence difference difference mae mae converge increase adequate feature partly limitation next subsection introduction player player normally position abstraction player feature deep blue increase play blue rest subsequently lose well capture level playing versus algorithmic indicate range many readily solve aid program exist often involve rather solution plan readily aid computer technology put emphasis force planning possibility leave direction human knowledge discriminate player record play use engine yield success believe methodology present fundamental need far would capture concept classify player match since domain game try game go conclude profile agent record ac describe style record base individual temporal aid engine architecture encourage attempt learn discriminate try white playing discuss limitation possible present applicable domain keyword record player game player player affect position generally tackle computer discriminate player review difference machine occur difference actual outcome feasibility write master strength human world less technical description machine program demonstrate learn enable self effort chinese self consume game record strong player evaluation temporal minimax record game pattern move prediction record game principle interact record exist generally accomplish fast learn include meta play result game module play look player reformulate play black piece high human already program computer already human play mainly increase rely far classification suggest temporal cite attempt computer program scenario attempt learn evaluation function course consideration likely improve method could weak correspond prediction td decay factor force accelerate momentum describe section world learn weight guess encourage fundamental player point feature probably strong seek maintain piece player concept difficult formulate translate algorithmic conclude remark component playing program example component play minimax game play world go level employ advance promise improve concentrate essence player weight record useful discuss incorporate relate structure addition record know player novel subsection accelerate training momentum generality evaluation define game sum value n measure unit material balance evaluation move play play version program tuning convert apply sigmoid choose adjust assume initially white move white perspective position white move white move game record word result record follow small rate logistic set recommend literature experiment choose rate subsection update normalise preferred fix material balance logic position advantage material position factor measure material dominate interesting note rule td opponent move make move possibility player program choose self become program move make player
current define possibly randomize measurable function observation history play feedback full double definition abstract potentially represent infinite metric metric call algorithmic represent result metric oracle access suffice space classic notion guarantee whether admit whereas arm instance instance whereas instance need guarantee apart several study background bayesian formulation payoff maximize payoff expectation mdp play difficult formulation passive science include formulation offline mdp action formulation probabilistic payoff adversary minimize strategy set lie subset payoff question idea lipschitz mab adversarial version mab well naive stochastic invariant view problem lipschitz payoff estimate payoff distant arm whereas linear mab arm crucial mab mainly fix payoff advance make present joint prove algorithmic complementary infinite space result oracle space reduce low topological self particular use covered course consider payoff payoff arm independent expectation bandit reward collect algorithm round round notation denote expected regret analogously open radius ball contain finitely every limit cauchy dx I cauchy identify formally subspace cover complete vice let family intersection topology element throughout refer small topology open ball topology singleton topological empty least order statement equivalent axiom use concept extend beyond infinity require notion namely von definition necessary material found prove lipschitz tractable much exist distribution problem expert first existence perfect useful metric binary center parent correspond ball parent could necessarily center radius perfect construct pick root construct perfect point define payoff fix let large enough tree nod child one node complete leaf child belong ball probability measure function first sign choose sign function child independently associate payoff hold I generalize lipschitz problem metric implicitly payoff metric give triple apply technique exposition one idea bandit similar formulate formulate expert payoff indistinguishable learn consider lipschitz along borel measure feasible kk x tuple ensemble borel algebra mutually expert mab bandit payoff remark theorem small among node probability measure induce complete node three let follow exist event random many event happen metric lipschitz mab double feedback exposition expert version lipschitz expert metric topological function segment order denote rely strategy since value compact open set contain call maximal segment open complement therefore attain towards play small ball significantly large strategy well via follow lie within output ball subroutine input call receive cover consist point let collection ball call take round sufficiently problem optimal consider function large least return us notation run rt chernoff let clean ball show imply suffice rt sx kt cover claim consider lipschitz non use proceed doubly length length tractable exploration subroutine subroutine play end fix total reward phase share exist let duration part exploration return phase proceed phase round run exploration complete incur round infinite compact diameter payoff baseline payoff baseline r mab constant intuitively ability payoff ball formalize ball round induce event divergence technique detail proof claim x ir r via intuitive less space oracle need metric covering space point space point six say finite finite metric suppose topological order iii arbitrary initial contain true isolate initial segment exist xx reveal mab expert cover provide tractable every even consider metric cover n section consider subroutine oracle receive play rx winner winner exist else point complete sufficiently strategy subroutine notation algorithm rt chernoff bounds clean payoff optimal rank isolate indeed open set optimal pick enough contain indeed dominate claim strategy complete cover compact metric cover radius run armed center ball mab sense look tune tune correspond fairly natural metric ball sufficiently rational denominator radius cover true support contain close contain near move let space hamming cube distance pair probability radius unit move distance least obtain follow existence asymptotically binary hamming lemma easy bind imply cardinality point least element least subset cardinality pair point imply cover expert call mab problem use space finite parameterized run phase round play phase pick break tie complete term completeness explain regret lipschitz feedback let phase bound event guess total accumulate phase claim feedback problem function version guarantee via involved analysis space include upper bind involved expert expect payoff phase let set choice hold feedback guess proof use separately essentially union many efficient bind chernoff slack chernoff follow root empty internal level singleton child diameter definition cover tree cover break algorithm say clean consider chernoff lb c lb sufficiently turn determine slack interestingly right ignore incurred phase clean phase base phase diameter induction prove pick break fix covering clean root arbitrary refined tree metric feedback tractable tractable suitably idea naive see lipschitz theorem g uniform plug theorem characterization regret except lipschitz full many upper bind proceed efficient repeatedly pack nonempty exist contain disjoint positive covering let maximal center disjoint every ball every ball cover packing recursively construct set consist finitely open ball equal positive ball ball disjoint ball radius ball let ib ib ball mapping absolutely infinite one ensure one problem expert distribution payoff function sampling distribution lipschitz subset sampling sampling random independently analogy notion complete infinite nested ball specify expectation payoff finish lower I bt low q tractable incorporate via lipschitz sufficiently feedback point sample average break tie arbitrarily lipschitz expert log rather fix denote arbitrary ordinal depth contain existence suitable decomposition every equal infimum let depth maximal ordinal ordinal via union ball let closure report cover return arbitrary covering net successive construct net depth union successive net b phase round phase strategy call throughout phase good guess previous end feedback arbitrarily phase cover construct roughly construct net point let feedback oracle specify chernoff lipschitz expert whose depth use covering point roughly version regret phase clean high clean depth argument show sufficiently large depth phase clean good guess within optimum reason depth ta final section lipschitz metric mab expert metric copy abuse expert lipschitz property never select metric payoff mab mab algorithm conversely algorithm tractable lipschitz mab playing time draw report let instance mab payoff expectation algorithm behavior identical payoff apart query infinitely success theoretically impossible output receive call kl omit let ft tractable metric completion remark lemma bound algorithmic direction lemma directly type less elegant payoff denote tb tn pick belong value set relation expression equal denote random count account least algorithm finitely kl x choice payoff give play payoff define two imply non integer obtain q last time select strategy play playing role sake convenience equivalence implication arbitrary countable iii space circular perfect perfect subspace ball leaf path root nest ball empty intersection call distinct leave correspond distinct tree disjoint thus point ball ordinal cardinality point recursion isolate isolate nonempty empty exceed index ordering open construct order ii imply order contain order know open imply point separate arbitrary implication fact induce open initial open topological order property perfect element well initial segment open topology perfect infinite complete direction nsf air office scientific microsoft research fellowship foundation fellowship arm bandit mab classical arm priori payoff certain imply logarithmic lipschitz mab finite bandit problem generalization infinite regret lipschitz mab either bound perhaps coincide compact connect technique online notion perfect set lipschitz mab term exhibit give nearly regret space form metric characterize tailor expert show sense completion compact category descriptor algorithm abstract general keyword armed expert metric multi henceforth year exploration sequential k payoff mab commonly evaluate payoff play one strategy range experimental armed growth time decade begin seminal subsequent application mab set bind apply making assumption payoff bandit trivial mab payoff become subject intensive mab information consist upper situation maker access similarity similar payoff information model define imply payoff mab et feedback essentially space precede real interval regret dimensionality metric upper exponent depend regret tight infinite isometry factor picture regret metric exist work provide space finite metric metric study ask metric achieve metric issue concrete
definite fact define zero actually include would subsection transform distribution contribute reduction depicted request statistical decision estimator define square x poisson density parameter suppose estimator call tp ik poisson gamma obtain apply view server analysis software numerator time increase range weighting regard effect www real www traffic access www request minute except period server date date describe likelihood px x px px previously follow lk analytical maximum however sub plot show mle date server real www processed evaluate request model point mle request b table show interval horizontal axis solid solid histogram estimate request respectively dot propose stationary plot server server model server table request third expect table result server h server max limit accord mle become show cause maximum non stationarity www aic server aic stationary exception aic small server traffic server drastically decrease request day traffic time server traffic strong www traffic data viewpoint model request closely stationary special stationarity stationarity www traffic forecasting planning www extent strongly mle differ server square error large cause small figure square around table estimate server server suggest length maximum estimation sufficient request concern request server derive expected value reduce squared point effect would strong model superior thus advantage h c aic stationary c server stationary show forecasting www traffic steady term theory calculate server stationarity express vary constant pointed consider traffic stationary real www traffic real also prove www traffic forecasting strong validity classical stationary poisson value point interval mean traffic advantage support grant aid scientific research project remark research institute engineering university department traffic forecasting past calculation focus wide www traffic deal traffic time viewpoint forecast arithmetic calculation www traffic theoretical empirical wide web www traffic engineering environment problem stable operation look analysis software analog web www server count file page intuition case forecasting point researcher traffic suggest lot wide suitable traffic structure express forecast assumption wide statistical substituting suitable assumption future bayesian alternative bayesian new parameter forecast posterior forecasting problem bioinformatics traffic distribution poisson estimator variation inferential depict assume time cc px vary cc conditionally regard kind definition x vary variance distribution vary model vary degree include stationary another
colour sharp expect solid colour region point interested demonstrate smoothing parameter estimate make wish simple residual behave residual propose I image surface gaussian response space simulate left note estimate top show vertex connect spaced covariate sake choose global bandwidth exhibit signal practically flat bottom choose identify location show easily pose constrain I subject j tucker require existence lagrange multiplier j otherwise negativity active acyclic system j lagrange multiplier system minimize event subsection kf reach l requirement region give region sense remove happen suppose swap f f km j c f f j change clearly since c c f l f equality merge therefore active without increase never twice terminate york regularization taylor pp application run string numerically multiscale multidimensional smoothing van locally regression spline e nonlinear removal r lasso extend take vertex familiar article discuss penalize total result challenge new algorithm include graphical discrete variation penalize regression fully automatic smoothing statistical contain sort graphical mapping variation focus penalize thought consider explanatory often sort graphical rise covariate graph term assume realization value complexity regression response grey pixel connect pixel image display image graph regression neighbourhood noiseless shown discuss penalize datum rough observation value vertex measure difference estimate measurement therefore penalize f plus vertex different vertex edge usual denote treat order pair direct completely vertex make split edge motivating nonparametric continuous natural first adjacent second observation convenient shorthand variation extended dimension analysis think pixel pixel neighbourhood pixel suggest picture variation plus van de smoothing allow smoothing procedure minimization complexity norm shrinkage lasso graph apply regularization every spaced create square et algorithm idea active method give minimize convex function minimum global minimum minima convex strictly unique share consist acyclic active optimization entire contain vertex denote subset hold example join together vertex hold since acyclic remove edge associate region j edge acyclic appendix string describe van consider algorithm value define f proceed reduce change active occur change f k reach target region might meet would decrease change merge join share acyclic minimizer region meet vertex event break therefore k edge join ensure acyclic occur must whether remove region take swap order edge possible event occur remove l f complexity analysis simplicity image vertex generic mainly combination stop edge active set add change active many repeat decrease monotonically remove active active active without region also check condition acyclic calculation work sub gradually small describe grow simplicity integer adapt satisfied stage p p consider grow look follow look vertex follow square continue connected implementation allow therefore connect rectangle vertex furthermore edge active total computational must constraint check perform calculation datum flat
learning half learn subset recursively contain one label make classification root leave transform label draw multiclass leaf node subtree give chain start root follow gives even classifier yield multiclass binary tree multiclass find correspond let fix chance give example predictor choose multiclass predictor choose label suffer error rate imply classifier node predict filter algorithm illustrate elimination label structure round pair round round win round pair predict winner classifier train stage subtree root example form round filter example reach second tree identify node accord training match multiclass binary label set yy ns nf nf f root idea multiclass associate upon learning importance underlie sensitive simple classifier analyze complexity reduction oracle call search time back toward read bit computation sensitive cost testing require per specify transform sensitive multiclass example importance weighted label implicitly cost weighted analysis transformation allow add single classifier cost sensitive importance weight instead pick random create weight accord use quantify thus multiple deal classifier leave close problematic iterative technique cyclic classifier node regard imply induce algorithm transform sensitive multiclass classification rejection binary time relevant quantity importance weight form two label binary regret core theorem relate result sensitive path analysis boost distribution call oracle round original term cost multiclass form choose prove corollary multiclass classifier multiclass depth importance multiclass induced therefore cost bad cost tight use weight x draw sensitive predict accord induce sensitive induction weighted classifier subtree subtree class importance regret c whether without generality predictor subtree root inductive hypothesis trivially satisfied tree label subtree subtree possibility leave second prove induction induction inductive paper definition theorem make use filter tree evaluate vector node winner immediate claim provide respective use hypothesis l r c li li rs r proof take mistake recall denote small importance weight since expect importance hold respective input even odd split quantity let subtree choose inequality last proof three case rewrite follow fourth side inductive hand side third three term lc lc lc yield ts l complete tight tree binary pair except multiclass ratio variant filter significant practice essentially label compute test pair give node namely achievable use often maximize classifier tree publicly multiclass split isolated letter speech handwritten digit handwritten result split split pair pair filter different multiclass dominate relatively tree filter computation tree learner row bold although tree build section multiclass previous operate phase consist elimination pair label per round elimination binary tree elimination substantial first pair mechanism lose eliminate root final elimination phase select winner elimination subtree leave subtree phase leave elimination importance depend play winner versa label set preference amongst choice elimination blue one red elimination node importance depth game importance match elimination play bind computational elimination simplification weight control computation importance importance training unlabeled multiclass induce small power distribution classifier case achievable second case good label regret use subtree part proof tree invariant subtree winner phase elimination time w ar inductive desire finally maximum apply depth depth bound case error fourth depth relaxation relaxation allow elimination broken version important elimination single elimination depth relaxation elimination odd label odd play round thus compare label remain chernoff coin kk verify computationally factor formula compare elimination phase importance I put everything elimination depth add phase eliminate low hold somewhat powerful adversary weight equivalently constraint transform elimination repeat comparison majority vote depth adversary regret say adversary round make force bad outcome denote multiclass classifier induce use deterministic binary outcome pick algorithm query adversary assign adversary nothing suffer yield generality appear round adversary win label multiply number round adversary adversary regret roughly maximal adversary choice k winner outcome adversary incur answer win round break arbitrarily ask label total must exist round answer consistently choose winner unbounded reference abstract classification select powerful adversary new depth small classification multiclass instance goal accord approach multiclass reduction way key multiclass binary informally good refined technique analyze multiclass formally achievable loss suboptimal create class classifier predict binary classifier label answer classifier reduction
posterior incorporate likely return argument prior reflect mostly instead assessment band rejection basis acknowledgment partially la paris project technology choice discuss approach include assessment mean error model calibrate balance inference make manner original modification valuable distribution translate poisson iid integer value feature converge even proper necessary necessarily location therefore mostly conversely concentrated support use obviously remainder paper assess validity marginal likelihood model still ad hoc relate argue thing difference solely prior abc bayes assessment bayesian aspect consequence inferential machine current gain compare directly derive simulated poisson abc produce evaluation parametric equally version eqn counterpart exploit use device poor produce every consuming obviously moderate argue former rejection subsection seem rejection bind additional analyse product
quadratic countable regression inverse application give correlate let let put concentration technical defer hold proof rewrite follow symmetric derive respective orthonormal eigen stand p lemma consequence notation project bernstein serve sharp selection criterion penalization exclude broad bernstein variable concentration stochastic key importance statistical getting penalization sharp proving bound control uniformly statistic sharp concentration considerable interest analogue bernstein overview advance discover refined topic
short memory converge towards range estimator fast bring parameter fast completely dependent estimator method ml ls correlate change estimator let generality argument define huber solution n way property follow n moreover expansion imply assumption v huber strongly convergence prove solution without et al consequence theorem may neighbourhood consider account relation readily argument r I n term straightforwardly term take behaviour pn pn relation convergence neighbourhood generality relation eq pn yield function making imply hand imply bound positive var var obtain arbitrarily bound probability contain belong compact since show exist p p strong q triangular eq schwarz account writing two solution hand argument obtain account obtain relation imply proposition fr consider regressor stationary estimate property classical include convergence rate point asymptotic classification secondary compact denote regressor variance consider study statistic l wide distinguish error account condition error relate parametric vast development consider square process mean coefficient condition ml case paper limit change restrictive numerous sequence tx rr consider contrary memory error regressor parametric point vast error long x functional fractional brownian ls estimator concern cause suggest spurious point none absolute regime change define minimizer respect introduce point determine method past estimator behaviour change et et regressor gaussian change difficulty especially function convergence totally obtain consider notice change towards value classic make long science exchange memory another sp daily stock although contain serial absolute serial lag need throughout symmetric continuously differentiable cl vector slowly sequence long range lx value measurable vary infinity slowly vary reader refer long process process slowly tv residual consider notation parameter previous least zero strong exist arbitrarily solution obviously equation define equation twice paper go depend estimator purpose calculate partial choosing suitably close play moreover consistency differentiable assumption remain probability consequence solution system intervention realize accord fix neighbourhood order supplementary
chain define estimator clear enough therefore estimator surely e small markov chain recall denote markov aic aic bic clearly contain much concern consideration great simulation generation matrix advance hundred markov suppose concern converge right depend complexity order case keep example behave well tool create schwarz number exist et chain make behave aic inconsistent establish bic relatively sample corollary mathematic mathematics department mathematics review relevant information definition explore markov contain estimator already establish aic stochastic space transition hold I integer n decade great order chain j schwarz information aic impact statistical evaluation problem autoregressive process move process chain derive kullback leibler discrepancy tool model maximum later matter aic present mainly bic consistent bic estimator sample natural admit chain order identification though behaviour variable multinomial distribution sample finally propose establish aic bic outcome review estimator elaborate law iterate logarithm situation chain sample describe exploratory simulation discrepancy intuitively odd study sometimes density support ft u triangular sense eq formally fp p test relate square thorough multiple stationary unknown value v q homogeneous derive process irreducible mc see irreducible consequently ergodic exist satisfy q likewise sake simplicity return positive random extract mc sample divergence iterate convergence ergodic markov chain ng ng q almost well begin complementary exist k definition limit j ni accordance previous n
e subsection shall generative make normalize max alternative perform discrete py parameter via extend glm take corpus posterior distribution factorize problem extension class cm develop co descent tb input distribution step solve plug dual optimize optimize row vector derivative discrimination one label datum model px bb optimization keep slack want proof ignore corollary conclude I normal algebra another optimum solve primal state corollary supervised utilize document discover document supervise topic model categorical response variable discrimination utilize margin model arguably max estimation direct supervise supervised demonstrate advantage topic movie topic maximum discrimination max topic latent allocation gain collection discovering capture semantic collection latent topic represent vocabulary document variable represent task tool otherwise lda unsupervise incorporate corpus online user usually post rating score rating image may discover secondary dominant user goal unlabele unsupervise prominent perhaps orthogonal goal latent serious alternative well corpus incorporate gain major supervise report classification lda document review rating score variant supervise design different model predict aspect associate paper loss supervision arbitrary model level differ well exist train maximize lda use fed classifier document rating movie discover sub prediction develop supervise lda report task later later discriminative share goal differ train train joint response likelihood learn latent contrast stage procedure first discover discrimination dirichlet margin vector optimize objective special partially observe discrimination structure hide undirected discover latent classification learned sense discovery latent max yield representation suitable apply topic g lda correlated topic principle lda underlying implement variational comparable lda property stems directly optimize suffer normalization make learning model introduce basic efficient variational discuss generalization present classification present conclude direction conditional paper lda discrimination couple max define use lda separate lda building variational allocation proportion document document draw proportion topic response vocabulary problem follow draw response proportion document define pz document posterior exact hide intractable variational solution variational e assumption like efficiently detail pz approximate inference approximate independence assumption inference efficiently change type model lda delta I conditional although mle great success max arguably empirically svms recognition allow deal integrate advantage procedure latent discover task real machine margin supervise unsupervised lda model exposition basic vector upon machine publish brief regression find deviation response datum flat choice find function problem slack trade norm amount empirical loss insensitive qp solve formulation lagrange support package routine bayesian possible represent prediction constraint margin sequel dirichlet extension elegant max variable extension latent discover semantic document collection principle assignment unsupervise accordingly integration parameter sample directly optimize intractable optimize upper random define bind kullback kl divergence integrate problem cm multiplier slack version topic constraint insensitive loss want hand correctly sufficient explain well e minimize variational margin estimation topic discovery couple expectation latent representation suitable constrain intractable obtain mean variational form factorize free variational parameter small topic topic iteratively solve step unknown e step constraint topic since alg last lagrangian argument infer infer specifically rule tb solve dual update time document element essential lie firstly expect version topic co involve secondly max margin lie around document last affect representation document therefore representation row vector diag lagrange multiplier plug partial derivative get e diag optimum k co also normal cholesky laplace effect prior programming qp solver may recent development corollary achieve cholesky decomposition prior prior similarly prior qp qp solver efficient leverage development corollary formulate exist specifically cholesky matrix let u primal problem formulate unknown equation discover representation apply lda naive unsupervise g use unsupervised latent topic representation dimensional representation couple optimal g movie discover low representation sub present unsupervise inter play discriminative apply learn integrated learning cm bias implicitly independence assumption lda setting suggest less affected margin choose prior depend latent representation independent model less affected topic rule coupling topic representation coupling lead inferior see qp reformulate leverage recent either primal un normalize partition involve reasonable margin specify lda show use normalize beneficial appear discrete brevity multi easily similarly class stacking equivalently write fy derive model q distribution develop lda discover latent topic however impossible dual prediction latent easily state supervise intractable except normal variational method high expansion max loss response instead I corpu similar regression integrate variational variational upper dy dy fundamentally expectation latent well kl term regularizer l last iteratively optimize rule omit explain insight since factorize perform unsupervised reflect discover topic example boundary e lagrange multiplier act bias model discover latent representation tend example word document latent discriminative classification optimum optimum regularization effect prior normal shift dual solve svm present margin lda supervise apply formally discrimination topic lda g likelihood slack general em underlie topic bayesian contain recent discrimination extend multiple mutual exploit latent extension promise use lda unsupervised lda topic empirically compare integration statistic collection marginal likelihood slack regression classification implement compare section qualitative quantitative text model regression c class c per class graphic image db file file mail software file window mail format bit format file graphic power water rs air circuit nuclear loop circuit mail neutral signal email temperature people people attack article center people price mail mb disk offer package mail controller mb condition disk video mail controller email email remove relate topic explore unsupervised pair grouping separation document embed category mix embed examine discover figure top topic moreover distribution average latent yield decay discrimination fact effect seem discover topic detail regard discrimination topic per topic graphic document salient lda distribution case discover provide evaluation lda document build baseline evaluate relative ratio build gibbs binary svm threshold whether max margin utilize build via run experiment final ratio number per category set align since margin tune regularization change perform multi category package solve sub learn align close discover topic max margin believe slight movie approximately unsupervise implement denote document input evaluation criterion pr response slightly topic representation alone highly separable integration max margin discover discriminative representation topic document stay eq decrease behavior suggest discover separable obvious improvement rule fourth term discovery get pr bad r classification classification svm partial lda find vision etc problem result classification important issue method high statistic concept maximize margin generative handle spirit supervise fully generative minimize field differ leverage supervise include review label lda discrimination provide max style structured network application principle dirichlet version discover topic document collection prediction variant predict mutual dependency globally consistent prediction scenario annotation neighboring tend smooth machine token align discrimination use supervise principle optimize integration predictive suitable variational
q spectral associate optimal define stationary eq section bound fx uniformly ergodic simple chain start point lead effort improve decay valid appear computable ergodicity one literature specifically appear function drift simplify statement entail q exploit explicit simplify write block proposition modify make reasonably appendix mention c instead cite easy variance satisfie indeed result put everything theorem need bind imply analogously final vx iii compute vx vx vx f vx obtain effort bind one experiment describe prove actual error normal play role unknown parameter uninformative improper determine location scale gibbs whose since notation symbol gibbs draw conditional small transition let rhs regard first student degree therefore sampler rescale student degree freedom drift let quantity analogously give take assumption student attain respect eq interesting parameter interest bayes zero mean q report repetition experiment name henceforth formula analytical table inequality quite sharp mse primary large satisfactory proceed drift corollary computable drift examine influence proposition depend correspond grey assume equal actual figure analogous bound black grey good contrast inequality depend c value bottleneck approach drift theorem specific derive norm present drift computable chain alternative might work well successfully space reference aim obtain concern square general application relate vast setting give finite relate exponential inequality explicit derive effort constant give turn come phenomenon dependent inequality martingale use motivated valid metric expression ix iy apply well functional uniformly ergodic chain chain detail refer reversible bound bound available result apply integral regularity stationary turn exponentially seem dimension inequality functional technique apply e geometrically possible inequality geometrically cl truncation constant rate chain one assume drift need focus either total translate burn stationarity asymptotic always example section avoid could chain often validate g lead introduce bias design confidence chain often trade rigorous heuristic refer mcmc argue essentially asymptotic difficulty estimation markov chain chain context heavily lot markov chain consistent geometrically ergodic markov satisfy condition condition non overlap batch markov chain condition geometrically markov condition ergodicity drift drift require check typically ensure clearly algorithmic require whereas variance estimator conclude upper acknowledgement discussion early grateful anonymous constructive comment eq decompose shorter visit notation let easy distribution vx consequently begin eq partially support science high education et department statistic university cv also continuous block markov trajectory consider term ergodic geometrically possibly high respect monte idea simulate markov converge ergodic average reliable long must run prescribe use distribute block chain starting propose sequential length trajectory promise total trajectory cycle provide identify introduce simulation tool quantitative bound mcmc aim end split independent inequality proof classical result process identity median trick lead far express computable uniformly ergodic asymptotic chain turn small motivate towards unknown drift parameter drift imply ergodicity purpose build derive variance uniformly assumption identical moreover quantity practitioner connection one benchmark technology hierarchical bayesian variance area simple regard analytic assess full also construct goal estimator combine chebyshev small trick concern far independent boost independent chain estimate chebyshev odd deviation require bound moment precisely choose describe conjunction scheme behave like reasonably less choose big enough make run chain absolute constant tight familiar asymptotic
observable variable trace protocol limited security protocol identify protocol intend crowd anonymous david start project fall suggestion use fact make privacy mutually vote retain hence vote protocol periodic random vote protocol contract message protocol achieve relation observable datum protocol make observable correctness protocol verification protocol study protocol formal exploration verify express logic security protocol model mdps check temporal logic logic issue randomize biased reveal information crowd might adversary guess message project protocol quantify security protocol create environment pseudo generator implementation detect random protocol implementation attack randomization source randomization check hold protocol black reason protocol could hardware core able correctness observe input output inspire reverse genetic network molecular process form feedback architecture discover among randomized trace work observable trace randomized protocol measure channel technique probabilistic example available www contribution statistical probabilistic security protocol protocol mdp trace aware exist security protocol probabilistic organize mention limitation work protocol mutual well several effort make towards security protocol summarize provide protocol measure lack notion assume model channel nature security protocol worst different quantitative flow space guarantee guess posterior probabilistic relation observable guess guess knowledge flow kullback leibler di al attack see conduct validate similar approach al error protocol find approach protocol work pi calculus check probabilistic checking check extension check formal finding circuit exhibit specify give technique protocol abstract implementation code black analysis protocol rest summarize measure channel mutual trace observable basically bayesian describe trace estimate learn htp protocol analyze channel number process observable protocol ensure across maximum across channel thus channel capacity take maximize algebraic decrease freedom always use expression still channel randomized characterize need protocol choose estimate knowledge except would accomplish maximize information marginal conditional analytically maximum attain conditional bayesian dependency variable graphical way join maintain give observable depend variable bayesian protocol need network trace previously genetic adapt available knowledge dependency variable reduce possible structure begin identify observable mutual identifying node edge proceed node similar start maximum receive algorithm size record increment empty learn learn maximum unknown distribution maximum need special case approach probabilistic checking code infer code h ps ss need capacity mutual information detail co maximum ab technique terminate answer correspond maximum channel capacity protocol validate statistical technique compare checking protocol example choose mdps available pay coin say exclusive exclusive channel maintain guarantee rely try conduct protocol experiment vary increment capacity protocol probabilistic checking infer plot estimate channel channel fair coin protocol fair coin channel attain close coin
assume also regard model well separate attain value beta level fit good classification obtain mean straight along beta whereas produces take figure beta identify histogram concern beta heterogeneous respect independent see heterogeneity secondly exhibit negative confirm incidence ht cccc elliptical competitive generalization model remark flexible powerful useful fitting respect inference valid well suited student context procedure parameter recent robust research another issue em behaviour condition mixture regression point initialization simulation either preliminary numerical pointed reduce covariance implement g provide decision surface general illustrate point consider surface follow case surface circular circular surface eq figure equation remark surface elliptical acknowledgement comment valuable suggestion thank proposition prop lemma prop cluster modeling regard joint property case mixture mixture regression far base student distribution long tail observation simulate mixture model flexible approach wide phenomenon unobserved heterogeneity work sample arise identify conditional density variable refer unconditional otherwise refer g know mixture expert machine biology explanatory multinomial logistic focus approach response variable explanatory model medium technology digital refer moreover propose market segment derivation literature quite indirect statistical view assumption special secondly student long normal theoretical illustrate numerical base simulated organize framework sec discuss simulated conclusion surface cluster modeling introduce response weight hence broad sense generalize idea give type characterize relation parameter type form indirect mathematical tool concentrate th depend conditional since density assume gaussian g gp g g mapping denote particular relationship assumption mixture pg finally random value value assume gp g yy g g write argument lead linear include quite group secondly th mix consider probability result mixture regression augment multinomial satisfied multivariate gaussian assumption gaussian every follow multinomial logistic p density complete immediately posterior get complete joint side side neural network diagnostic list table relationship assumption none g traditional mixture py eq student provide robust normal tail observation asset pricing variate random parameter definite denote mahalanobis gamma chi degree freedom throughout location g g q refer moreover vector random g g classification plot classify accord cc classification classify illustrate different situation datum parameter b gb unit properly model attain plot classify accord symbol classify group denote classify third distribution group cccc identify matter reason classification classify true classification classify present concern noisy datum fit base unit mark secondly reduce whole group step follow strategy group mahalanobis distance maximum likelihood outli classify classified group otherwise denote chi distribution forward outli minimum determinant scope concern parameter reduce student classify group concern unit list different gb distribution variance augment rectangle noise tt tt outperform tt recognize large small small true outli cccc cccc outli error misclassification student gaussian student misclassification correspondence gaussian set generate gb divide group previous generate rectangle unit outperform misclassification recognize misclassification fit matter identify cluster tt point misclassification ccc student outli outli cccc outli case student cccc outli outli confusion square misclassification student student small misclassification bivariate simulate concern accord sample rectangle add unit summarize outperform tt misclassification attain small square student gaussian ccc outli cccc outli student
use encourage estimate simultaneously consistency consider type approximate mention large relaxation rank subject trace tight manifold norm goal explore method namely nesterov outperform code beta require less latter one organize subsection provide simplification programming saddle point nesterov aforementione second interior aforementioned cone variant nesterov smooth finally conclude remark notation symbol euclidean denote dimension diagonal whose diagonal let semidefinite interior let denote minimal eigenvalue z n denote standard norm norm xu u tx qr x z consist functional endow denote vector define lipschitz respect differentiable subsection eigenvalue presentation identity symmetric scalar statement I equality due fact possible follow hold identity scalar identity note eliminate conclude dual strictly conclude identity view duality w moreover fact perform variable last arrange integer prove page immediate immediately lemma integer eq identity every scalar identity subsection subsection restrict reduce row subsection observe matrix appear many similar type clearly assume orthonormal diagonal f ta tb tu optimal relaxation optimal hence formulation subsection fact quadratic optimal f quadratic strongly convex strongly conclude unique cone respectively reformulate nn r l easily proposition reformulate nn gx define smooth saddle suitably variant nesterov subsection introduce notation optimal immediately relation addition max possible develop solve nesterov scheme min first computationally superior report paragraph problem lipschitz continuous subsection nesterov solve yield nearly report paragraph subsection saddle suitably nesterov subsection result let scalar penalize coincide saddle point suppose let objective assumption together immediately saddle point min namely scheme base lipschitz subsection detail subsection nesterov yield hence section review nesterov smooth solve convex minimization subsection detail variant nesterov formulation specifically problem review nesterov smooth cp respective cp nonempty make follow regard function every strictly b differentiable well imply differentiable solution proposition dual namely optimal finally recently nesterov smooth notice type subproblem variant follow need prox subproblem modulus immediately note complexity corollary discussion arithmetic operation iteration variant bound expensive variant method nesterov solve maximization subproblem fact see scheme iteration nesterov subproblem subproblem real scalar solving value decomposition eigenvalue h nf nf tf zero easy see solution problem efficiently terminate variant nesterov objective nesterov subsection solving address prox problem formulation discussion also scalar specify aim lipschitz help form fix easy convex modulus respect proposition assumption b specify prox set variant nesterov let fix minimum achieve last equality identity allow due identity modulus view nesterov dual result smooth prox find exceed precede variant nesterov iteration relate original smooth prox function apply exceed view find arithmetic per nesterov consist overall arithmetic require nesterov actual maximization order detail subsection paragraph nesterov efficiently together optimal solving terminate variant nesterov properly dual dual computed q report compare nesterov smooth method version beta instance use experiment first generate value cpu gb variant nesterov discuss subsection interior version cone write worth interface call task suitably program represent product nonnegative terminate terminate performance randomly table number seven amount column computational instance variant nesterov smooth method less instance memory memory rr rr c optimization trace multivariate explore variant smooth computing penalize least generate version former memory efficient drawback handle turn exist alternative problem e dimension qr triangular handle subsection show nesterov lagrangian subproblem replace upper triangular matrix later hard subproblem presentation discuss constrain problem unconstraine penalty exact penalty strong consequence make function value nonempty region statement every solution gx imply statement let statement inequality every hold conclude inequality statement moreover gx fx f hold threshold compute instead yield view slight h page yield optimal solution version solution feasible h ix x fact feasible every feasible accomplish lagrangian subproblem different lagrange idea acknowledgement thank anonymous numerous comment pt pt support grant author support part grant grant author part nsf grant dms estimation nice finite compute however variant nesterov
region covariate nan hypothesis curve plot empirical horizontal intercept empirical simulate intercept case label case restriction precede case figure intercept power reach also error power identical scale balance case identical symmetric normality test case w violate violate asymmetric test sensitive power estimate value covariate adjusted residual overall regression line treatment specific interval might treatment prefer adjusted conclusion use significance treatment reject usual k reject test treatment reject tool covariate similar residual underlie among iv range treatment residual extremely covariate corollary main theorem conjecture remark university mail phone adjust control covariate particular homogeneity variance adjust residual adjust line entire ignore level compare appropriate adjust residual appropriate remove covariate specific power extensive normality size covariate adjusted treatment treatment show sample covariate mean power nonparametric covariate distribution variance heterogeneous different functional power test covariate residual cluster relative factor covariate cluster exhibit suggest covariate treatment factor homogeneity isometry may factor often strong external remove influence quantitative effect adopt covariate article affect relationship factor covariate experimental choose technique instead covariate available assign treatment covariate etc several influence covariate statistical biological remove covariate compare ratio site technique ratio residual iii analysis ratio g effect recommend still ratio covariate ratio remove covariate covariate relationship relationship isometry response covariate nonlinear intercept ratio introduce heterogeneity assumption homogeneity ratio give spurious recommend remove covariate remove research set residual refer henceforth residual remove covariate recommend ratio covariate adjust widely show superior ratio residual argue residual totally overall pool residual analysis wrong iii well recommend tool rather assumption treatment group covariate demonstrate balance sample design otherwise especially covariate inference common group hence recommendation favor article wrong residual also covariate adjusted empirical power monte nonparametric test covariate adjust residual entirely sense fully covariate continuous categorical see covariate additionally fact unbalanced heterogeneity group alternative outperform influence present covariate covariate adjusted residual detailed nan hypothese condition simulation use power discussion provide model adjust residual provide single suppose factor replicate value level line covariate intercept slope error term regression analysis stand identically treatment factor ij ij nan eq reject covariate depend reach parallel parallel otherwise article parallel slope parallel line q slope treatment equality treatment different intercept overall level residual value equal treatment assumption adjust residual r residual residual identically parameterize covariate treatment parallel treatment covariate residual difference error homogeneity necessarily assume extension group k treatment notice parametric section distributional equality test adjust compare treatment parallel hypothesis test without covariate adjust respectively parallelism sufficient hypothesis test respectively stand equal estimate residual rewrite average treatment covariate treatment n r assumption take expectation rewrite equation pair condition imply equivalence hypothesis slope covariate overall response slope ir iy slope ij xx side iff hold equivalent treatment hypothesis test statistic square treatment mean error degree df since treatment covariate adjust square covariate adjust residual calculate ir mse r n ti test r adjustment source inference df literature become statistic hypothesis f f ff ff level score similar reach likewise see large increase regression entire line adjust residual covariate mean get parallel base df calculate approximation k normality loss slope arbitrarily intercept response generate error distribution error introduce generate covariate treatment sample size replicate summarize slope apart power increase choice increment simulation treatment specific line expect occur approximately pilot size error replicate replicate covariate variance interval replicate variance error iid double parameter parameter pdf x x df log location parameter pdf fx heterogeneity variance variance dependent variance iid treatment iid treatment case treatment choice variance case roughly difference consumption consumption non covariate consumption covariate consumption heterogeneity adjust residual assign restriction different generation symmetric around notice case scale variance non normality case distributional vs asymmetric term investigate generality replication heterogeneity normality variance might naturally distinct partially covariate covariate mild overlap covariate covariate uniformly generate treatment treatment difference figure cluster first covariate covariate note different treatment covariate treatment choice research consumption randomly select treatment hence treatment middle treatment mild comparison present simulation size relationship treatment difference detect record difference record nominal base mc detect
present subsection nn conditional probability p median well associate divide recorded panel panel record associate sis record sis scad lc c pt demonstrate difficulty minimum regression oracle know statistically difficulty significant oracle setting take see magnitude get ratio oracle difficulty evident value font indicate lasso scad estimate lc pt table lc e c pt le correlation correlation comparable increase size sis scad scenario setting sis scad fail correlation large set interestingly help increase signal notable lasso scad third sis reasonably well normally scenario scenario size correspondingly reflect linear lasso scad logistic magnitude job screen irrelevant scad regression l pt c pt lc l rank generalize show possess marginal screening surrogate screen utility screening option variable signal estimation example meanwhile sure vanish false method current cover link leave broadly applicable outcome covariate arbitrary therefore besides rich regression transformation censor pursuit interesting topic extension number covariate marginal procedure highly jointly marginally weak screen lead interesting future choose important sis preference fdr also employ final choose interesting scope current follow contraction van value rademacher sequence element value rademacher q positive font mean table e define tail idea involve bound application n contraction side cauchy schwarz jensen expectation bound e n since combine p nk k small utilize definition minimizer regard set leave n na bound triangular theorem b ft contradiction constant lipschitz continuity x b score conclusion sufficiently interval function deduce n nj condition b ga r condition consequently strictly increase obvious generality jointly normally distribute independent first side chebyshev satisfied tail j j mi n term tail schwarz take bind complete key proof exceed n b expansion eq interior x bound sufficiently tail eq b b put side conclude positive entail q bound prove case x j c n bound hand equal hence negative continue cauchy schwarz prove signal taylor nm nk nm choose tend one prove obvious consequently tend minimum x k k k g theorem easily see exception tend exception set exception exception negligible taylor last consequently exception tail q exception negligible probability eq conclude contraction negligible acknowledgment conduct song constructive presentation paper corollary remark part nsf grant dms dms play increasingly important research fan marginal correlation possess response propose rank maximum linear fan special possess screen property vanishing possesse sure screen surprisingly applicability spectrum quantify extent screening depend covariate true parameter study addition useful learn scientific research brain study phenotype height million snps disease classification profile grow rapidly demand lot challenge statistical also dimensionality accumulation computer comprehensive reference therein small predictor contribute lead prominent role statistical propose bridge scad penalty method concentrate property learn problem simultaneous challenge computational algorithmic screening sis select deal aforementioned challenge relate show possess independence screen sis screening technique sis ordinary argument heavily easily even limit significantly categorical ordinary heavily explicit expression call research sis model less ranking regression intercept preserve marginal preserve marginal well former surprisingly marginal marginal utility interesting method assess select likelihood ordinary covariate impose aspect sis normality set obvious generalize current framework sis third generalize ranking ranking base fitting type misspecification drop establish tail traditional asymptotic misspecification interest practical variable marginally jointly marginal develop iteratively screen selection procedure former focus linear applicability sis surrogate propose sis procedure sort ranking whereas sure key maximum noise technical difficulty monotonic transform generalize sis exponential likelihood sis present formulate sis summary section scalar family dispersion easily derivative consider dimensional intercept copy covariate whenever note take standardize rank base standardized denote model covariate regression robust instability np version regression denote predefine threshold learn rank coefficient independence learn dramatically possibly hundred although implication pass sure device mle use self response depend sufficiently convex fisher positive moreover control noise relative uniform convergence rate sure screening method former control select concavity minimum b update condition parameter j l nm hold linear linear regression poisson second condition exponentially lemma exponential family bound give convergence sure property screening depend use positive p nm nk fan bernoulli constant bound ordinary weak fan permit whereas handle covariate sure screen property number reduce independence answer simple see one probability tend ii consistency q regression deal euclidean nonnegative differ condition exist nk nm explain interestingly screen property depend correlate indeed percent discover negligible order fan condition result fan condition screen generalize suggest sort view build increment screen equivalent possess screening formulate screening sort independence rank marginal magnitude screening common computation procedure optimization two parameter computation traditional utilize magnitude incorporate whole increment magnitude current comparable level implication convexity otherwise still need discuss beyond scope sure hold screening increment increment follow purpose selection consistency hold minimum
pa pa pa pa pa pa pa pa pa pa pa pa pa pa pa remark hence pa pa pa note p property random variable side stochastically independence exist k j pa pa pa conditional almost sure problem pa iv follow triangular influence influence I suffice kk nr function random result definition acyclic commonly among arise biological moreover may order reduce estimate network paper propose adjacency lasso penalty grow size achieve simulated datum example efficient study compact joint variable conceptual type direct causal relationship relate direct also bayesian base acyclic edge direct direct cycle belief important application study cell pathway play np early greedy search hill super large impractical intensive particularly skeleton causal relationship setting order gene expression present graph conditional undirected graph zero concentration penalization use node explore estimating concentration lasso consistency frobenius norm regularize result estimation matrix precision covariance penalization cholesky covariance matrix order interpretation cholesky estimating skeleton natural theoretic adjacency offer considerable improvement variable selection consistent adaptive consistently estimate usual assumption theoretical evidence mechanism network method gaussian simulation order sensitive permutation association amongst direct parent undirecte undirected use adjacency whose entry specifically regardless infer illustration result new original graph change probability start skeleton true cm eps equivalence challenge estimate variable conditional remove graph parent node reveal graph parent illustrate suppose normally covariance connect variable direct acyclic graph parent variation association simplification let represent eq coefficient normal simple simple latent variance x establish influence adjacency establish relationship skeleton latent give entry depend regardless joint result datum non without scale one precision control penalty involve therefore solution edge lasso penalty find latent graph formulate order triangular estimate solution problem lasso obtained facilitate modification original lasso estimate adaptive penalty differentiable reformulate use number triangular prevent nonzero optimization non negativity solve algorithm scale applicable hundred penalty denote see solve optimization row I equivalent fact n reformulate regularize square project section connection neighborhood set order w estimate suffice solve estimation square range r package summarize comparison computational complexity introduction exponential node surprising pc restriction although considerable improvement estimating gene pathway exhibit iterative require lasso solve comprise covariate coordinate matrix graph calculate include overlap number estimation adaptive lasso regular compare cpu well complexity pc neighborhood pc adaptive penalty accord cpu graph plot demonstrate high pc repetition gb ram adjacency asymptotic property type estimate researcher estimate section matrix overlap property estimator focus asymptotic estimating nonzero adjacency establish structure price main lasso require adaptive estimation lasso adjacency lasso exist iv well adjacency depend choice tuning include validation bayesian choice propose tuning state control probability distinct set next every consist node tune respectively general tune false probability graph lasso goal easy requirement recommend prevent generate low triangular generator control well size edge difference performance represent graph equal drawback dependency node goodness coefficient commonly classification false respectively fit worst compare establish report pc value investigate lambda image gray obtain calculate specific offset effect present observation gray precision observation base base false positive although computationally use gene network direct h along gray edge simulation ham lasso parameter lasso give propose likelihood outperform size may undirected network skeleton pc ordering determine estimation partially complete accord additional simulation considerable hand magnitude decrease comparison remain observe addition significantly power true create consideration similar result observe exclude performance tuning confirm focus aspect estimation distinction plot suggest vary variation positive deviation base time large estimation normal correctly study simulation distribution freedom performance similar penalize additional simplifie disadvantage complex expect ordering variable play significant well unknown generate three dense pc cause recognize correspondingly degree structure crucial lasso adaptive carry flow human system pathway establish perturbation cell molecular know include protein analyze moderate false negative rate lasso give include undirected edge enforce see close play control expression e provide whole datum application goal connection whole genome gene present use method relatively attribute successfully reveal combine consider pc true connection lasso drop adaptive penalty small true edge positive network positive structure lasso penalty estimation adjacency
extreme distribution algorithm algorithm derivative sometimes derivative censor complicated avoid disadvantage visualization graphic graphical reduce iteration complete procedure censor type extreme estimation transformation illustrate use type call equation eq fact point unlike extreme technique monotone decrease prove ng extreme value solution obtain deduce mle consider maxima km software book file fit iteration extreme decrease existence r I guarantee existence uniqueness observation observation sample meanwhile relation q uniqueness mle q censor replace r censor testing fail lead solution reduce censor I censor time fix iteration mle q argument precede decrease point mle type sample exist unique fix type ii censor distribution newton compare iteration newton fix newton take iteration take take converge axiom theorem conclusion theorem theorem exercise theorem likelihood estimation parameter extreme paper newton require unlike
represent option predictive often need common study efficacy medical pearson enable conservative hypothesis coherent decision would probabilistic estimator give negligible concern level terminology establish confidence consistently indicator indicator usual statistical associated significance x side next proposition represent function asymptotically xx interior converge x yield prove proposition consistency smooth suitably transform likelihood ratio statistic consistent estimator regularity indicator side discussion frequentist bring consistency frequentist coherence establish ability hypothesis frequentist flexible distribution require need confidence posterior require set interest dimensional composite neither consistent specifically convergence interior manuscript lead clarity thank discussion coherence provide university thm remark thm example department road odd accord measure call loss frequentist posterior frequentist posterior confidence automatic reduction apply case result frequentist coherent axiom theoretic theoretic cite support unlike interval hypothesis truth keyword attain coherence confidence utility region test utility motivation interpretation observe confidence almost confidence repeat result probability lie coverage rate report match addition matching enable leverage inference lie basis coverage rate predictive probability model location model location scale yield asymptotic model g hierarchical mixture model achieve necessarily asymptotically suggest function integrate nuisance attain bayesian rate advance vision objective universal angle matching prior inference raise goal motivate formula distribution confidence benefit inference interval conservative appropriate value either largely effect various shrinkage likewise know tend avoid interval valid interval parameter confidence contain true give support coherent inference available fair odd condition conservative fail reflect rely extent maker conservative control pre post confidence fisher approximate fisher particular instead careful actually observe frequentist formalize extend level lie confidence rate repeat understand achieve often substantially true lie confidence reasoning frequency level notable inductive logic often effective decision know hour whether particular accurately absence relevant reading car indicate hold equally level confidence hypothesis employ framework applicability inexact third recent arbitrary general motivate extend include theory additive odd depend theory scenario price hypothesis confidence level price probability posterior summary concept frequentist posterior decision state probability completely idea foundation decision truth framework frequentist reporting interval hypothesis determine level interval situation circumstance frequentist posterior reduce exact automatic confidence decision loss measure coherence axiom compatible case frequentist framework example example side assign region assess practical scientific pearson symbol angular tuple pair index parameter quantity sample generality unless nuisance except otherwise note kolmogorov measure measure measurable borel complement field unnecessary value usage latter call space triple estimator nominal coefficient set xx measure turn nest likewise provide sum level equation confidence estimator eq kp x mutual consider hypothesis analogously call accordingly hypothesis confidence strongly ensure confidence measure field framework choice estimator induce extend structure credible improper taking advantage advantage likelihood function distribute accord nest confidence upper tailed yield equality student nest special upper cumulative probability c estimator valid set induce equation p drop difference confidence hypothesis observation case confidence level correct confidence integer equal binomial reasoning process ideal field low odd determine confidence coherence odd upper low probability probability decision interpretation would pay small price express function call family specify evidence satisfy axiom avoid make framework coherent price gamble extend measure induce turn upper initial price agent loss generalize make decision unbounde behavior amount average preferred decision restrict dominate multi statistic use decision preferred yield member member member multi alternative practical broken additional consideration argue dominate rather need x undesirable eliminate replace implication application assessment science arguably valuable apply report inferential role play many value become identity origin radius outside radius chi cumulative cdf justify approximation among conclusion hypothesis implication work confidence scalar enable approximation relate justification level branch maxima estimate characteristic bootstrap introduction fundamental bootstrapping coverage rate justification continue merely neither latter test available approximation acceptable lie confidence value consider minimal scientific significance application researcher datum increasingly minimal biological significance gene effect confidence hypothesis confidence hypothesis practical application minimize confidence estimation bayesian bayesian frequentist confidence posterior posterior expect minimize p prove frequentist mean prove frequentist mode maximize analog xx like distribution confidence appropriate prediction bootstrap value bag prediction bag study uncertainty method determine rely versus illustrate use compatible update confidence theory differ fundamentally dominant statistic broadly prior invariance nonetheless theory make demonstrate framework frequentist require distribution correspond observe kolmogorov suppose decision must assumption vector mapping quantity write respectively successive name conditional odd prior odd determine odd actual observation current learn point bayes sampling upon whenever prediction frequentist check poor reflect initial decision avoid reduce expect loss single confidence measure expect coherence concept place law probability none define kolmogorov replace theorem statement illustration agent whose self agent version unless none ever finitely distribution agree probabilistic logic conditional either parameter update statistical support understand mostly david transformation book know use odd principle occur consider agent odd book argument consider coherence argument requirement distinguished game rule incoherent book coherence bayesian temporal theorems coherent theory thus represent value quantity ground minimal concerned conditioning dynamic coherence maximization case light distinction coherence confidence hypothesis reasonable odd distribution case place contrary conditional hold incorrect work consider conditional bayesian temporal non light subsequent bayesian posterior uniqueness inference coherent give inference frequentist selection prior intend criterion scalar tail confidence prove consistency property condition level composite truth confidence decision cdf value likewise distribute nan side test value pair central name scientific order asymptotic prefer avoid distinguish kolmogorov measure incomplete theory incoherence distinguish value hypothesis
envelope accept probability algorithm independently compute rejection area trial accept assume notational convenience scaling unnormalized constant clearly envelope mixture inversion probability compute envelope c independently return acceptance coincide acceptance use every reject low c ce xx optimize acceptance change frequently fortunately insensitive various note apply avoid author associate valuable comment bivariate conditional bivariate conditional work simulate conditional simulation exponential conditional specification conditional receive attention concern actually suggest convenient configuration acceptance efficient approach g readily
long convex work j series inequality accurate accurate solution accurate furthermore identical minimize change application inequality plug accurate setting k claim argument problem svms discuss algorithm improve subroutine problem wherein bound briefly maximum solve training hyperplane use notation identify empty suffice solution svm produce let margin substituting size point notable vector guarantee define inside convex origin equivalently look small finding polytope similar polytope exhaustive list start version problem rewrite dual read polytope problem minor modification since whose geometric purely optimization viewpoint algorithm treatment appendix competitive version set polytope rotation multiplicative please ask stem variable derivative substitute kkt root note monotonically monotonically increase nonsmooth adjacent sort sign root time sort make fs loop terminate iteration linear total sum aggregation slope offset sm fu support svms recent interest geometric idea iy hyperplane margin separation class write yield qp j one dual feature boundary boundary space live associated svms context become ic therefore iteration compute name core vector rate convergence hope convergence aim efficacy performance algorithm random average guarantee multiplicative choose reproduce report fair comparison time relative refer cccc cccc bc bc bc comparable furthermore usually take believe benefit mm theorem lemma corollary conjecture theorem theorem assumption zhang minimum smallest contain produce radius yield greedy polytope problem polytope polytope present face polytope heavily duality argument n ball small radius diverse statistic graphic wide dependence svms million dimension significant interest art extensively concept small ball expand build greedy every build current stop away current include continue know problem algorithm achieve denote value approximation simple show scale approximation effort minimum convex polytope set find polytope shape set rotation polytope apply two machine find hyperplane computing distance polytope algorithm one study iteration follow section notation duality future find appendix bold bold letter denote entry euclidean dot n definition f q accurate standard concept convex use sequel strongly brevity gradient brevity lipschitz gradient duality section respectively cast convex nesterov subset map function standard certain respect necessarily smooth aim c prox verify smooth constraint use transpose maintain figure conjunction derive follow duality eq word duality gap idea answer question find point maintain desirable allowing update efficiently cast convex answer htbp plot node right right right black thick give set nd cast problem reformulate n I set nesterov minimize efficient towards cauchy satisfy turn prox notational convenience place algorithm recursive plug immediately gap lie ball radius conservative plug accurate ensure require map conjunction find accurate time computation cast qp linear appendix simple algebraic cast quadratic qp constraint bottleneck apply exist point radius give
compatible constraint bayes note practice space px side implement constrained information outcome thus impose imply bp p receive form maximize lead receive second piece constraint update process current new updating lead might exist possibility simultaneous maximize simultaneously fortunately check consistency b update posterior decide clear treat constraint distribution however state expect indeed really retain posterior decide old validity refinement family correctly reflect restrictive processed updating remain comment understand process failure lead reflect background kind assign box ball box amount kind box inform company know kind ball information getting box allow randomly ball box get perhaps open box look format outcome represent sample total let original outcome would get start eq normalization yield normalization lagrange multiplier determine mathematical wish select ball multinomial distribution use model use flat integral form sake joint rewrite wish piece information kronecker delta infer particular box constraint apply whole refer actual box take expect become process familiar bayes recognize use familiar derive reader reproduce standard contrast process different general inform company know information get old ball know getting allow select ball box since piece processed simultaneously maximize normalization simultaneously piece look like sequential crucial updating multiplier multiplier satisfie purpose section second wish current theory pose problem one summarize different different company know expected type ball time many ball box select ball box let ball problem determine get description intend key see identically distribution type addition closure interior become asymptotic essentially say true entropy lagrange multipli sample avg expect use think kind method version label information take box arrive maximize proper multipli plot represent compute ball produce close however drawback represent expectation multinomial produce one perhaps almost indicate underlying probability asymptotic use incorporate special allow go would emphasize method constraint inference argue contrary constraint regard case template currently need assumption fluctuation analogous moment recover rule acknowledge discussion c create detail dropping differ force calculation nest sum take symbol equal technical worth sum second involve lastly sum first newton currently less feasible nest numerically good physics apply economic situation entropy economic information relative example detail template deviation advantage inference much idea physics economic paper economic purpose shannon entropy statistical assigning probability regardless idea assigning assign allow method expect apply moment address large justify use justify simplify bit issue data maximum entropy update form moment resemble produce produce compatibility far address individually one datum comment discuss whether process sequentially conclusion accordingly lead inference potential economic similar two ill behave q dirac delta require lagrange impose usual normalization include function gx constraint emphasize satisfied posterior bayesian proceed want close vary derivative lagrange constraint constraint
complete elsewhere statement first mat ern universal quadratic toeplitz apply analysis maximum check integral say value root small equation two derivative ml maximizer interval interior regularity condition well fulfil mat ern know result regularity series argument b asymptotic square follow surprising small nearly full reach close common limit mse estimation delta law full efficiency large mean error v b efficiency error similarity statement zhang asymptotic discuss introduction adjust discuss wang reference therein recently develop also well estimate generally equivalence ml estimate theorem concern know variance rather restrictive law denote obtain consistent say full parameter restrictive study refer believe reason least good estimation mat ern reference herein equivalently law already directly ml result large particular case implication possible extension grid mind decrease datum likely accurately log determinant worth similarly obtain true square error one line candidate say generality firstly check collect interest ern lemma denote g g equation fx dx fx dx part boundedness enough let filter integral claim proof observe g dx dominant j w j take un lemma prove un lemma claim h h w g algebraic schwarz paragraph g w sign w w easily obtain close remain g g g g boundedness scale ann correlation toeplitz system long zhang likelihood ann spatial statistic series near randomized base noisy single http mat ern correlation prediction http explicit estimate c k mit texts stein data theory krige spline observational conference fix journal spectral fields optimization zhang inconsistent interpolation model zhang toward framework spatial zhang hybrid mathematical call energy ml multidimensional gaussian belong mat ern family regularity white noise square range analog grid toward close analog benefit implementation discuss obtain observe dense regular mat ern commonly use stein correspond known autocorrelation stationary ern process autocorrelation c interpret range correlation independently drop concerned dense grid two successive ideal without correlate follow less analysis useful insight approximation article setting measurement equivalently call actually restrictive since course digit include equal due ill may possibly rescale observation vector whose k k td stein expression effectively parameter especially costly eliminate search maximizer zhang also recently costly likelihood range maximize respect say explicit score idea even wrong remain du wang stein zhang obtain firstly role correct otherwise secondly propose maximization estimating case equation r n energy underlie sample gibbs candidate equation estimate equation new parameter derivative r w r base first plausible idea proposal instead fix coincide thus second known case estimate justification isotropic field time lebesgue list article context much et series exist implementation chen al rather strong insight capability complex limited lattice grid use mat ern randomize principle mat ern family restrict spherical mat ern notation adopt thought framework wrong quite behave toward monotonic plugging quantify provide indeed asymptotic reach efficiency large
proof adopt analytic notation function log denote notation paragraph entropy express cc else continuity know variational alternate introduce support outline later support inequality lead prove support general omit fix supremum full achieve since shift even scalar add result assume differentiable class form denote treatment test potential testing include channel code detection normal represent unknown hoeffding alphabet support sequence hoeffding cardinality weak result hand chi degree report weak elaborate section decay irrespective argue bias address observation draw divergence decay address later guide constraint application alarm probability hoeffding variable moderate false alarm predict simulation hoeffding alphabet show compare quantity alarm simulation central see note give difficult instance nc integral divergence ball hoeffding address issue universal partitioning alphabet extend hoeffding sense apply conclude yy instead xx n trivial suppose class yy partition incorporate regard alternate ratio alternate may anomalous desire finite alphabet letter shannon expect optimal letter bound away sequence distribution symbol length symbol else sigma algebra symbol satisfie else establish cardinality alphabet conclude hoeffding reduce paper paper consider approximation problem xx known express eq onto exponential di divergence express project recursion current scheme robust section test ball analogously q next establish basic ball collection g intersection zero supremum support hyperplane obtain h f follow exponent constant satisfy q adopt criterion evaluation hoeffde notably evolve type limit exponent exponent alarm exponent pearson supremum exponent hoeffding describe knowledge yet optimality choice proposition support eq likelihood test equality exponent clear follow achieve h proposition separate mean disjoint theorem prove approximation alternate several equivalent relate likelihood representation infimum rhs interpretation identity supremum infimum minimizer consequently identity conclusion optimizer know correspondence generalized analyze let ml estimate n value assume attain solve reverse define divergence maxima identity test use restricted denote optimization use divergence evaluate divergence reverse establish characterization projection achieve must vanish first equivalent r r convexity follow minimize establish reverse infimum achieve onto hyperplane reverse onto family geometry linear suppose supremum achieve r linear states support furthermore minimize test asymptotically finite consider solve composite exponent achieve discussion follow proposition moreover f dl f r conclusion solves establish easily composite corollary alternate refer detail prior class coincide strictly log class choose suppose use consistently follow identity statistic establish involve specialized establish specific support open supremum achieve independence satisfy basis lemma alternate q denote identify asymptotic bias alternate depend cardinality far f ii ii sufficient follow function class function part linear part ii linear suppose linear first f r second limit impose assumption ensure assumption whenever furthermore eq theorem derive interpretation call threshold alarm hoeffding threshold section divergence hoeffding uniqueness contain span constant assumption alphabet coincide thus application prove follow appendix theorem part denote divergence define suppose f f apply observation assume decomposition central variance dominate bias dominate term xx contain interior suppose together compact contain interior hessian asymptotic condition directional h decay eq gradient directional dimensional chi square variable freedom denote define condition optimality value obtain generality assume observation marginal law concern appear z nz concatenation requirement satisfied follow lemma neighborhood function coincide function satisfied substitute definite function open open uniquely respect consequently results derivative derivative uniformly u representation random give g gd derive limiting term apply g conclude since decomposition limit cauchy together prove apply third decomposition first generalization characterize variance statement define analogous verify rest hold establish note supremum linear ii establish follow r hoeffding much theorem informally approximation denote distributional approximation large chi square variance interpret result degradation parameterize alternate increase intuitive reasoning hoeffding alphabet choose chose basis test alarm misclassification correct roc curve vary hoeffding increase test suggest divergence hoeffding class match summarize suggest hoeffding exponent priori alternate belong testing incorporate reduce variance hoeffding dimensionality phase ensure alternate corollary make approach universal testing test applicable satisfy central research synthesis adaptation extension markovian extension detection synthesis report address computational report exact optimization tractable building surveillance acknowledgement thank ar behavior continuous version appear identity follow follow simple place
dms grant mh eq q respectively henceforth nonzero regularize estimator model learn much reference note concerned case set interest purpose similar proposition often certain weighted type estimator norm type estimator attain counterpart amenable unable attain precision omit focus require idea vector vector respectively regularize satisfy inequality maximum least illustrate focus use two bound select notational ease verify case always let eq also moment coherence next comment tail error typically value condition domain restrictive typically restriction proceed constant restriction cf need see probability ease follow derive low j b nx put inequality group far choice continue put suppose later indeed make union large outside lemma note assumption combine get complete e right great contradict assumption suitable mle respectively condition bound regularize almost brevity omit get condition let family borel let condition gx inequality interval length continuously extend open contain x get case state regularize include purely second necessarily contain analytic become hard k proposition trivial constraint order nc treatment analytic identically reflect nonlinear unknown become
point large indirect measure indirect might figure map normalize distance viterbi noise intensity intensity behave differently small map superior ml range intensity complicated although perform similarly intensity theoretically examine map estimation consider intensity estimation agree vanish agreement upon intensity operational regime define like interesting slightly linearly stable intensity less stability concern range application carefully development analytical average figure furthermore think semi estimation knowledge remarkably modify interesting beyond process map like behavior observe binary attribute hide dynamical exact acknowledgement thank sciences institute support u nf institute mail sciences institute usa edu maximum posteriori hide markov energy spin focus finite value exponentially reflect zero entropy finally noise intensity reduce phase ising order various bioinformatics economic many hmms amount state corrupted posteriori approach find maximize map readily viterbi extensive estimation surprisingly clear state insufficient understanding infer complete picture know method sequence simple employ comparison average pr hmm relation computer science physics way average cost pr pr logarithm respectively temperature symmetric hmm rich sequence linearly intermediate many solution problem reflect entropy ise intensity poor furthermore regime phase ise phase transition general section also discuss concrete binary paper discussion respectively write probability influence describe far assume generate offer method generate p equally sense typical set overall converge nearly delta strong dominate prior distribution put source viterbi possible repeat ergodicity argument solution observe ise due employ see moment another quantity observe overlap discrete parameterize distribution unbiased error refer refer depend act realization composite likewise combine spin model external govern spin spin interaction constant uniquely determined probability constant situation spin spin align note situation field uncorrelated markovian xt partition terminology statistical physics temperature fix pick state minimize configuration equal ising express free define map account overlap sequence hamming limit equal eq spin act spin temperature limit repeat express recursion govern depend value take assume infinite check positive integer never limit add mix original identical merely make realization composite take depending recursion turn task characteristic study next return free write take entropy kronecker probability typical physical extensive act spin spin amount temperature note check recursion binary fig respectively reader easily generalize graph cc cc cc cc transition add bar compose hence actual element serve indicate general read see fig matrix tensor also block zero augment markov go value amount change matrix sparse treat quantity process process moment overlap former free redundant probability symmetry lower relevant indicate normalize half energy one confirm note depend formalism require moment overlap see inequality realization assume transition regime contribute write latter lead continuity energy stationary probability maximum ml entropy uniquely stress
hx n bb invertible central wishart density eq polynomial polynomial hermitian integer write negative integer eigenvalue polynomial notational simplicity symmetric degree vary coefficient expansion replace differential hermitian verify coincide polynomial hermitian following give symmetric k equation hermitian case case three hermitian let coefficient j include one partition polynomial c yy table symmetric cc cc cc ccccc definite put differential invariant invertible notational convenience real symmetric differential invertible consider differential hand put call plane prove fy assume fy md upper z dx analytic get theorem absolutely laplace absolutely I mx ba yy k hermitian summation special absolutely terminate hermitian matrix suppose assume lemma transform x ng z q z hand analytic wishart distribution generalize case aa w dd differential form q give density eigenvalue present denote summation partition function corollary put corollary large resp eq resp resp resp take grateful comment point lemma cm polynomial keyword wishart matrix mathematics china china deduce formulae matrix distribution small wishart polynomial real early introduce density wishart tool appear polynomial argument definition appear involve partial easy polynomial division division argument argument formula wishart derive polynomial function eigenvalue denote complex division matrix put hermitian eigenvalue say aa na na ij ia determinant volume dx dx dx ss constant invertible define invertible follow come paper element eq dt ss td st
asymptotic fulfil easy fulfil ergodic fulfil strictly function correspond limit process therefore composite statistic consistent estimator obvious contribution modify statistic equality x contradict alternative equation consistent easily alternative eq kullback alternative chi square test find interesting test ergodic de universit du france goodness fit diffusion hypothesis suppose trend coefficient asymptotic von test particularly transformation modification composite basic play special role bridge real I hypothesis function diversity test come er von family er von hx f metric kolmogorov statistic allow statistic tend similar ergodic diffusion suppose hypothesis concern mean basic hypothesis trajectory differential trend hypothesis satisfy bound solution condition fulfil q ergodic law simplify exposition stationary asymptotic interested asymptotic moreover test test q hypothesis constant test belong moreover test special propose similar goodness observation study chen reference therein goal test direct kolmogorov hypothesis er von test estimator kolmogorov test hx relation statistic invariant asymptotically equivalent estimator course choice make therefore q process work certain let median testing trend fix side alternative value first von normality class density mild condition normality therefore test simplified use avoid q wiener introduce condition fulfil hypothesis time density suppose stationary eq law central limit distribution multidimensional law last double sided wiener formally follow distribution wiener property ergodic integrable estimate q strictly monotone decrease note first term alternative course correspond integral similar goal choose q vanishing
close lasso within lasso multiply run active likely support associate assume cn polynomially thus grow pattern union procedure homotopy method development method solution compute computing quantity path unique variable become bootstrappe pair different replication bootstrappe run avoid lasso objective penalty quadratic path within follow homotopy make per homotopy complexity put bootstrappe residual replication bootstrappe even require active propose cm cm cm white correspond equal rest consistency bottom consistency similar know bootstrappe design normal covariance satisfy one lead lasso figure replication variable value regularization noise bootstrappe bootstrappe residual report replication exactly present illustration tend fast pattern relevant proposition top plot satisfied path since bootstrap lead behavior resample knowledge need bootstrappe slightly lasso cm consistency bottom marginal probability select way obtain design middle right top satisfied consistency plot create consistency region bootstrappe early regularization I value bootstrappe residual replication increase essentially inconsistent variable leave intersection replication eq zero black resample know generating bootstrapping except middle relevant design design design simulation bootstrappe consistency multiply marginal study general scheme see bootstrappe behave differently resample noise residual resample behave correctly compare top currently various replication bottom plot analysis lasso general high bootstrap linear bootstrap residual work bring variable enhance thank also bag random forest minor soon union bind z optimality condition noiseless sufficient eq need sign j j j j c satisfied occur great p follow p long p probability covariance complement rank strictly n result inequality pattern absence zeros eq q j j thus allow well weak consider term upper bound use upper well infinity elementary standard constraint notation eq optimal full w c soft triangle shape bound moreover design constant normalize q apply j q asymptotically normal get apply c select variable type upper c hc use appendix ac concentration appendix variable hoeffde distribution normal use nz follow upper c first select thus lead allow lead desire j follow reasoning include inequality upper j j appendix eq exclude bootstrap satisfied give great u regard c combine outline j consider eq lead constraint x n follow non iy iy k k n q thus eq extra moreover hoeffding need n variable set relevant variable lemma select q use appendix q assume p n approach proposition proportional great full analysis outline together note lemma j c j j z reasoning c z large rest proof along line affect pattern q sign pattern lead desire union thank work support france project cm lasso lasso decay regularization correct specific decay enter select run replication support lasso selection novel setting provably attract lot machine processing refer effort dedicate efficiently homotopy regularization know loading selection I load certain correlation variable irrelevant light lasso design dependent step procedure main contribution propose approach resample focus detailed asymptotic pattern estimation extend number variable much select enter variable tend one latter would estimate irrelevant enter support sufficiently eliminate resample dedicated availability bootstrap support consistency regular support bootstrap rather consensus scheme agree case provably consistent also potential additional bootstrap bootstrapping bootstrapping type bootstrap moreover empirical setting bootstrappe lead consistent bootstrappe residual currently unable bootstrappe residual high prove residual run order new low fine new regularization properly version bootstrappe run homotopy bootstrappe residual design time lar section example follow extend denote denote matrix singular element eigenvalue subset vector element index include index consider form setting depend ratio potentially large high fix design identical distribution invertible normalize constant loading setting section extension lasso population e regular include moreover sign equivalent consistency tend infinity derive non bound compare due review power exclusive regularization path many finer particular asymptotic regularization illustrate zero generate sharp situation notably simply exactly correct tend see assume homotopy enough provide noiseless global minimum desirable regular exponentially pattern limit unstable limit exactly hinge sign away hinge path tend slow sign sign noiseless sign first away hinge regularization path eq equal occur consistency sign tend statement probability select correct pattern regime regime tend rate agree probability tend pattern pattern tend consistent probability appendix note early design ps fs assume ps universal inequality detail behavior small one tend previous zero otherwise next section tend tend stability relevant variable non trivial could fast tend sign norm obtain n simply fast constant appendix sign give currently explore possibility share aspect consistent induce tend zero tend infinity hope satisfy least way resample next finer allow presence without consider consistent upper certain relevant j n probability selection selection claim tend zero tend irrelevant unstable several copy piece two bootstrap distribution replication set give original restrict estimate active probability incorrect pattern denote generic p include irrelevant original replication term natural replication feasible us copy design active proportional remove irrelevant variable scale incorrect selection exponentially fast multiple copy split enough piece happen main goal show use bootstrap copy cut piece small matrix strictly fc piece partition present two section together replication consider I n nk select twice note replication show running procedure q large enough bind incorrect b b design constant scale polynomially currently try bind extra replication remove variable overall incorrect tend exponential copy minimax improve research finally explore way intersection appear replication important loading however scope alternative resample residual adapt replication consistency resample scheme differ slightly
two specie restrict common far large population interestingly study consistent gene flow nonetheless support population applicability model single model estimate attribute origin exclude check mutation pattern match use remove software total simulation per report acceptance variation acceptance simulation glm smoothing reject favor among discussion still computation tackle innovation algorithm adjustment term frequently relationship summary summary advantage tolerance value distribution glm spirit advantage glm likelihood glm glm approach framework glm type sampler mcmc glm great glm set simplicity implementation standard package show factor population structure among strongly significantly think exactly never summary namely attribute previously arise almost impossible simulate value fall model acknowledgement author david recently realistic population genetic calculate analytically change abc key innovation adjustment allow large computation realistic magnitude adjustment allow integration factor methodology population ever powerful computer refinement gibb become tool scientific past bayesian distribution turn alternative hard core discussion issue create dna determine quantity bayes often fact realistic population genetic stochastic sampling simulate summary number set capture rejection question scale mutation extend multiple summary von vector statistic space tolerance zero around truncate accept b n f dirac center distribution process truncate hard q give exactly truncated make guess parametric posterior full localization exhibit simpler easy sample reasonable time respective save whose lie list retain yield tolerance small unknown formula complex curse raise acceptance rate approximation partially process et variant metropolis hasting term directly sequential monte iteration method rough within ball retain basically summary statistic local implicitly vector suggest many density posterior density closely center true empirical statistic adjustment prior distribution posterior posterior prior actually vanish moreover clear abc glm literature unfortunately share statistic truncate constant account local linearity statistic summary sm empirical put distribution sharp get curve might several explain introduction normal covariance normally multivariate already square quite insensitive influence estimate dropping exhaustive treatment book observe truncate perform proof multiplicative exceed impractical calculate marginal integration univariate component normalize analytically numerical speaking posterior element selection principal method nearly impossible curse logit model hand readily bayes factor determine marginal density check rejection estimate aid statistic count fraction fall center glm run q probability favor model comparison obtain rejection glm analytically posterior infer mutation site sequence bp tolerance level estimation always report replication assess infer analytical calculation measure curve equal vanish uniform figure abc glm rejection algorithm tolerance value rejection posterior observe tolerance rejection abc
development circuit evolve fire relatively minute day node episode occurrence ratio dramatically development medium provide despite large firing day day fire rate episode connection include increase methodology track development show dramatically medium h display overlap occurrence episode discover functional connectivity spike connection frequent episode inter event discover serial connectivity method allow episode strong interaction among delay cause propagation delay chemical building number occurrence serial episode show occurrence episode directly among demonstrate spike discovery frequent pattern discovery fix consideration strength context symbolic time episode motivation dependency among symbolic source conditional event characterization strength present application delay general delay record connection delay event currently episode episode connectivity address future dr simulator counting report general center university ann structure activity multi challenge repeat activity group neuron neural temporal framework counting occurrence propose statistically significant counting strength connection resolution previously pattern strength present pattern discover neuron train array episode occurrence statistical interact time characteristic electrical spike sequence spike refer multi spike train contain fire spike individual activity group datum identify network neuron underlie array tool lead collection multi spike neuron critical understand region act currently amount temporal efficiently analyze appropriate technique intensity stimulus aim pairwise small neuron frequent episode discovery temporal firing spike train temporal mining deal symbolic stream specifically sequence event analysis multi event spike neuron spike interested collection event prescribe firing follow fire episode constraint delay compute frequent episode precisely pattern approach major implement algorithm episode statistically threshold random connectivity episode overlap episode implementation mining work recurrence relation result characterization distribution moment section discuss connectivity strength simulate demonstrate usefulness employ delay neuron fire pattern serial extended encode spike fire rate suggest produce spatio temporal discover frequent episode piece connect subsequent episode discover reconstruction popular detecting repeat shift spike train early currently neuron temporal circuit involve suggest base ability simultaneously neuron infer firing computing involve frequent view time order denote denote type finite denote event type collect spike pattern episode general partially order consider order sequence event type serial episode order serial episode event appropriate episode neuron neuron occurrence episode serial datum specific firing processing discover frequent episode tractable episode focus episode episode non time period occurrence illustration episode say every counting algorithm style wise discovery datum episode pass count episode count frequent episode use count candidate episode efficient count occurrence apart consideration computational efficiency frequency spike serial episode would episode base count occurrence occur repeatedly frequent serial episode structure frequent episode time discuss set episode frequent count neuron want count indicate restrict difficult threshold episode limit assumption interesting notion look threshold indicate probability introduce refer time bin delay neuron delay conditional weak know hypothesis use detect episode simple episode occurrence observe time period neuron interest time word connection overlap fire furthermore identically success hence occurrence binomial trial occurrence bin occurrence occurrence interval binomial success probability readily take side independence decrease expect vary parameter span episode episode occur characterization mass moment function use exist inverse time w moment numerically mass cumulative good normal range easy note except obtain variance replication independent hz firing rate quantile plot consider approximation reasonable refinement compute threshold count skewness fire hz suppose conduct episode neuron independently fire use interval test hypothesis demonstrate variety strength occurrence map true confidence replication connection coverage mp arbitrary episode expectation expression depend neuron episode become dependent demonstrate combine statistical efficiently discover strength complicated firing neuron mining episode occurrence episode event constraint map count episode occur episode fire episode would respective count divide datum functional delay strength ratio mining interested connection episode ratio frequent episode candidate count much node permutation node episode frequent true connection embed node rank connection count node eliminate
describe bayesian imply thing rule stop irrelevant entirely collect analyze contribution hypothesis ratio use successive ratio martingale martingale martingale reasonable independently red realization martingale false dotted dotted line evidence martingale nan hypothesis martingale compute role easily martingale show figure test martingale even though b logarithmic figure unbounded trial reader notice martingale around reject calibration conditional increment variance increment iterate tend almost surely eventually obtain hypothesis want conclusion conclusion correct choose measure conclusion conclusion nan test leave corner observe figure figure generate sequence slowly converge show enough reject two false hypothesis identically independently martingale test moderately conservative test stop whether positive precede paragraph test matter rule select nevertheless report value follow follow matter know advance asymptotically tend reach conventional size define q take value stop chance stop interpret report guarantee advance reject choose close possible significance satisfie define require wider determined much wide question know advance interval slowly sequence weak detail stand thus posterior always observation ratio spirit interpret reject rejection stop really correct hypothesis fully mind resource decide sequential take shall seem stop statement consider irrelevant treat statistical advance come e discussion report continue naive hypothesis law happen term pointed happen something without know frequentist mainly notion adequate purpose hypothesis fix consider hypothesis martingale accord decomposition discard compare essential hypothesis modification introduce principal often say process member process exists stopping stop popular standard vi expectation adapt strictly decomposition test therefore purpose test martingale dominate martingale local continuous essential admit brownian motion vi local martingale martingale vector harmonic nevertheless fail detailed calculation martingale replace purpose martingale represent martingale belong dl dl times integrable integrable belong dl martingale sense martingale let martingale arbitrarily dl vi stopping dl appendix first observe iterate version euler theorem pt iterate entropy equality rewrite far rewrite finally write log kind hypothesis alternative finally detail roughly size tendency article shape discussion gr lead correction improvement section grateful development type de france department mathematical logic mechanic department computer science university unite inverse value interpret bayes factor large attain way eliminate inverse characterization increase function eliminate regard simple well relationship nonnegative visual initially evidence value begin martingale mean risk cumulative lot look make look well relate familiar support alternative say value stopping leave great value far understand complementary answer martingale testing every perhaps shrink shall say exist martingale randomness treat martingale dynamic measure establish interesting reader familiar mathematical notion answer converse exist scale parallel supremum martingale bayes factor bottom probably people picture bring algorithmic randomness reader familiar field historical discussion formalism theory reader mathematical section coin fair depict prove devoted mathematical introduce wide conservative test explain explain maximal current tool carry thesis start capital capital begin strategy tends infinity zero zero martingale idea predict event subsequently role randomness historical perspective behavior tool mathematic subsequently important technical tool mathematical time survival mathematical slow concept test nothing reject establish pearson pearson hypothesis likelihood reciprocal ratio density say continue reciprocal martingale directly goal nothing reciprocal role become increase importance bayesian start often call bayes ratio multiply ratio factor informally statistical probability justify rejection differently report hypothesis reject pearson pearson probability advance merely report significance adopt point define bayes value version narrow wider equality version attain conservative version conservative conservative narrow recall triplet value measurable random take tt interested formalize convenient specify martingale order index whenever adapt measurable resp martingale algebra almost surely automatically limit surely see vi satisfie namely contain subset space complete usual se vi early define value prove test page vi call relate factor discuss informally derivative satisfy bb conversely whenever nonnegative measurable relate concept consistent call usually measurable set satisfy also satisfy treat tie carefully satisfying generation precise statement assertion former assertion whose reciprocal give factor include completeness practical arbitrarily correspond statement almost bayes bayes martingale moreover measurable test convergence necessarily expectation right evy page martingale imply ta final stop vi supremum true supremum point test inverse test artificial construction directly practical help give explanation discrete level merely test equal whether whether martingale start capital last amount precise martingale supremum construct capital capital everything check time nontrivial integrable algebra complement algebra generate either borel subset borel check suffices borel subset equality rewrite strictly increase characterize increase say strictly dominate admissible admissible prove give perhaps borel algebra bayes check hold c fp formula random fp fp inequality stochastically inequality part equation give produce admissible primarily interested behavior take significantly etc example let increase martingale independently demonstrate convention say martingale martingale admissible continuous start suppose increase bayes fx ty fx ty statement argument show check martingale modification precise equip complete
series devise markovian scheme thus usual sampling simulating conditioning truly move derive ar good calibration particle figure evolution particle observation autocorrelation graph configuration around particle particle around observation grows show particle evolution simulation plot index could possibly occur large simulation higher demand iteration particle hour severe rate value row write http simply calculation gibbs advantage comment loop paper mention well deep extent finite horizon nature noise mechanism rao denominator eqn past obvious information provide fortunately value care selecting evaluate bias stochastic volatility example et al sufficient reasonable unable likely explanation good implementation may optimal nest perform hour report meaningful paper b theoretical paper complicated evaluation setting offer state comprise trajectory enough remove limitation smc come complete smc ask resample rao proposal distribution technical
shift pool especially yield research effort towards multiple comparison problem kind fitting package implement suggestion multiple comparison arise frequently study participant interest etc arguably yield simplest comparison correction pose burden type correction reference american control practical powerful multiple discovery testing report department stanford analysis stein generalizations american rates bayesian journal biology hill use hierarchical american rates american operating characteristic extension false discovery rate grant liu identify microarray bioinformatics sometimes journal american comparison health trial health journal american stein fourth berkeley ed california thresholding possibly sparse parent biology teacher effectiveness york city economics education nan journal american association estimating journal american association center education c office impact student american economic review may estimation randomize experiment journal statistic potential assessment education journal behavioral w inference gibbs j multiple encounter try state journal validity study journal resample testing adjustment york v v pt www hill http www edu pt apply find inference setting seem challenge paradigm correction moreover multiple entirely hierarchical building partial pooling toward typically keep center adjust multiple make interval wider equivalently adjust width yield efficient low comparison concern bayesian comparison statistical nearly social physical find simultaneously evaluate question compare point compare several different compare indicator rate state examine effect examine impact program outcome comparison conclude set test even nothing go additional serious propose review relate comparison concern test may identify statistically fact real paper perspective rarely believe multiple insufficient modeling correspond within appropriately shift toward often shrinkage classical adjust wide adjust interval appropriately sense interval say make likely adjustment detect difference many problem examine many comparison significance paradigm question simply put problem put burden goal procedure realistic comparison illustrative solution outline correction several relatively use classical health development intervention birth service home intensive care program randomization take within site birth experimental slightly complicate purpose randomize block eight overall effect give composition site program implementation varied site like know statistically concern make false risk sometimes arise interested whether effect site might look like j program way represent site allow significance site hypothesis incorrectly test least raise perform independent eight site significance would chance reject popular evaluate test specifically work calculate divide assume example overall significance translate test wide confidence estimate along correct nominal significance reject intervention site adjust reject nan hypothesis expense type reject claim average false reduce researcher goal dependence variety correction bootstrapping method test example focus reduce rate instead false rate control rather rate powerful paradigm rate independence test test fdr make field would expect vast quantity effect examine al science application likely thousand hypothesis less truly zero distinction formally control rate already perform variable group statistically significant difference perspective significance yield correction probably helpful would one significance alternatively sometimes specify perform expectation similar comparison tie hypothesis moreover fail test goal propose circumstance present perspective implication argue perspective simply perspective primarily establish site effect concern accept fact ever importance test similarly occur accept truly group shall discuss statistically serious concern might fact phenomenon treatment new york fact reverse analysis concern compare different none might want effect actually near likely instance statistically actually plot estimator distribution great produce less uncertainty deviation examine automatically increase something go help view within error substantially simple score parameter allow intercept across seem think realization addition specify kind could real power notably intercept model pooling treatment site site often figure plot next interval correction horizontal dash line display horizontal solid quickly statistically significant effect site estimate really reflect shifted estimate predictor partially pool fit group surface mean pool sense intuitive uncertainty estimate population treatment site true certainly really inference essence use effect estimate site actually ignore site allow site site sense ignore find site site level great site get less uncertainty site trust illustrate run decrease site sample result display increased bootstrapping result right way reject hypothesis procedure ignore comparison raw graph actually though clearly nothing piece default graphical obtain average mathematic student state average student take average distribution parameter case ambiguity claim confidence make little paradigm examine parameter fit simulate effect state purpose classical cutoff state simulation claim plot significantly one blue classical summary confidence central comparison classical set extreme true evidence correction evidence comparison comparison correction ccccc treatment raw e one additional correct deviation treatment evidence little pool toward none comparison close statistically discuss chapter meta new notable analysis effect estimate effect get standard eight order sort situation might multiple actual raw happen statistically study bayesian estimate likelihood mode zero bayes pool toward insight deviation plausible simulate eight value eight separate distribution group analysis compute count statistically estimate time error number correct simulation statistically sign correct avoid statement bayesian get sign correct essentially multiple comparison way estimate statistically comparison bayesian occur school classical inference inference correspondingly statistically happens repeat simulation treatment statistically statistically whether classical price pay claim huge child york city assess factor determine effectiveness finding teacher effect moderately deviation level system researcher use approximate model variation teacher effect teacher effect persistent background decade teacher broadly school like award leave problematic aside analysis rarely ever address involve comparison try get compare thousand distinguish teacher individual therefore concerned type fact analysis health find likely rate attractive participant survey statistically significant versus plausible comparison physical survey use measure five point scale attractive vs category possibility compare people compare compare comparison summary wave wave wave statistically study multiple comparison bad idea percentage correction would change significant properly multiple either proportion measure grain pattern beyond importantly sample simply literature analysis people risk type reasonably uninformative prior heavy tail effect reveal information effect well percentage program multi site expand birth weight birth group differently intervention weight additionally treatment across group effect site child birth weight low birth allow intercept site hyperparameter plot set serve group arise site birth child stable treatment correct birth child sample site conclusion analyse none close zero adjusted width four include close researcher attempt impact many eventually positive significant chance require big conceptual natural describe effect think draw knowledge big trivial
computer university correspondence edu cm edu distribution class necessarily follow normal positive independent chi square variable chi square especially two genome estimation particularly second em infer em need substantial burden eliminate mathematical practical extensive study two statistic find typical gene weight square association association complex disease evidence single interact create super allele trait et al design statistic goodness quadratic sample statistics chi nan chi distribution permutation power al lin attempt specific advanced similarity normal normal inaccurate sample statistic assume normality test write follow discuss systematically central alternative statistic square comparison procedure permutation permutation increase test significance permutation procedure intensive estimating typical power significance power must calculate apply association genome many order account comparison additional arise permutation current generation study un trait association estimate computational arise category rare phase intensive distribution solve et rare merge common thereby reduce efficient fast permutation moreover prune rare situation consideration apparent way forward generalize paper asymptotic value power need permutation assess robustness use illustrative display positive case definite assume symmetry study plot distribution distribution likewise assess performance distance al need study unknown population count notation k ks hypothesis focus multinomial therefore variance asymptotically positive orthogonal tr z r w w asymptotic assumption hypothesis let represent let semi chi chi approximate adjust statistic chi freedom base approximation chi chi tr quantile reject level formula indicate coefficient chi approximation calculate major inaccurate high dimensionality ii chi square positive definite similarity similarity consecutive subsequence similarity formula simple carlo simulation random use base formula procedure much simple fast alternatively eigenvalue positive negative estimate chi difference variable may monte carlo technique describe study find asymptotic md b kk discussion case singular statistic appendix multivariate positive definite definite central shift square singular easy verify propose shift chi quadratic idea liu al able fit list formula refer square liu df df df critical note find value rejection usually correspond replace estimate alternatively q quantile automatically conclusion et al eigenvalue approximation power singular diagonal aa sd distribution singular dimensionality convert non illustrative statistic al actually long necessarily z weight moreover therefore freedom singular conclusion show chi chi square chi illustrate deviation claim hypothesis normal previous discussion follow chi large different normal inaccurate difference rate approximation demonstrate propose yx calculation central write x x software integrate source file approximate quantile size specific power file http edu software set et ii table complete length counting measure proportion seven simulation approximation nan hypothesis examine plot million simulation axis small million significance million combination level chi chi procedure proportion true one hour ghz ram estimate however second procedure stay summarize table approximation prefer simple high list approximation evident provide probability estimate value large mean million permutation give conclusion base date examine examine purpose simulation note good approximation hypothesis figure examine kolmogorov true effect nan moreover situation usually formula case value examine approximation tail parameter fairly moderate power claim normal rate substantially show variance suitable figure chi small size increase also become acceptable figure far compare normal square kolmogorov von combinations second illustration observe chi distance especially candidate distribution around phase information unknown distinct category matching length test indicate significant measure error unknown counting measure additional require significance approximation describe easily calculate required population separately em package start use em refine calculation minute ghz gb ram finish single calculation procedure discussion analytic quadratic statistic use test well efficient way calculate specifically show quadratic combination chi square distribution situation square distribution liu degenerate generally speak approximation deal nevertheless latter moderate probability matrix computationally less intensive similarity perform dimension decompose appear well property hard accurately population recognize lead error testing et several develop structure al focus population square would similarity conduct association genome association fast estimate genome region manually limitation length define method explore acknowledgement support grant institute medical center ny suggestion student statistical association trait correction test statistic principal hessian letters b genomic association mf improve relate chi american estimation molecular frequency population frequency available via http project package liu h zhang chi negative quadratic form central statistic lin base association test effect population genetic association nature nj I na wide association jk na genetic marker american journal association american human de population ga trait linkage complex trait human chen zhang http weighted journal american association similarity goodness fit zhang variance kl trace application independent random hypothesis start chi approximation shift see diag ta liu liu al formula quadratic form degenerate multivariate c kolmogorov piecewise let keep generate kolmogorov check von formula
qp challenge accordingly reliably precise solution fairly accuracy future rule method interesting efficient deal gradient descent hard coordinate might decompose iteratively hinge leave thank constructive comment soon helpful rewrite mkl follow turn know lagrangian express lagrangian sufficiently neighborhood proof assertion remark mm ac new mkl type iteratively solve qp proximal cost minimization roughly proportional active aim scale mkl include net efficient exist powerful rkh choice mkl combination assume function source combine weight set example heterogeneous numerical text link principled manner challenge kernel give put evaluate mkl various programming sdp order solve sdp heavy propose kernel update nice tuned solver program approach utilize cut plane weight instability solution sequence mkl drawback call improvement rather scale show behavior solve linear qp newton solve qp qp grow article mkl norm formulation introduce also formulation sparse estimation efficient view efficient scale call thousand original presentation primal mkl problem believe compare interested svm lp qp minimize cost kernel thousand train less second recognize initially think point often outperform kernel accordingly instead well mkl uniform combination elastic smoothly connect paper norm allow mkl review slightly kernel combination general organize mkl weight section extension algorithm primal application proximal carry efficiently exploit mkl section mkl elastic net mkl formulation special elastic mkl efficient fast block summarize super appendix matlab software regularization block mkl direct extension formulation square version later belong output usual setting gram fix consider rkh kernel sec also rkh accordingly hinge loss mkl learn objective without get serious way prevent increase explicitly follow inequality arithmetic take formulation motivate squared block constraint instead solution map let minimizer formulation minimize square formulation regularization subdifferential convert mkl notation attain problem nm nm simplicity rewrite k eq section extension proximal twice differentiable discuss minimization algorithm iteratively minimize proximity proximity adaptively objective last proximity try keep solution proximal seem mkl primal proximal mkl minimize original mkl necessarily interpret gradient subgradient equation zero q subgradient learn unique scalar solution q converge optimal linearly proof mkl dual efficiently variable result apply mkl threshold proximal mkl minimizer update tb mkl augment prox prox repeat lagrangian mkl lagrangian multiplier maximize mkl lagrangian respect problem lagrangian minimization lagrangian see note minimization respect conjugate second conjugate hinge loss illustration regularizer envelope soft operation algebra convex positive semidefinite generalization eqs proximal ignore problem eq dimension conjugate whereas smooth every iteration follow hessian efficient require correspond sparsity make formulation dual region prox point interior intermediate hessian objective minimization call mkl update correspond lagrangian technique search minimization inner use unbounded attain boundary situation separately line last net mkl uniform combination formulation separable handle question technique conjugate immediately formulation equivalent mkl mkl mkl table discussion relation rkh obtain optimizer formulation correspond multiplication regularizer exponent exponent mkl mkl mkl mkl original iteration need proximity operator regularizer envelope proximity follow note along due elastic mkl regularizer write proximity obtain therefore conjugate q regularizer convex obtain eq envelope conjugate regularizer cauchy schwarz envelope modification case envelope become manner generalization straightforward need prox matlab general loss block dual primal dual proximity convex regularization smooth outer dual show elastic primal q expression rewrite follow differentiable regularization penalty function primal compute primal q kkt therefore primal differentiable e loss resolve elastic net block norm elastic net mkl cx net kernel solve norm calculation conjugate formulae exist basically formulation say constraint give rkhs corresponding consider instead average scheme convert one inner easily publicly efficient updating update differ totally update descent envelope envelope fx fy fx xx proximal update envelope increment next fortunately envelope smooth optimization discussion envelope optimization norm mkl regularization norm mkl formulation reason utilize bfgs special case set mkl regularization confirm binary uci repository logistic loss report elastic describe optimization regularization candidate gaussian e normalize experimental choose repeat run convert duality tolerance compute dual multipli project domain dual objective mx mx j dual kernel publicly available code solver program inside performance summarize addition standard show tend fast factor factor increase active dataset iterative procedure accuracy regularization mkl hinge seem perform elastic varie logistic hinge strong hinge rapidly accuracy nearly identical hinge regularization kernel net much block decrease regularization uci objective current refer outer line search gradient obviously outer outer iteration kernel become intrinsic small drastically light per require qp heavy relative duality gap cpu dataset see drop fast support decrease duality gap order duality gap gradually drop early
jk jk jk jk jk q feasible recall magnitude birth make modify appropriately live location calculate nearby location assume generate deal issue monotonically indeed dominate location gain dominate value solution unnecessary thus large estimate present study investigate four namely block arise function simulate spaced signal add take replication posterior distribution add value parameter deviation trial establish wavelet estimator reconstruct signal cross false asymmetric wavelet block haar wavelet discrete cross area interaction ss bt false estimator noise replicate rr rr bt fdr ex fdr ex bt fdr goodness measure replication clear moderate procedure naturally cluster perform noise reasonably level future number point location modify tail behaviour behind behaviour adjust could also speed inclusion minor software include would develop method software implement describe request thank discussion reconstruct noisy rather make possibility correlate scale frequency lead structure analytically past approach regression eq spaced interval noise normally zero approach transform large behave transform distribute distribute combine remove transform small coefficient perform large evenly discard discard threshold wavelet propose median wavelet coefficient give zero coefficient capture wavelet allow prior extension area basic disadvantage long possible chain sure chain simulation stationary introduce exact section discuss coupling past compare conclusion discuss area produce cluster moderately order pattern introduce left interaction process neighbourhood set neighbourhood follow compact metric configuration suppose finite borel regular class compact case rest technical trivially subject true wavelet zero wavelet transform constant formulation prior discrete location variance wavelet nonzero view replace allow integer assume lattice concentrate zero observe pure support wavelet choose periodic equally simulation begin past follow stationary space stationary return I practice show goal go back finite space couple wish spatial point q respect rate value monotonic process evolve birth death birth death configuration constrain birth birth let birth death start death started accept evolve follow also equivalently sample require back time keep rejection lattice modify normal possible integrate lattice performing rewrite jk jk dd dd jk jk simulate process mark lattice process amenable use slight abuse third subset dominate intensity
track expert forecaster sequence switch limited formulate tracking hard maker tuple measure switch interval tuple average meta reference moreover generation tuple achieve version decision maker observe outcome remains consider bandit problem become always imply exponentially forecaster satisfie average implement compute essentially invert detail study multi maker round computed manner incur norm constraint first forecaster full maker loss correspond task extra maker model characterize task among course hard could example paragraph inspire loss incur may example loss whenever performance make markovian define big round action make single replace supremum loss need explain modify unnecessary multiply loss care include min addition measure incur round additional hard constraint design task forecaster implementation discretization take back previously index decision maker give opponent choose maker maker hard one eq must contain shift definition shift contain denote shift take aim maker minimize cumulative eq pick denote uniformly ideal forecaster distribution draw application give simultaneous round forecaster continuity forecaster regret bound inequality take convention bind obtain inequality appear achieve infimum regret exist take element whose cumulative infimum order action differ length action define close simplex take uniform neighbor distribution take value put together n n simulation propose although would typically markov pair form time convention chain uniform index take simulate simulate exact state programming discuss take approximate space efficient resort sequential monte broadly know ideally suit approximate formulae expectation function law large particle importance particle interact particle property population approximation target constant go carefully design importance step characteristic section approximate approximate version forecaster describe partitioning except last fix action aggregate super task able precisely restrict element start simultaneous action compatible partition super task loss satisfy forecaster run shift ensure choice complexity yield comparable moderate easily bandit super task unable unable computable position constraint put action position keep track row diagonal another trick berkeley use issue trick task half view es sup paris france paris en discuss maker simultaneously task related impose maker restriction tractable introduce efficient select short discuss bandit additive loss task considerable attention multi simultaneously relationship maker choose simultaneously repeat task put simultaneous motivating consider company several customer customer order company loss receive offer consideration suggest offer customer age company select batch offer budget company offer send customer response send simultaneously action budget constraint maker accumulate good problem play repeatedly game consider nash equilibria address feasibility game play reference parallel enforce constraint loss game loss difficulty requirement maker choose restriction maker survey show graph tracking decision maker restrict set limited bandit decision maker instead observe game learn finally infinitely maker function natural finitely discuss closely relate concentrate average feedback complexity order maker deal simultaneously index finite action space n action integer real associate task repeatedly round maker choose choose index assume among task outcome maker outcome vector completely arbitrary maker compare maker important maker restriction simultaneous action forecaster allow vector maker aim regret difference cumulative cumulative action allow basic maker vector outcome maker reduce expert history maker treat task maintain forecast least bandit solve restriction need satisfy structural satisfied well meaningful like basic maker freedom maker maker budget round make thing one integer value become value meta problem exhibit forecaster implementation proportional cardinality precisely denote simultaneous instance cumulative loss put mass eq tune direct application denote cardinality complexity prohibitive cardinality example shift lower exactly shift finally draw action accord reduce online short e maker assign round loss path exponentially dynamic edge graph draw online short example order short state hide mean satisfied thing four simple action space sequence one define underlie shift action spend view example markovian follow x put differently indicates impose already stay thing concrete rely transition transition transition eq one ready graph multi online short sequel round acyclic correspond task two associated round
total divergence minimize eq conditional rewrite pm ta ta hypothesis new agent weight update bayes suggest system arise fact system one illustrate statistical ask likelihood give pdf q likelihood informative change thus belief would mathematically imply independence result produce next obtain datum look almost exception posterior eq q obtain simulated datum order separate converge differ design toy mixture bayesian rule ta bias towards observe act furthermore uniform pm interaction instantaneous instantaneous value instantaneous deviation result imply either result whereas correct propose causal calculus agent bayesian mixture I integrate agent heart adaptive agent na lead output observation crucially vanish intervention present unique agent environment propose representation ai superposition previously expert like architecture stochastic find amongst usage principle ai main derivation inference rule divergence potential application would take similar bayes control translate stochastically formulate bayes relationship investigation engineering cb pz uk intelligence formalize sequence output possible actually compatible world obtain leibl world uncertain pure stream agent intervention calculus calculus model adaptive behavior stream allow approach control behavior intervention calculus bayesian kullback environment consider intelligence environment system exchange symbol symbol environment sequence perfectly tailor environment face robot endow behavioral problem act primitive distribution interaction uniquely determine define coupling system rise describe stream specify valid model true coupling stream produce observation history stream role observation sequence output stream stream
filter alternatively estimate systematic performance currently explore problem compress gaussian describe measure analogously consider minimum ht ht significantly transform sparse fewer suitable satisfy isometry recover optimization q effort cs generalization include also specifically causal discrete system impulse response signal neither recover uniquely turn belong dimensional ar isometry orthonormal build intuition practically relevant piecewise integral operator act sparse long orthonormal usually variation compress sense develop alternative filter impulse mathematically familiar toeplitz consequently toeplitz rip tool indeed property toeplitz idea stable idea deal turn every finite system order filter employ convolution leave pose duality speak indeed measurement rip projection provably toeplitz scenario random construction organize mathematical section describe filter state reader understand main proof address blind deconvolution section technique namely decode causal reconstruct autoregressive non autoregressive known driving assume vector task compress ar driving measurement ar standard setup ar efficiently variable toeplitz preserve speak assume shift version effect shift particularly suitable notational purpose submatrix compose rearrange equation use simplify project reason choose iid rip rip constant toeplitz construction projection projection toeplitz entry th order confusion true train refer possible decode similarly omp directly apply regard original signal still mind contaminate noise algorithm take equation follow minimization find derive recover invertible paper solve minimization equation propagation state isometry quantity obey cardinality direct sake completion unique unknown need process e impulse ar spike define jk spike amplitude satisfy drive equation drive spike necessary spike consecutive solve random benefit random naturally toeplitz exploit consider blind deconvolution reconstruct coefficient problem blind deconvolution simplify compressed identity well completely version ar matrix solve problem show state technical denote noiseless define comprise comprise u condition small tx cp practice generally persistent converge compressed say energy relaxed simplify condition let scenario scenario bernoulli sum hand spike align signal phenomenon illustrate experiment ar blue curve see blue spike db sign spike sign spike condition satisfied also assumption snr small spike ensure spike zero lasso estimator hard analyze clear kkt choice theorem provide ar autoregressive contain transform process past depend input equation toeplitz triangular side assume decoder follow process note toeplitz toeplitz kkt condition unique applying condition convert toeplitz equivalent invertible last prove situation decoding lie neither spike train combination iterative comprise step iteration use require minima switch rewrite eq final iteration round round step update equation fast rate hand small practical implementation early stage stage figure illustrate un un round round update choose image pixel neighboring consider causal ar causal ar impulse impulse process discriminate process subtle difference deal condition boundary case follow random since boundary submatrix comprise multiply side follow p simplify p p minimization lasso p p u complicated way view problem perturbation noisy p unfortunately consider namely sense eq u write formulation proof sign ensure primal pair construct obeys give choose value magnitude ar contain place simple contain idea case note assumption automatically drive loss assume root function ar tell furthermore summation j j ty convexity primal dual ensure give violate spike next break whole satisfied check exist condition construct theorem inequality rr verify clearly follow denote begins already impulse cause almost finally correspond reasonably prove ty ty ty exist construct fix I give know last satisfie sign p jx u magnitude determined sign general choose r similar kkt respect need check inequality ty ty ty ty ty next multiplying compute yy ty py pe pe pe ip ie show correct I ip ip ie I follow I hold probability equality proof omit simplify formula submatrix comprise comprise row index simplify matrix comprise row index index comprise index comprise row permutation rewrite check I p note ty p ty matrix ty ty get ty ty ty ty ty ty ty ty ty ty I simplification ty ty ty
generalize nice feature update well result generalizing prediction nash nash algorithm profile guarantee agent strategy representation anonymous game still fundamental know agent must choose agent something unlikely action another guarantee choice payoff action would round payoff equilibrium slow furthermore guarantee distribution equilibrium strategy converge learner nash game use stage rather equilibrium search strategy use move move equilibrium give require history examine procedure frequency guarantee game game tend observe agent payoff payoff base learn fast body empirical pricing game grid converge would explanation game rapidly equilibrium practice anonymous game straightforward tuple infinite choose simplicity interpretation agent mixed action second interpretation notion say agent action mix profile choose action want fraction end action action profile process profile limit something payoff payoff result perform payoff rather agent payoff change agent perform action agent follow p sa ensure anonymous require notion continuity distance difference agent utility anonymous round agent player game opponent game payoff choose action payoff play utility game action maximize good amount difficulty determine allow action denote well well try agent start action action agent apply say probability note notion merely sequence say approximate dynamic every determine learner anonymous converge stage need game well short sequence action distinction irrelevant small show property degenerate strategy agent want know payoff payoff divide game stage almost play explore choose next agent maintain stable environment explore originally learn reasonable show agent learn stage round need select mixed action agent strategy play explore uniformly previous know payoff distribution agent know action I I convenience agent stage ai ts stage learner observe statement run triple agent choose receive notion tuple g stage learn anonymous profile error learner fraction stage action stage play expectation average exponentially close true expect via standard hoeffding variable correctly anonymous game converge game learner agent rate exploration close change strategy fraction fraction action symmetric pure incorporate explore play explore distribution exist play width depth close example reverse closeness notion specify distinction close calculate game agent actually follow lemma sufficiently lipschitz give requirement acceptable lemma acceptable well least learn well nash profile specifically nash actually nash equilibrium three practical converge quickly system robust know payoff requirement guarantee stage argue case robustness fraction assume error due noise agent leave issue increase equilibrium population learner allow within round apply slowly learn implement regret converge nash equilibrium game converge closely stage make two tell get random payoff match observation distribution prediction lead particularly game payoff due mistake decrease agent experimental illustrate behavior learner include describe payoff payoff match payoff adjust game symmetric report contribution strategy contribute collective agent contribute contribute game strategy agent second implementation learner describe stage learner result strategy distance average payoff nearby close notion play distance action length mistake presentation graph similar population agent population agent mistake mistake cause successfully although mistake strategy agent converge equilibrium mistake rare equilibrium due exploration also fraction action action cause asymptotic equilibrium allow far depend tight require converge equilibrium nash equilibrium agent determine stage convergence tight ten result payoff determine strategy short stage stage convergence mistake agent successfully regret learner payoff game payoff end random take approximately design result system collect ensure enough expected payoff learn game statistical typical result show natural learn interesting issue exploration agent stage average payoff receive certainly guarantee arbitrarily give stationary strict exponentially recent estimate value new base stage spirit play game possibility tolerance mistake equilibrium notably utility add mistake time fraction agent play could leave system reasonable newly agent follow treat mistake tell newly quickly agent include agent utility function hold possible type infinite agent interval care define agent pay service internal seek currently extend game stage agent control next stage slowly purpose typically need hundred low exploration mean need explore pair learning problem guarantee another agent explore action payoff agent game well behave concern well game state game stage action converge agent equilibrium strategy sufficiently equilibrium dynamic near learn play try agent simply well utility sufficiently strategy hold agent make mistake treat entire stage cause mistake scalable
sample facebook sample specifically use pick error discard repeat process require sequentially evenly allocate explicitly facebook allocate sequential nevertheless rejection allocate proposition yield sample allocate social allocate allocate consecutive pick element valid w w easy indeed accept pr completeness interpret sample uniform allocate space implement meet measurement facebook assign simply knowledge actual select exist support facebook time furthermore acceptance experiment long facebook emphasize compare ground appendix consider practically static facebook duration day thank confirm facebook take privacy bit node setting comparison sample across day facebook grow growth day growth fact analysis therefore growth facebook negligible evaluate facebook run rw collect next day visit change day collection approximately space define degree summary day day absolute additionally day importantly way growth day former node magnitude variance essentially interaction latter consider become important however problematic vs experimental facebook far take broad internet topology case conclusion topology life internet topology table library long require sample achieve facebook measurement life internet topology play major design degree counter intuitively visit rw instead visit leave therefore round stay bias rw measure node fix number random introduce variance end rw simplicity ignore clearly rw choose uniformly contrast stay round drastically limit count consecutive independent calculation description email email large www graph sample length estimate median triangle green circle actual repetition node kolmogorov ks statistic vertical bottom corner part whose connection therefore exploit test toy graph clique rw error deviation bar comparable size rw topology carefully carry strongly estimate process switch clique clique say inside behave enter typically stay good se eventually rw chance return significantly leave systematically outperform rw intuition confirm fig case fig outperform life topology require translate bandwidth gain recommend internet topology com uci edu edu develop obtain property candidate weight walk walk rw substantially biased result offline assessment process show adequate facebook collect knowledge user publicly service facebook internet internet facebook popular million member membership twitter context population user population user medium site per month account spend online spend email website facebook visit website internet google minute day google top top regard traffic ranking facebook internet example collect study strategy engineering perspective enable design social user activity improve user storage may understand traffic activity traffic engineering serve potential influential user social trust collaborative spam enable online effectively party service well become interest rise number characterization attempt understanding study complete view collect social university company privacy envelope need social million encode bit friend topological node attribute amount least access service requirement view limit impossible fully case necessary overhead instead desirable property precise population inference ability property list individual sample make principled difficult primitive obtaining systematically reweighte uniformity user social appropriately use implementation contribution contribution consider widely bias towards highly characterize consider random rw sampling lead bias quantify correct appropriate hasting walk stationary user technique past facebook collect uniform facebook select facebook bit serve method collect rw analysis former ready latter practical second markov carlo adequate safe stop sample third compare aforementioned technique facebook highly non trivial various access limitation implementation representative publicly available request facebook privacy property describe methodology candidate include high facebook evaluate efficiency quality facebook section conclude elaborate point obtain rejection temporal facebook sampling vs work investigate quality efficiency ii characterization relate work walk traversal sample replacement visit visit method visit node depth ff basic technique research popularity incomplete node however towards artificial topology confirm bias sample online social statistical estimate suggest possible compute random distribution walk graph well excellent sampling wide www towards high bias analyze correct classical chain bias metropolis describe apply select representative alternatively collect result recently improvement walk jump walk weight walk unbiased rw weight baseline complement formal diagnostic test use several knowledge close property evolve state implement multiple chain recently use demonstrate walk analyze corrected practice bias walk agree term network study context namely facebook publicly available twitter main throughout treatment unbiased temporal include evolution yahoo social social evaluate dynamic sampler assume social b facebook context asset study rejection truth uniform allocation star induce evaluate paper summary challenge researcher face collect datum analyze online small study collect weakly use study facebook collect graph graph facebook region collect user profile friend large appropriate facebook wide percentage user privacy conduct york difference example value coefficient follow also dataset facebook property collect social work examine setting facebook privacy common additional privacy neighborhood demonstrate accurately large body service mention community show network present drive content video popularity distribution collect range usage file characteristic fm music site feature conference appear paper material sample truth section empirical assumption social graph static appendix purpose characterization facebook additional comprehensive model discuss extent undirected facebook direct collect publicly ignore isolated facebook thank contrast fm approach remain duration b facebook support mean identity neighbor mechanism uniform generally user social graph frequency attribute setting user us property degree cluster section last base node therefore random towards characterize node edge possibly representative entire facebook global attribute combine process social iteratively visit node discover neighbor proceed choose visit implement new iteration visit select node distance start incomplete densely graph walk move rw inherently bias connect probability node twice visit rw property correlate bias weighting later visit unique belong discrete interest node transition metropolis chain carlo mcmc directly achieve exactly distribution paper initial stopping meet uniformly stay move accept towards reject move towards high truth assess graph measure system truth typically fortunately facebook exception time bit interest random allocate user regardless actual allocate evenly across space completeness derive appendix refer although node facebook allocate facebook user user retrieve per attempt consist string infeasible bit interestingly complete facebook change bit information usage facebook obtain uniform baseline truth primitive believe design multiple walk convergence walk explore graph convergence walk reduce accurate advantage multiple view valid inference convergence diagnostic develop answer least question sample discard start collect approach long discard initial burn burn measurement day sampling decide target request account server activate type friend continue address backtrack exist degree upon discard consistent assumption exclude publicly isolate want rw node collect sample independent repetition returning visit rw ran collect collect collect sample burn rw collect total rw repetition partially especially collect show rw observe still overlap rw probably uniformly user friend isolate consistent statement result characteristic collect unique k neighbor period cluster avg million unique facebook evaluate efficiency quality walk interest period exclude independent x sample exclude burn estimation sample summarize chain apply yet inspection trace base discard seed formal resource section property burn walk sample convergence diagnostic membership space iteration z see diagnostic diagnostic walk walk membership new follow user metric score considerably pick spike new york membership likely walk york particularly iteration drop test walk discard total evaluation remain rw make exclude burn length way analyze confidence analysis formal convergence assess approximate equilibrium top property bottom discard burn convergence attain determined burn facebook connect random seed visual inspection reach drastically fig walk individually combine degree walk iteration likely get average seem walk additionally average rw stability iteration clear interest number inspection need sample per walk walk resource walk another allow correlation consecutive sample process examine estimation percentage rather average entire iteration even equilibrium reach sometimes break bottom iteration perform well despite membership york generate visit walk mcmc advantage space collect bottleneck perform rather storage post effect collect indeed collect information visit friend due correlation consecutive essentially sub node compare technique iteration estimate truth kl kullback capture accounting calculate kolmogorov ks vertical distribution respect usage ks online importantly grind cc cc cc represent real bandwidth mcmc use choose metric principle metric want estimate need mcmc converge respect slow several metric membership uncorrelated use easily metric relevant estimation network burn cover metric interest degree result metric three node degree strongly converge uniform walk stay poorly take long reach network collect representative fig confirm expectation perform example york ny error I np kn give walk deviation walk population present estimate namely degree size essentially baseline c rw chain identical rw degree represent order magnitude rw rw much notice rw shape lead wrong perform true metric network result present poorly coverage rw bias possibly degree well albeit high degree c cdf look distribution topology facebook facebook assign bit happen user profile add friend expect rw usage big difference usage rw cover uniformly rw clearly shift origin shift probably sharp facebook first bit old degree correlate degree indeed show facebook user history website phase initially auto assign stanford assign introduce degree rw explain shift finding recommendation comparison demonstrate facebook property simple rw rw directly uncorrelate node degree move generation traversal community principle correct rw chain time appropriately remarkably ingredient employ formal diagnostic several show reasonable believe implementation walk sample adequate subsequent another ingredient start use single improve duration efficient less finding partly due partly slow avoid present simulation yield subsequent heavily biased weight correct intend facebook ready intend people ease desire target distribution
space generator sort speech recognition engine speech equivalently trajectorie invertible order recognize recover bss coordinate train engine identify bss global permutation global determined data match produce bss finding pair human procedure separate source describe separate statistically independent component component separability source coordinate system product correlation range follow vanish velocity correlation form consequently system diagonal respectively order prove block transform correspond prove belong block vanish correlation factor block scalar must alone although derive imply permutation coordinate must index respectively function alone consequence bss compute velocity algebra find satisfie choice range partition datum index group contain choice choice step subspace space plotted subspace compute function map embed therefore related coordinate system source separability must invertible also manner must separability source perform exist linearly bss source choice lie denote subspace another linearly relate source follow determine source linearly mention span derivative set pdf occur series objective bss comprise statistically measure function equal product bss permutation component wise transformation separability certain locally invariant derive local velocity constraint constraint illustrate record device often evolve simultaneously situation necessary separate signal knowledge considerable blind source variable linearly although bss human usual objective bss mixture find transform datum define special system total locate location usual bss statistically product bss permutation wise transformation however weak suffer problem solution mix source reference issue uniqueness fraction velocity trajectory within location early bss pdf component strong separability note recover side former one fact guarantee bss virtue physical interact generators interest pdf induce geometry second velocity bss metric compute first respect mathematically bss suffer space deal require densely calculate derivative accurately current correlation advantageous require speech separation experiment iii record single minute differential method differ independence one exploit usual time unlike mix derive constructive parametric employ without neural differentiable unlike class describe source variable simultaneous speech record implication discuss multidimensional source bss procedure section scalar datum combination velocity scalar invariant space impose scalar necessary simply satisfy source coordinate explicitly transform factorization construct velocity trajectory segment neighborhood denote possible factor velocity moment formal computed average velocity subscript vanish identically next let use corresponding factor definite always diagonal therefore coordinate scalar must equal alone derive coordinate separability true coordinate system coordinate function likewise function separability bss compute velocity correlation find continuous satisfie compute triplet vary plot lie single require separability manifold coordinate point six coordinate system statement understand manner invertible relate manner separable describe mix variable bss invertible transformation coordinate x proportional denote proportional source rescale bss derivative determine relate partial coordinate linearly source varied manifold speech record single human human impulse extract unknown bss differential simulate simulate cavity represent series spike separate hz hz impulse amplitude impulse slowly vary smoothly latter statistically energie db pre short use energy hz frame nonlinear function two datum determine redundant component inspection within ambient produce hide dynamical degree freedom redundant eliminate dimensional neighborhood establish sound coordinate bss ii statistically step bss procedure compute function
minus kind include dependency arise examine dependent crp connection derive gibbs setting study corpus show exchangeability distance sequential traditional crp original dirichlet process flexible text vision biology advance scalable approximate dp valuable modern mixture describe chinese crp partition distribution structure describe sequence customer chinese customer concentration crp customer belong flexible crp exchangeable permutation essential connect crp mixture dirichlet process distribution dp mixture parameter exchangeable crp exchangeable equivalent dp assumption cluster time collection news article article nearby tend non exchangeability crp assignment version base dependent dependent arrange recover crp dependent crp infinite allow crp represent partition table assignment crp connect table distance crp connect customer assignment arise model customer representation gibbs tool cluster datum traditional crp appear research model crp partition crp customer partition dependent crp customer crp prevent reverse note present crp employ nonparametric include place collection draw dirichlet covariate induce customer value equal draw respective exchangeable condition covariate distance dependent alternative modeling exchangeability difference affect property class crp distribution include special distance crp crp assign product review process far crp section develop distance crp fully observe customer assignment algorithm dependent crp assumption dependent well also alternative formulation crp fast sample one original customer choose connect another link customer assignment htp cc chinese restaurant crp chinese restaurant table enter restaurant randomly configuration table traditional crp customer table customer assignment customer table customer sequentially customer table exchangeable invariant plan term customer customer table product customer customer table illustrate figure assignment index customer customer measurement customer crp draw customer distance notice customer assignment customer assignment customer customer customer restriction sequential mention customer table customer customer see might finally customer assignment cycle customer customer cycle assign customer assignment expressive determine measurement customer time encourage customer customer encourage customer proximity present eq normalization requirement present version crp write crp many set distance exchangeable appropriate exchangeability decay customer result decay take satisfies example consider customer decay customer distance window partition crp previous requirement guarantee customer customer define window decay logistic decay recover examine detail express traditional specifically crp recover marginal customer assignment proportional already customer precisely crp derive draw decay include crp setting crp customer nearby emphasize lead partition crp particular customer customer property capture way precise characterization marginally might goal modeling preference reflect likely share marginally model common preference unobserved might prefer marginally preference regardless whether discover city status city disease marginally require computed ratio contrast factorization easy computation marginal invariance marginally invariant choose computational crp illustrate crp data terminology collection word fix vocabulary document language modeling document crp iid base place table customer assignment exhibit share dirichlet smoothed language alternative also set crp assume occur document formally decay distance draw assignment assign customer assignment induce assignment indicate draw traditional crp emphasize crp customer successively previous dp nearby endow draw set opposed assignment document sequential crp structure cluster lag external date distance covariate mixture process sequentially long generally dependent crp mixture provide dirichlet mixture kind setting integrate parameter proportion distance dependent crp amenable sampling formalism regression mixture distance unlike gibb integrate different crp nearby nearby crp consider node distance far pair group impossible group mixture crp nonparametric interpretable draw crp mixture generally mixture invariance distance generally capture capture assumption appropriate model posterior exploratory datum intractable compute crp place combinatorial number customer strategy approximate markov chain mcmc observe setting section distance distance fully chain crp consider assignment figure denote hyperparameter factor let draw variable markov crp gibbs iteratively draw crp observation remove link customer examine describe remove affect partition assignment gibbs sampler scenario highlight way condition table join customer link two split happen table remove customer customer point case remain case effect link customer another change customer self change term partition factor table gibbs correspond partition link self join table might link finally join gibbs sampler term type type four could table simply representative customer indicator ensure observation index customer table mixture probability table parameter draw collapse sampler conjugate pair integral setting compute new rely posterior customer inner customer assignment sequential arbitrary sequential future point case customer thus unchanged use approximate distance occur middle discovery change assignment thus new know advance sampler leave customer unchanged unobserved simply ignore crp marginally distance crp necessarily crp distance crp marginally invariant marginally crp prior partition measure dependent suppose covariate latent draw parent dependence formally measure measure conditionally implicitly point equal cluster marginally crp dependent crp crp crp situation invariance dependent obtain marginal modeling mixture setting text set explore decay distance connect crp well fit text fully observe gibbs dp mixture crp dp mixture customer fast sampler bayes dependent versus document denote traditional crp denote crp distance crp collection journal article model assess sampler visually autocorrelation chain factor distance crp traditional crp indicate dependent crp crp bayes document various function crp decay function hierarchical base shape curve setting tb corpus bar across sample dependent crp outperform traditional crp decay decay traditional crp test corpora month article analyze contain corpora article datum three news paper th year retain hold article likelihood article well early hold corpus crp best corpus logistic decay crp exchangeable crp mixture well exchangeable crp model dependent crp emphasize gibbs sequential collection paper connection window decay enforce customer another customer refer immediately treat undirected subset abstract citation color graph assign connect color repeat figure note group possible connect crp crp emphasize crp crp simple cluster emphasize concept flexibility crp choice explore long window treat modeling image distance express flexible crp sampler inference dp collapse dp e algorithm applicable conjugate iteratively customer assignment collapse crp sampler customer cluster assignment customer posterior sampler two likelihood either crp traditional add set amount constant lie computation sampler example traditional sampler sampler local optima tb illustrate identity decay red sampler dependent crp identity crp log crp representation leave panel corpus show corpus indicate well local crp fast sampler document collection crp uniform dirichlet illustrate traditional crp gibbs iteration proportional posterior posterior likelihood constant traditional crp sampler local optima corpus chinese restaurant partition crp derive gibbs examine text dependent development image dirichlet fix correspond variational worth explore approximate acknowledgment david nsf foundation google fa thank anonymous comment
mu mu repeatedly chain start remain mode less original gibbs random walk logit hastings acceptance logit exchangeable necessarily parameter exchangeable give imply exchangeable applie modification achieve share get neighbourhood converge go main second simulate convergent slowly iteration application mixture book prior complete I trick gibbs sampler generate ij difficulty move around compare mode guarantee simulation reproduce appear show normal likelihood eq partition single allocate particular reduce go bound therefore unbounded code behaviour mu seq seq mu surface like col color exhibit illustration unbounded mixture go average behavior walk reduce target avoid pick rather ratio need choice power computation sample computation impossible explain need constant twice numerator denominator acceptance twice power vanish reason derive mcmc detailed balance balance generalise represent distribution q marginal similarly finite respective generic importance approximate posterior set need distribution support least correspond compute importance obviously efficiency importance impossible reversible acceptance move balance proposal relate acceptance markov kernel reverse reverse move exercise rigorous derivation perfectly kernel forward backward marginal exercise distribute identically truly process sequence random variance stationary bt process moreover tw bt bt necessary moreover I student series evaluate replace book degree new proportional indeed indicate proportional eq integrate jacobian expand unconstrained next band around causal process autoregressive polynomial causal root circle plane symmetry root causality empirical sphere change plot root outside root n col program look triangular shape analytical look either root root eq together region amount equivalently causality q region triangle q therefore pm eq well book value posterior integral integrable parameter bound remain integrable derive root relation set sentence first book expand root recurrence process proposal prior prior acceptance book hasting ratio extend reversible jump modification consider death move root new program acceptance eq convention conclude distribution normal distribution covariance proportional distribution proportional costly require derive recursive construct single step whole argument compute give conditional distribution exercise formal construct account obviously horizon correlation ability far horizon marginal distribution deduce identifiability eq reason switch write therefore double close integral prediction formula book obvious development ji fx r lead px r fx near illustration relation sufficient new close decrease point near neighbor h five experiment evaluate carlo produce neighborhood diag n neighborhood sum summarize size figure neighborhood joint discuss general fy fy fy fy fy k extension solve compatible e joint distribution conditional joint despite formal agreement conditional joint mass compatible distribution make use size joint pseudo define satisfie therefore unfortunately get differ distribution since line conditional support marginal marginal exercise replace exercise conditional never product support marginal support conditional clique neighborhood structure regular clique draw member square clique neighborhood structure conditional deduce ise mrf development exercise exactly neighborhood array determine normalize array summation summation indicator I array array neighbor structure array exponential summation neighborhood deduce index odd wang exercise replace former exercise array four near neighbor deduce update whole simulate pixel even pixel index simple case wang obvious book graph node index node update node odd powerful stage gibbs computational array color exercise normalizing summing term exponential involve sum even neighborhood normalize face establish associate conditional deduce exercise initial conditional multinomial proposal proportional another possibility select propose multinomial q efficient purely show wang exercise obvious modify number sub interpolation pair program txt correction boundary estimator minimize lead solution mode similar completely basically look permutation class arbitrary therefore obvious pick minimize allocate choose scheme reach configuration converge produce experimental checking risk representation optimization run obvious since integral basis detail two distribution cdf integrable lebesgue negative condition derive positive semi test seq help try seq lm simulate generate lm lead residual std pr intercept degree freedom r df therefore simulate function meet check write mean nan na complete else complete pairwise complete else deduce knowledge write four density exploit x z z z think could classify reason propose illustration find claim minor high claim tail normal extreme http www reproduce histogram conduct report relative file book create file histogram inference follow region whether strongly differ define sample histogram density ad smooth col col lead roughly normal different two joint conditional show quadratic solution eq q normal geometric family distribution book fit representation fit failure also fit fit exponential component define show update imply therefore iid show depend statistic see conjugate update e obviously statistic give posterior varie q model gamma family jacobian check exponential impossible iid derive distribution prior jacobian student marginal q distribution eq get prior show model jeffreys prior location pz therefore constant jeffreys long space change location therefore jacobian negative result jeffreys q get fisher density integral devise parameterize improper matter sufficient go fast go cauchy go exponential prior associate posterior expect decision x go x go go recall denominator use simplify without term appear numerator monte mean appropriate compare precision carlo density like nu nu pi nu nu nu sigma mu b n exp comparison seq b precision converge h comparison bayes happen importance evaluate integral eq importance variance exponential polynomial q integral less exercise dominate matter importance like nu nu df col output distribution large high dispersion log importance weight student density density factor deduce miss marginal q ratio integral expectation respectively posterior bridge infinity converge dirac regularity identifiability constraint converge distribute cdf uniform property purpose available rank transpose r tx deduce happen dimension x xx x linearly dependent decompose expression eq check via solve linear lm note x identity x xx establish correct virtue form coverage probability mean regression explanatory x deduce mn posterior show exercise x student degree freedom location equal exercise restrict represent matrix hypothesis k satisfy constraint mean combination expectation actually write format notational conjugate apply sense factor matter differ nc c student difference prior consequence exercise predictive distribution derive integrate produce student n nc xx x exercise nc coverage exercise distribution eigenvector deduce determinant obvious I z generate vector whole x x n marginal distribution n exercise predictive exercise indeed jeffreys nothing prior exercise mn file txt provide suffice instance ty cc ty cc ty converge obviously vector explanatory predictive student exercise distribute produce irreducible work axis center necessarily chain depend disk bt sampler iteration jump positive disk density conditional exist density distribution irrelevant eq role check value gibbs mu program lack influence start need loop mu seq mu col triple machine show c series lead posterior take fix prior equivalent since bank jeffreys corresponding file txt bank dataset available bank bank book call residual median std pr intercept code adjust statistic exercise jeffreys sufficient give statistic sense update rather due fact link last covariate figure book bank via prior right auto variable ny I check mu flat call good compare bank difference unobserve py pz simple inverse exercise cdf define irrelevant back flat give kx introduction gibbs code file function smooth converge fast modification move smooth transition compare bank dataset txt bank probit intercept flat right auto distribution proper create enough control nonetheless traditional control define bank bayes hypothesis bayes k simulation multivariate normal suggest direct txt bf probit multivariate contain bf probit divide approximate factor hypotheses jacobian deduce transform density jacobian determinant make multiply square logit kp exercise jacobian examine sufficient exercise probit logit little asymptotic density bank logit bayes compare exercise exercise estimate file txt prior full sample log bf logit bf logit strongly probit exercise except twice factor support contingency probability dirichlet deduce associate q variable apply multinomial contingency comes replace restrict contingency four matter since build variable zero since pick factor exclude model exercise term major control regressor rank go least whole exponential term go quantity show improper equal relevance converge normalise available close equal median rather intuitive late number binomial irrelevant code posterior post equal median would produce stage distribution size deduce expectation expectation eq capture day keep track number say give expectation derivation estimator proportional n quantity increase statistic extend capture episode individual number case capture episode converge proportional extension capture consider capture future extend capture different episode observe likelihood therefore number individual another stage mark mark recover mark give contain lose mark complete second partition tag loose obviously possible summation acceptable term must keep simplified form reproduce switch exercise modify file book posterior change direct prefer metropolis step modify conditional proposal thus simulate modify nc p nc prop n p n prop nc book constraint r integrate cost full conditional exercise sum remain gibbs full conditional simulate distribution simulation standard appear consume former write complexity elementary individual life adopt history constraint likelihood case accounting constraint computation replace marginal deduce normalize constant surface therefore marginal give simulation output matter normalise gx
subgraph temperature introduce individual factor edge leave subgraph world section field eliminate without loss subgraph unlike ise random like subgraph state index edge edge maximally configuration path random independently assign node cluster behind different component mixture edge indicate might well index mixture develop constant relate easy calculate go proof insight remarkable relationship correspondence configuration physical section proof subgraph world importantly graph could idea fully hence subgraph draw mixture distribution ise use parameter value many proceed formulate variable random variable reverse give begin choose choose desire draw directly draw possible proceeding stage stage provide choose choice normalize constant binary probability divide yield side multiply term right side turning complete easily suffice utilize fix start edge nod leaf exist maintain degree requirement preserve receive choice minus degree yield equivalently therefore e complete random wang devise chain fast index generate draw ability move random subgraph wang subgraphs subgraphs cluster show subgraph f j f I return subgraph draw section maximal forest edge weight let forest connect let configuration configuration along node onto condition equal condition word draw plus combine case value line correct choose way task remove nonempty leave leave forest either first search either forest leave subgraph provide obtain spin let consistent pe pe pe mean e combine z e desire coupling past algorithm perfectly immediately simulation world prove study theorem huber computer college edu via family say simulation utilize extra randomness whose runtime flip come draw simulation input size simulation wang approach ise edge call wang draw turn view ise third assign call subgraph remainder organize discuss show subgraph direction subgraph draw convert similarly convert single draw wang approximately ise answer direct relationship discuss early create reduction create series drawback multiple configuration draw vice perfect ising initially propose extensively phase transition application either refer spin spin configuration weight function strength ise factor directly control say
make partly choice lead third qr analyze post post ordinary quantile penalize qr qr true perform well qr qr true subset qr design interest post qr international economic growth contribute penalize technique comparison implicitly index notation f n f n na na ba formulate primitive regression underlie quantile eq compact quantile index quantile function population minimize asymmetric absolute minimizer analog high setting ordinary inconsistent motivate remove least zero penalization lead penalize quantile estimator penalty quantile depend penalize asymptotic problem programming define optimal solve polynomial avoid curse dimensionality selection estimator post qr ordinary quantile quantile specifically penalize remove model work estimator oracle unlikely post well choice introduce independently distribute conditional quantile quantity recommend c recommend account sample recommendation parameter contract implementation practical variance cross also choose allow behind lead precisely slightly qr recommend select dominate noise suitably rescale criterion function value general subgradient regression index omit indexing ease exposition continuously constant conditional quantile away impose regressor quantile condition jacobian u condition impose assumption vector empty restrict eigenvalue condition primitive condition bound impact appear concept restrict set eq invoke post analyze lasso se highlight usefulness condition brevity correlate normal estimating one eigenvalue away constant design satisfy eq normality smooth obeys state general primitive specified remark coefficient hold n chernoff small design hold concave design regressor consider estimate location differentiable uniformly bound zero design linear coefficient un hoeffding inequality approach small population design away nonlinear impact replace primitive form suffice minimum eigenvalue restrict condition quantile analog rate design prove least control modulus conditional evaluate overall penalty restrict identifiability rate follow lastly require fast mild going condition appear precisely turn behaved design interest concave hand hold bind invoke se analyze se less penalize dimension unnecessary outside characterize control quantile quadratic function neighborhood covariate se b unnecessary component support design assumption straightforwardly compare qr choice oracle polynomially result restriction polynomial model penalize dimension select estimator correctly minimal provide thresholded penalize apply ordinary obeys component obey growth eq singleton qr indeed post qr qr relatively permit qr well due qr select obeys select contain converge post qr case perfect selection rate post qr singleton drop necessarily happen coefficient separate additional regressor coherence regressor dimension polynomial order well single penalize quantile penalize paper post regression apart perfect oracle regression qr analogous qr different fundamental excess loss function logistic principle regression problem assume penalize hold design regressor imply qr sparsity property drive qr start characterization determine via equation universal establish rate qr preliminary penalty specify belong least inspire analogous relation condition derive show control norm restrict control quality quadratic preliminary derive bound error q combination argument contraction fundamentally principle alone neighborhood define intrinsic uniformity armed qr assume condition level least provide obeys condition derive penalize intrinsic norm uniformly range recommend rate qr depend significant strength summarize quantile extreme quantile slow convergence obtain role rely proof quadratic nature control quality refer discussion design proof shape geometry restrict classical argument insight follow event uniformly event preserve n u f p upper growth bind obtain quantile fundamentally sparsity link quantile regression gradient highly piece wise control sparsity rely empirical process argument gradient crucially exploit result eigenvalue pre probability initial restrict pn design interest k main eq obeys eq states initial control crucial penalize estimator corollary theory supplementary possibly need qr growth virtue lemma hold upper bind u turn selection qr thresholded estimator analogous regression say separate support one sided support unnecessary coefficient hard eliminate unnecessary hard characterize zero quite unlikely practice certainly example motivate post allow selection unnecessary post estimator rely crucially identifiability u u obey dimensional theorem establish post selection error post qr condition assume hold bound hold obeys growth describe qr inspection proof reveal qr fast qr fail contain rate qr hold recall design theory material appendix calculation overview intuition qr component allow qr still case succeed u unnecessary component obey post fast qr extreme high post qr refer reader note last eq imply optimality u u identifiability growth access practical propose monte international economic sample conduct monte penalize post oracle post canonical quantile select quantile course outside carlo focus penalize risk error identically distribute equal penalize estimator panel select much design penalize always miss regressor coefficient regressor notably despite partial failure post perform report right penalize rarely support penalize estimator select theoretical theorem stochastic agree summarize recover penalize quantile zero drastically improve penalize quantile reduce notably estimator never always regressor post risk perform identically ideal expect penalize well make estimator find estimation performance post mean qr post qr oracle design qr post penalize qr qr penalize international economic growth primarily lee consist panel national period analysis total observation goal covariate central growth initial per growth rate per growth hypothesis typically fast rich thus hypothesis point reject positive characteristic characteristic include education science market predict initial covariate analysis severe rely hoc case previous finding simple resolve lower large million specifically median resolve important turn perform covariate drive penalization lead separate slowly decrease order market exchange characterize political instability additional several report reader discussion ordinary median confidence estimate confidence interval work work additional select covariate coefficient interval conclusion quantile middle range believe empirical finding growth inferential finding agree report parameter per black market political instability political restriction consumption education exchange school population school population black market political instability ratio real consumption net education school school ratio education education year secondary population instability restriction education exchange rate high school secondary school complete high education education proportion year secondary growth population school population nominal nominal k rademacher independent condition j envelope obeys ball empirical hoeffding I choose make side display condition therefore diag probability least furthermore u u u n event lemma probability lemma population claim proceed large eq proceed radius criterion quadratic scalar expectation mean proof error divide main argument tail note lemma greater far p nc e p q ix ib k k k p k p n ij hold last inequality law iterate hold inequality intermediate x ix elementary hold p inequality inequality characterize sparsity eq variable uncorrelated play sparsity sign property conditional uniformly complementary problem show part row I x linearly independent one basic solution wu un number quantile equality establish conditional feasible polytope moreover imply therefore dual generate finite intersection row event absolutely basic degenerate empirical bind cauchy probability supremum side establish note p contradiction convenient rank score dual solve ij u j non provide next step inequality triangle probability c linearization control linearization definition cauchy schwarz next control define shall preliminary number function class q bound universal constant bound proof argument supplementary material supplementary brevity restriction universal since restriction total cover statement control uniform sparse derivation early semi definite positive proof inclusion converse u thresholded inclusion u x c e nf z z nf c nf n n nz ny n n n proof interest condition mild reader
least side consider lead least least sufficient statistic count marginal respectively prove marginal statistic likelihood mixture try describe model study light classic fact basic variety describe describe vary describe non zero coordinate set vary describe equation repeat fixing describe simplex base join diagonal model form element scalar order second gives add lp lr mc ic mf g g vanish entry diagonal term variable appear appear I ip kp k ir add variable kp ir ir ir kp p proof proceed I p k ip kp kp kp k computation find polynomial model true di department definition theorem contingency encode explore element non number simplex contingency space j geometric structure algebraic polynomial p vanish negativity complexity algebraic geometry must intersect negativity algebraic study involve one take probability contingency log algebra algebra introduce geometric model structural exact basis negativity issue basis lattice basis effect special contingency table agreement medical science result discuss due statistical mathematical definition describe object especially show vanishing differ boundary simplex basic emphasis independence model effect model show special diagonal effect model common behavior diagonal cell diagonal contribution present diagonal study geometry model contingency categorical therefore product x statistical define algebraic cell define form power encode simplex log probability know obtain ideal polynomial ideal pure binomial I I define integer move pure binomial part finite connect contingency table path count notion move basis generate ideal deduce generator ideal contrary next implication independence role px px py independence suitable non negativity reflect negativity namely equation negative rank probability must formulae easy write implicit markov basis independence polynomial ideal say corresponding implicit simplex notation q define define sub space description issue find two field vector literature see relatively easy parameter obtain result model distinct q minimal form move move distinct generator define intersect hyperplane framework definition diagonal model matrix normalization geometry model give write polynomial easy check equation polynomial detail connection investigate negativity condition impose example respectively construction belong
minimize leibler eq uniquely strictly partition pseudo estimator root give objective scad pn large satisfie pn large enough completely center large minimizer helpful fit concern nuisance parameter property derive standard bic consider lemma argument weak kullback I maximum likelihood true different former maximum likelihood penalize fit fan wu pn combine complete yield identify establishe criterion penalize minimize criterion bic adaptive consistent adaptive lasso tuning sequence accord fan wu tend edge estimator nonzero partial similar derive scad let defined penalty pn enough term last imply choose radius light study fit work minimize adaptive study conduct likelihood consistency result bic cross commonly fold validation disjoint index subject cross validation score calculate optimum graphical structure ar model sparse graphical employ point connect distance entry covariance ensure penalize scad lasso penalties validation criterion specificity define true false positive negative true false positive classifier deviation different inverse covariance sample exceed setting reveal different assess method size tend glasso fan wu method initial obtain examine tuning via table specificity coefficient simulate set standard graphical adaptive consistently well outperform increase advantage scad scad bic yield specificity ar specificity sensitivity scad specificity sensitivity adaptive lasso per specificity confirm tend infinity specificity penalize sensitivity sensitivity structure compare fold edge per versus consistently specificity validation simulation overall bic exhibit large cross time intensive compute cm conclusion investigate tuning selection graphical establish true scad bic condition research science grant hold wu bic cv bic bic cv bic scad penalty cv cv scad york fan variable fan wu exploration scad covariance li network shrinkage tune smoothly white lin li discussion cm mathematics york mail mathematics york mail pt wu department york pt mail mathematics york pt mail cm pc pc cm pc thm axiom section graphical plus minus minus abstract graphical independence maximum smoothly absolute fan li penalty article establish bayesian criterion bic penalize penalty lead empirical performance bic demonstrate advantageous tuning selection study phrase oracle introduction relationship zero among covariance exactly vector multivariate denote covariance indicate represent vertex corresponding coordinate represent dependency relationship identify simultaneously address likelihood penalize unstable another standard approach perform forward elimination however hard furthermore computational search computationally propose quadratic lin implement inherent wise box quadratic solve show graphical lasso fast convenient tackle wu propose deviation scad approximation penalty result method lead penalize penalty lin coordinate bias estimation bias fan scad usually fan li hinge scad quadratic knot origin ensure penalty heavily penalty scad method select correct produces know true namely adaptive impose regard penalty efficiently implement lasso scad fan wu li iterative optimize denote
asymptotic efficiency segregation sr r ar r investigation underlie order analysis investigation asymptotic critical segregation sr ar investigate article implement monte nan monte investigation segregation calculate mc nr stand estimate imply carlo replicate mc th use segregation alternative percentile alternative underlying case monte yield mc notice also skewed monte carlo segregation depict density estimate may consider power test proximity present carlo monte carlo underlying severe segregation case compare carlo critical segregation case top two case replicate levels monte carlo sample analyzing factor base critical monte investigation critical corresponding table empirical level moderate appropriate significance severe segregation high furthermore segregation alternative size circle line triangle critical alternative underlie bottom underlie significance investigation estimate carlo mc nr mc nr stand similar imply small power monte carlo empirical power mc mc case separation much carlo replicate estimate monte mc density skewness experiment bottom right dash value power estimate case severe suggest ht monte carlo critical leave c underlie c carlo critical critical value critical power close yield severe association version level solid triangle critical association underlie power finite assumed let triangle triangle wish h segregation alternative relative underlie underlie construct use version density nr ac asymptotic rr jensen hold segregation alternative vertex allow mn nr nr n nr nr rw nr denominator complete maximum suggest density nr nr nr I give rr nan iff segregation alternative underlie case version nr nr equivalent limit however finite infinite nr nr nr j r nr multiple triangle asymptotic normality nr segregation association figure accord segregation association ht segregation realization segregation leave association great except segregation realization except underlie association realization case nan segregation association segregation consider value consider underlie segregation segregation repeat realization segregation association figure result indicate point per enough estimate relative segregation large per triangle test suggest choice moderate alternative well segregation circle line triangle top circle line triangle dot critical underlying case bottom multiple right triangle size triangle necessarily update conditional unconditional triangle form simplex vertex edge q boundary assign arbitrarily opposite contain let euclidean distance vx vx polytope vertex pe rx rx rx rx rx ir pe rx transform polytope face preserve uniformity become boundary sphere particular simplex regular face underlie parametrize proportional proximity segregation association theoretic test spatial proportional property triangle respectively point density compare imply assume fix abundance imbalance perform randomization conditioning employ relative arc edge testing bivariate spatial consider graph demonstrate statistic employ asymptotic normality statistic test segregation nan two class triangle co type parametrization geometry independent triangle e likely parametrization segregation association alternative tend point pattern segregation expect large association parametrization reveal power segregation hand well performance association underlying monte randomization otherwise recommend furthermore testing segregation recommend association acknowledgment partially advanced project air office scientific contract f office grant proposition example mail edu tr introduce direct various proximity graph determine family parameterize statistic provide alternative arc employ relative analytic asymptotic statistic illustrate bivariate spatial segregation parameter efficiency asymptotic alternative dimension keyword efficiency randomness segregation statistic classification base testing bivariate spatial extensively population pattern point one investigate two class implication especially specie see article derive family spatial segregation randomness roughly segregation association frequently generality characteristic observation segregation specie class mathematical popularity tool provide move movement although landscape suited application movement conventional maintain reduce graph integrate geometric tie patch spatial dimension preserve relevant spatial graph usually lose see concept depend adjacency express allow spatial edge domain network model spatial interaction intra inter relationship quantify patch potential patch theoretic measure design spatial interaction instead generalized coefficient point arc bivariate neighbor place arc vertex give demonstrate good dominating find minimum dominating e multiple distribution number prove extend non parametrize arc pattern family calculate arc proportional two datum obtain arc arcs arc edge without underlie tool article scale demonstrate underlying graph central analytically difficulty encounter edge edge describe asymptotic power segregation association triangle provide extension proof defer appendix arc arc pair edge replace arc underlie graph refer uv uv ng represent number order proximity region z set arc ix x x jx x x n proximity comes represent denote x ix nx nx symmetric arcs random px x px px furthermore ji ij x nx nx h find joint nx h h density ij ix jx nx finite brevity similar underlie nx nx x note ij mass note h nx h h triangle define proportional triangle nr pe nr n pe nr rotation v bt preserve uniformity furthermore scale map triangle boundary edge distribution collection region preserve uniformity preserve edge proximity uniform henceforth proximity map recall px pe rx rx px rx occur two underlying section equation value r r r limit equivalently underlie natural relative density recall r n pe rx pe rx r pe rx rx pe rx pe rx rx see distribution relative edge small st rr rr indicate skewness underlie figure skewness skewness may derive asymptotic successively depict histogram vertical ht depict histogram base replicate line normal axis scale depict histogram carlo replicate indicate severe extreme ht depict histogram replicate indicate severe small skewness extreme vertical axis proportional underlie graph proportional edge nr nr nr pe pe rx pe rx x pe rx pe rx pe nr ii nr pe rx pe rx ip nr nr nr nr nr nr nr order segregation class tendency fall tendency near segregation let
accuracy beyond paper super fista fista clearly approach synthetic section benchmark detail measure server ram regression reason measure conjugate minimum duality algorithm compute criterion implement absolute duality duality iteration construction finally criterion fista additional whereas vector non g matlab inner solve matlab choose initial proximity conservative since appear soft multiplied seem intuitive formal argument detail variant equation l solve dual implement notice happen poor order undesirable proximity constraint modification without specifically use proximity al function rewrite initialize conservative set proximity condition counter increase factor note section e simply conservative equality computation duality vector fista matlab instead unnecessary gradient implement matlab implementation optimize algebra algorithm implement matlab code logistic implement regularize logistic regularizer diag newton method matlab file library bias include subsection first convergence size proximity sample label sign mm repeat ten confirm fista minimizer eq run correct obtain minimum trajectory multiply initial residual estimate tb theorems residual vs cpu residual v show run describe keep mean bound top leave residual theorem result result difference optimistic realistic analysis order reach quality fista iteration step panel fista value fast fista needs every spend much accurate second obtain computation precision higher clearly bias term panel residual cpu spend parameter variant increase factor increase residual primal linearly roughly report slightly concave super convergence probably information spend roughly achieve residual tb summarize plot cpu spend reach fast roughly scale parameter computational show cpu error stop iteration run converge run except solve less advantage demand advantage large without subsection run factor inner spend conservative set cpu show stack bar segment bar correspond outer one use roughly outer half hand slightly therefore outer note half iteration spend iteration fast conservative makes recommend figure total spend variant conservative proximity conservative previous except problem clearly outperform benchmark five bioinformatic provide validation combine split dataset regularize set well cpu format dense category graphic example contain goal treatment multiple patient dataset denote gene gene subject subject setting begin polynomial order iii triplet obtain standardized mean dense even fista keep design deviation standard deviation zero place cpu whole order separate regularization constant warm strategy regularization solution conservative initialize summarize spend second show bold see fast number tend accuracy fista typical contrast except grow reduce seem almost r dense sparse time cm tb normalize efficient tb regularization regularize minimization minimization generalize super augment lagrangian importantly check assume convex assume loss check regularizer check look projection onto result inner approximately compare need convexity obviously many argument primal lagrangian logistic rapid confirm simulated regularize logistic regularize fista synthetic dataset fast large number observation dense relationship inner outer computation improvement change make condition basically light fista category convergence another small iteration prominent member empirically show shrinkage first class effectively use analytically sparsity shown compute efficiently sparse include primal lagrangian split advanced thank helpful discussion center development mathematic proximal convex proximal right proximal operator convex follow elegant convex prox prox similarly give sum side eqs operation onto convex take ball radius regularizer soft operator therefore special ball attain envelope envelope envelope pair prox conjugate line envelope conjugate tb threshold regularizer indicator envelope consider inf quadratic envelope envelope differentiable completeness subgradient envelope prox prox envelope projection figure line step generalize allow minimizer bind close namely minimizer follow arithmetic geometric expression accordingly substitute complete follow term let prox ready analogue decompose residual follow b reduce follow arithmetic applying expression depend front bound inequality term hand expression line true last line minimizer show x f third attain l bind value cl mc denote delta primal gradient hessian diag diag z ic diag logit ik ik ik ik ic ic mc ik ik mc envelope function element wise p c proximity operator envelope regularizer n prox n j norm tr prox en prox en convergence recently estimation minimization theoretically super due model analysis lagrangian algorithm interpretation generalize wide extensively efficiency propose lagrangian super linearly estimation become common application bioinformatics rapid development tailor machine sparse minimization plus paper loss term regularizer differentiable various factor tool machine diversity arguably square signal reconstruction estimation variety wide logistic loss function square loss matrix design stack input g minimize design compare regularize apply context denoise sparse reference therein contrast focus design recently shrinkage see iterative lagrangian version algorithm proximal rigorously converge super linearly mild grow framework wide practically regularizers improve convergence augment structure sparse instead consider al efficiently exploit intermediate solution al play important role analysis primal section recently dual lagrangian paper review derive minimization special discuss section theoretically behaviour contrast simulated moreover compare recently regularize dataset variety finally given formulate regularization close proper sequel function close proper see continuous see twice equivalent hessian uniformly quadratic loss smooth hinge exclude quantify examine nf ff differentiable strict important study line regularizer notational convenience close information see regularizer w theorem dual indicator radius lagrangian primal variable multipli vector al note ordinary sequence primal respect carry involve separate follow vector outside domain obtain onto ball th soft way substitute soft soft processing minimizer q call slight abuse terminology turn minimization contribution review framework rigorous section minimization sequence number repeat e duality term proximity term next even original although carry e decrease obvious differentiable decompose small minimization short substitute constant omit right coupling right contain equation know shrinkage section bind precision use parametrize adjust wise maximum substitute eq maximization saddle concave denote maximizer general different max min naive final derive compute maximizer slightly derivation write turn envelope iteration minimize minimizer like inner section derive review proximity envelope specific function al function envelope prox regularizer choose similar slope optimize minimization next point highlight use adjust become tb mm derivation part handle term rest term discuss special qualitatively efficiency minimize simple discuss regularizers equation proximity operator regularizer conjugate regularizer envelope see converge linearly asymptotic note increase exponentially generate eqs continuous modulus compute inner weak stopping rather compute accordingly stop side increase approximate analogue objective theorem moreover eq super linearly theorem inequality see obtain weak obtain perform minimization let set assumption inner precision early safe unfortunately exchange criterion practice practical subsection discuss assumption term proximity may restrictive setting translate residual residual think locally strongly within bound quadratic q constant depend bound bound sure increase minimization contain use strong convexity around rapidly objective eigenvalue hessian term hold globally exist convex conjugate unique positive constant continuity imply minimizer guarantee become weak valid weak hold point constant asymptotic require proceed predict close super convergence complementary super convergence assumption number justify section study category comprise try overcome term category overcome pose separability efficiently minimize three constrained iii subgradient constrain rewrite auxiliary cone challenge auxiliary pg method compute project pg linearly pg overcome scale bfgs constraint lasso constrain ip basically generate call connect center ip well convergence upper non arithmetic geometric iteratively solve regularize weight technique study variational generalize jensen context kernel context challenge framework remain arbitrarily
logit boost logit theory include selection survey idea output small confusion subset minimizer empirical empirical estimator choice regular estimator piecewise span vector fourier basis algorithm smoothing square ball datum radius amount perform terminology choice convex mention cv instance local estimator focus length solve choose difficulty algorithm compute selection contrast function contrast poor statistical except target think generally dimension depend use contrast call reader find much deep insight give selection procedure main identification goal target selection build measure excess possible estimation call almost depend optimality procedure assess way framework efficient asymptotically optimal sometimes weak hold inequality remainder tend tend build adaptive belong procedure minimax model aim among typical selection build identification bic quality recover model algorithm true replace b identification consistency strong efficiency define framework exist former case bic aic aic bic prove model minimax rate sometimes share recent sketch particularly cv work procedure select dependent penalization exhaustive completely approach reader procedure book five come framework model procedure form asymptotically unbiased heuristic explain start theoretical inequality directly imply prove selection principle cross include section penalization approach principle classical penalization principle aic square bar contrast penalty prove several framework constant reference therein drawback penalty aic depend suboptimal overcome large computational possibly simple framework paper depend multiply level drive multiply several weight bias slightly penalization example procedure leave well noise grow unbiased risk principle penalization penalty oracle hold choose procedure take proved inequality typically identification bic framework soon factor infinity part picture since coincide show penalization also soon idea bootstrap tend procedure also category context context statistical risk roughly penalty complexity penalty call localization exhaustive cv procedure successively formally loo loo loo name n successively define loo computationally cv introduce alternative loo preliminary partitioning approximately successively formally much less loo loo size indeed incomplete design idea point idea index almost belong appear j bb choose randomly coincide split several time fix observe drawback risk closely penalization penalization introduce loo regression estimate risk estimator independent close see approximation estimator among use replace several apply loo loo bootstrap give bootstrap estimator modify theoretical behaviour area assess quality first split distinguished division primitive loo procedure evaluate rule primitive loo loo independently loo discuss loo selection understand cv purpose cv estimator cv cv make two evaluate biased imply risk particular risk rigorously decrease function near tend decrease rigorously precisely cv make grow list statistical behaviour cv confirm several estimator projection estimator design asymptotic expansion yield square spline expectation loo calculation picture expansion loo loo tail problem shift loo loo shift loo stay realistic biological compare bias loo bootstrap bias decrease loo fold bias nearly minimal bootstrap moderate sample ratio otherwise minimize nd report loo nd n call double cv small cv asymptotically regression equal penalize prove unbiased cv behave differently variance b several proportional around strongly unstable conversely trend stable notice bound regression upper deviation bootstrap loo tend variability cv maximal hold minimal loo complex vary quantification split complex sum split link furthermore variance cv behave differently optimal simulation detection confirm contrary usual performance kind knot prove correct asymptotically statistical regression oracle inequality estimator among square small constant performance cv bandwidth estimator contrast efficiency loo multivariate inequality efficiency kullback suffer target condition cv efficient framework classifier oracle compare penalty contrary still enjoy good nearly procedure st cv procedure naturally propertie statistical loo prove inequality describe loo several base particular classification paper identify procedure identification bic may cv loo efficient identification goal cv confirm result prove cv method inconsistent loo even asymptotically select tend context order selection compute cv numerically procedure specific result slightly hence loo variability consistency confirm put somewhat validation cv understand consider goal fast two inconsistent depend convergence framework measure cv see consistent selection prove rate consistency soon proportional negligible front gap condition good cv consider candidate simplify cv parametric strong imply cv loo condition experiment average cv second originally principle cv procedure dependent cv smoothed smoothed excellent smoothness density globally presence cv modify efficiency consistent change achieve choose robust regressor robust classical cv cv huber one model cv part play rest independent cv model selection procedure framework independent positively correlate use issue choose change bandwidth almost enough short dependence enjoy optimality long method seem several alternative propose cv modify term correct short framework partition cv bandwidth cv combination parametric dependency time datum dependent particular predict observation cv similar specific stationary cv difficult small therefore cv directly provide chapter add penalty prove oracle lead change selection first meta polynomially criterion trick cardinality contrary procedure cv consume naive usually intractable nevertheless cv framework greatly decrease cost cv density formula loo histogram recently result projection formula hold estimator risk polynomial square spline lead relate naive close projection closed formula partition write sum dynamic minimize risk piece detection cv form efficient avoid j formula discriminant loo expensive empirical formula loo user appropriately cv impossible framework cv induce behavior criterion choose cv particular model compare variance estimation snr well otherwise suboptimal snr keeping often take see identification model cv risk decrease split cv cv close link variability cv quantify minimal variability seem framework unless formula split hold loo optimal estimator final trade point account final user preference allow computational reduce aforementioned term hold question split choose proportion well define every scheme practitioner split every validation sample make compare splitting improvement take split like seem intuition explain similarly cv every point nevertheless cv procedure distinguish additional variability quantify empirically theoretically mild remain appropriate strategy uniquely split phenomena bias decrease training whereas loo high framework estimation minimal framework like report literature remain claim problem conclusion statistical instance depend loo perform risk asymptotically optimal whereas contrary snr small loo thank bias bias independently mainly goal selection drawback often consistently order provide contrary procedure candidate perhaps provide quantitative split split result direction prove cv method greatly cv generally make distinction order realistic show
mixture assign j positive specific cluster cluster model nonzero effect sparsity marginalization extend able zero induction zero discovery choice set nj dp prior prior concentration mass follow emphasize structure component dp measure entirely call dp mind nest dp group patient significance presentation empty base mean indicate similarly augmentation iterate step probabilitie j nu c p j j pc use ik draw distribution nonzero associate basically make algorithm completeness singleton b cn I accept singleton set pc c nx pc nx c used partially collapse integrate eq index draw approach obtain calculation step gibbs sample conditional step singleton prior high mass dirichlet typically extremely surprising draw implement accept solve successfully sequential pc nx condition k kp expression propose previously perform change prior draw sequential describe inference need label switching refer reader although distinguished variance extra indicate group variance sparsity many choose cause well omit shift microarray researcher shift distinguish study different gray distinguish attribute distinguish figure format posterior identify markov strong support cluster simulation burn inference show figure c attribute cluster fail identify single th discrimination encourage clearly signal attribute simple least exactly identify attribute finally simulate dp consistent example vector example simulation example increase attribute informative four cluster sample belong cluster cluster selection different error overlap true attribute joint besides correct table outperformed consider square perhaps favorable intend situation work situation square tend discovery distinguish subset tend note attribute subset select utility contain divide microarray within interval whose finally select gene end mcmc concern variable diagnostic run burn start markov another assign quantity agreement indicate dataset put conditional gene demonstrate cluster structure condition help deal switch sample allocate posterior misclassifie sample misclassifie know consist discover gene gene relevant cluster distinguish separate dirichlet shrinkage mass sequential mean study sample effective challenge choose approach modify sequential sampling plotted estimate procedure microarray heterogeneity genetic existence signal dirichlet simultaneous variable also distinguish formulation double usage dp make mcmc update model study gene expression markov monte microarray discovery decade simultaneous investigation potentially characterize distinguish known cancer obviously possess power different attract attention recently class come principled inference compare procedure largely heuristic important inference cluster proposal next formulate prior computation mcmc sampling
remain create bregman derive introduce term handle coincide unknown develop interior point truncate inner scale area compute soft every extremely light gradient solution naive size recently author propose augment dual minimization addition minimization converge inner precision method primal explicitly update lagrangian multiplier soft every iteration apply coefficient ip exploit organize algorithm present sec experimentally compare brief direction base surrogate surrogate lagrangian dual duality dual indicator duality hold maximum minimizer maximizer dual respectively multiplier associated constraint coefficient primal barrier barrier reduce ordinary lagrangian lagrangian duality f eliminate depend radius method barrier increase super see l maximizer kf strict accordingly dual lagrangian function everywhere except soft hessian hessian complexity active second derivative complementary multipli regular point newton factorization second conjugate truncate newton element backtrack objective test various state art algorithm interior algorithm size experiment mean increase increase keep choice singular value replace series ratio largest additionally approximately correspond experiment increase target experiment regularization decrease equal approximately computation element bottom ghz processors gb memory criterion variable primal term define definition objective eq dual tolerance inner choose large require barrier parameter affect behavior large gap manually choose problem guarantee super condition conjugate less large grow hessian datum poorly condition propose fast constant fig increase decrease robust efficiency keep barrier parameter well matrix law horizontal variable optimization framework dual sparse coefficient explicitly algorithm favorable condition solve million minute improve
leave unique h prove continuity external expectation converge local precisely ball radius try probability go rt g pp expectation take ss lk g rt cs il rt rt apply expectation reduce calculation expectation latter arise calculation long exercise outline acknowledgment work partially support fellowship nsf dms proposition corollary theorem conjecture remark electrical stanford university department electrical department stanford problem ise several propose limitation remain analyze complexity systematically precisely coincide graph random binary mechanic graphical vision spatial introduce statistical mechanic convention ise structural sake simplicity know double unbounded resource question precisely parameter denote probability sample change unbounded resource general graph pattern long range correlation complexity strongly graph correspond beyond critical low complexity appears phase coincide illustrate strength thresholding thresholding correlation denote dominate computation correlation straightforward exist choice graph bound range correlation graphical decay vertex happen graph family degree range characterize advanced range limitation result encode fix vertex neighborhood conditional rx change possible change fixing marginal allow set motivate local threshold candidate minimum empirical calculate neighbor minima maxima contribute consequence imply impractical set neighbor vertex vertex degree regularize logistic logistic fail indeed graph degree show necessary reconstruct notice incoherence picture difficult evaluate family restriction expand second term surprising relevant practical extensive simulation good ising model generate bias sign conservative temperature indeed graph model easy figure remove independently success vertex average empirically phenomenon threshold poorly irrespective ise grid evolution sufficient threshold prove auxiliary convenient notation submatrix index neighborhood since hereafter shorthand r x minimum omit context quantity throughout ss min min sense graph one edge node finally ss high rely ising prove uniformly sufficiently c min ss omit
j si jt false negative report happen follow chernoff bound e former solve especially tight binomial precise fig show exception slightly close report close hand would probability binomial mean chernoff illustrate time item algorithm trade negative alternative procedure efficiently eliminate accurately value measure threshold filter away technique wise hash sketch similarity decide likely filter could built maintain use hash lift overlap set hash large cost extra pass usage representation counting introduction subsequent time cosine information retrieval item weight extend instead user may informally go achieve maintain rate rate movie actor actor work rate movie relation actor actor cosine random confirm concentrated around case around possible behaviour due choose bring close surprising reporting report scenario omit dominated phase count mean fluctuation effect possibly threshold significantly change time see speedup come large transaction analysis suggest transaction speedup item transaction moderate give support usage range though locality hashing appear distinct item lsh comparison hash signature actor signature hash table range necessarily indicate lsh signature count connect actor base appear association mining system outline way thank software pointing observation lemma full version http www people paper mine strong mistake report base find association low item rely show variety high theoretically average transaction experiment mine speedup order significant mining association market setup sequence item customer canonical define association indeed capture exist lift cosine confidence closely overlap refer measure take aspect rule previous rely mean occurrence compute signature item similarity partially compare signature approach offer flexibility sense go directly item negative rigorously understand efficiency main focus many association paper usage recognize believe come carefully consider transaction pair small pair read ram modern able per second require occur frequently initially frequently enough potentially report support pruning focus mining elaborate contribution capacity fast device gb ram sequentially speed bit fusion offer read around million word even massive read challenging keep must million million per ghz cycle cycle item processing item transaction likely cpu rather I hash table cache mb core hash operation item cpu rather conclusion believe time carefully optimize count os pass cpu remain bottleneck efficient frequent item set refine transaction however similarity measure value degenerate count number usage pass least occur transaction item equivalent multiplying incidence vector transaction transaction appear algorithm sparse transaction get huge factor interest transaction frequent random transaction considerably count occurrence miss support association transaction relevant locality hashing occurrence signature occurrence sample possibly false negative sum plus initially read result form support similarity hash locality sensitive method describe show signature angle incidence positive negative signature item transaction show handle use locality deterministic signature database community threshold sometimes refer join described locality avoid employ signature serve signature take priori method join exhibit incidence transaction present measure lift find occur transaction probability support significantly I transaction transaction time sample infer false negative nearly input similarity item close hope similarity factor transaction mining negative give speedup magnitude present set work locality sensitive hashing transaction occurrence capture handle n special increase decrease similarity increase computable time fig measure lift cosine overlap find randomized probability return similarity factor reduce eliminate basic occur transaction pair expect strictly function indeed except occurrence define xx xx sort j occurrence transaction occurrence iterate transaction sort build either time relationship memory sufficient sort count hash consider ram table internal constant concrete c item transaction accord occurrence item transaction occur occur occur occur would output add particular binomial trial look si measure exactly sample standard ram latter external implementation transaction denote pair run report dominate analyze complexity sort transaction sort step spend loop assume j f consider spend proportional sample otherwise case transaction similarity expect well input happen dominate run scheme count large transaction require thorough report highly great similarity frequently could imagine greater choose small term find item
distinguish degeneracy map treat covariance agree reproduce shape degeneracy true estimate equation relation implication report single report ridge solution use ridge grid fit study initialization usually adequate good usually identifiable value degeneracy somewhat snr high degeneracy relevance contour degeneracy small error table degeneracy apparent band magnitude directly need provide degeneracy built estimate accurately magnitude prior galaxy outperform assess detail explore help sensitivity possibility model good metric combine suitable analytic many carlo thus probabilistic smoothly galaxy et parameter et forward principle use advantage hence many would like discussion simulated effort people respect whose effort would grateful team university purpose estimate modelling nonlinear interpolation template avoid use parameter treat weak approach uncertainty predict parameter goodness fit provide outlier parameter simulation ap machine zero cover surface temperature spectra error h accuracy star depend magnitude prior vary range still strong degeneracy parameter magnitude advance probability survey help reduce surveys star infer datum task galaxy physical evolution star population require via parameter temperature numerous parameter signal snr phenomena spectral type l band index generally narrow reasonably method pattern recognition use generally feature determination ap mapping ap spectra spectra variety name machine al galaxy company classification tree al linear function projection estimation example volume line index really place enable relationship label template star produce despite nonetheless try fit cause severe independent ap contrast generative transfer ap affect yet already try overcome et ball plus extension distance likelihood create solution way create label template close smoothly good within error covariance grid grow parametrization might ap template metric use mahalanobis scatter add mix impact mahalanobis loose sensitivity interpolation template template far consume also unnecessary generative forward template generative shall estimate assessment solution ap base idea interpolation spectra g star magnitude galaxy star priori span wide accurately want intrinsic integral processing comprise section latter report plot discussion http www outline terminology multivariate table summarize notation band refer general band pixel band spectrum counter band counter ap band sensitivity model band true band generative transfer explicit unknown ap template spectra generative generative forward nonlinear grid forward band demand continuous also calculate ap fit grid keep predict train basic newton forward fit detail fig remain square residual local ap calculate discrepancy predict offset n taylor reciprocal make toward estimate ap iterate vi stop basically iteration vi way spectrum stop fix ap large move function opposite limit likewise initialize true solution band several sensitivity multiply eq ap v matter model provide function derivative arbitrary work I find strong ap explain weak relative term explain fit weak minimize function reproduce weak ap little optimization ap separately strong case strong ap generalize ap ap value strong ap fit residual increment add illustrate strong bottom weak solid precise follow let ap ap band ap point strong red residual illustrate bottom panel semi regular grid weak ap easily fulfil grid weak value strong dimensionality apply evaluate near grid identify close component increment change component specify increment component component smooth combine axis arbitrary forward carry ap axis practical working mean describe purpose paper progress irrelevant constant logarithmic bring I band ap variance spectra covariance strong forward smoothing e conventional cubic spline drawback control knot spline apply smoothing control specify via result fit h unique maximum smooth fit however many overfitte component practical known author inverse model well similar datum something analytical derivative calculate select priori ap resolution choose ap update depend upon limit standardize rarely apply impact ap ap thus ap update noisy spectrum valid update standardize low limit arbitrarily ap undesirable code upper limit step offset incorrect limit rarely expect forward model good prediction ap grid ap estimate ap apply important assess via ap evaluation error include systematic mostly former distribution outlier vector transformation algebra x apply equation assume update estimate ap calculate goodness reduce q forward diag name large refer fit model naturally provide usually resort intensive sampling measure update take band snr band ap measurement proportional measurement performance improvement g illustrate thereby observe dispersion blue red vary nm nm spectra line broader remove snr model spectra retain pixel bp cover nm pixel cover nm pixel spectra experiment extensive library bp rp simulated generator library library former value k uniform unique grid incomplete reason fig range star combination star simulate ten parameter al band library step combine show five three weak estimate ignore contribute scatter case low curve spectrum highest high present limitation library offset clarity dash band calibrate publish classification currently spectra demonstrate spectra break nm dispersion rp dispersion plot varied hold variation weak little spectra course snr spectrum law end observation observation magnitude simulator background well spectra instead zero pixel g magnitude band define mirror rp sigma number spectra specific four distinct try determine vary case scatter forward represent split near half indicate g band rp adopt rise normalization offset spectra magnitude bp rp primarily area divide band procedure evaluate global ap range present measure grid temperature estimation ap star evaluate full evaluation tm strong ap either tm star tm star tm full grid evaluation tag ap star random evaluation star star evaluation ap training mean systematic twice statistically marginally magnitude standardized multiply fractional evaluation f tm g tm tm tm l tm tm tag tag problem band central nm point star ap point plot standardized unit prediction forward forward band fig weak describe smoothing fig forward band fit plus robust fig standardize varie compare weak ap band library ex leave top plot reduce goodness equation ap final adopt ap horizontal look sometimes rapid star long sometimes star depend noisy near away something adaptive encourage property cycle problem specific ap estimation ap residual minus statistic accurately significant systematic scatter spectra obviously acceptable accurately reliably distinguish subject inter table magnitude result nn limited density template grid report ap template grid noisy exact template weak star grid noise spectra leave time confirm grid limitation star evaluate full error average g report entirely model star priori variance limited line dominate ap gaussian fact clear tm grid tm tm assume already rough spread act scatter set level magnitude summary show star even full range precision twice g compare star within full star bottom different systematic star vanish turn performance quite lot little h see panel fig plot exceed limit grid section suggest include strong systematic see essentially star update predict corresponding grid ap report logical desirable report average ap act result implicitly star relaxed test train tm h precision galaxy identify poor star tm uncertainty residual lower positive residual black plot tm element uncertainty panel compare residual low uncertainty important forward comprise forward almost forward unique training grid point vary strong component build residual respect strong combine nothing else change two different problem weak ap important parametrization accommodate fig forward spline et fit band black plot standardize full tag cut show accurately variation summary applying small limit change star fortunately many star swap train broadly summary list near bottom significantly tm acceptable additional ap systematic trend correction make g star star star tm problem uncertainty h weak uncertainty g evaluation error tag star present analyse star accurately star surprising spectra simple signature
cb subgaussian theorem satisfie selector cb research foundation grant author grateful reading manuscript constructive significant presentation reference throughout present immediate implication shall admissible exist index conversely hold decompose correspond large large absolute fact hold assumption index large absolute integer factor similarly q obvious direction location large clear admissible immediately admissible hold ks ks inequality argument denote q location absolute argument observe obeys admissible k imply ks x definition preliminary also provide respect metric ball cover cover net main exploiting carry onto include completeness state proposition selector c design j nc apply union obtain immediately proposition condition optimality imply theorem hold part condition thus hold immediately crucially exploit condition selector apply proposition selector true feasible x optimality obey cone desire optimal hold see front let constraint crucially exploit random fundamental apply method tight compose gaussian similar small gaussian come tighter developed set extension follow derive cf therein lemma immediately plug lemma concentration measure normal call canonical vector np fa b gauss fa conjecture definition pt pt pt pt department kind restrict associate necessarily condition bind isometry define hence broad condition functional intrinsic low complexity implication keyword sparsity selector isometry subgaussian typical appear graphical modeling approximation recover noise throughout norm random shall section impose gram guarantee property selector nonparametric elaborate definition put model penalization pursuit factor convenience selector refer non lasso selector appropriately section column represent ready introduce restricted formalize selector purpose completeness say therein see condition tailor particular check absolute guarantee large body work principle condition strong restricted isometry constant submatrix extract integer isometry small quantity hold subject subgaussian compose column order extend family subgaussian ensemble hence behave certain introduce covariance column theorem constant eigenvalue specify section believe case random randomly orthonormal selector matrix entirely self exploit thresholding adjust select rely selector condition select significant conduct propose matrix eigenvalue impose need let vector subgaussian random vector basis copy vector note random copy strong context suppose integer number case check admissible coefficient integer admissible equivalently sense definition sufficient admissible necessarily paper I matrix normal hull cardinality denote throughout understand understand main result subgaussian ensemble class subset subject cone constraint even broad set coefficient contribution element need rip isotropic independent copy row least absolute immediate consequence euclidean guarantee small follow bind vector concentration around proposition see admissible location coefficient long clear generalize restrict property rip particular sparse lemma theorem identify canonical understand appear appear elsewhere
potential statistical significance partial stream future describe list episode block frequent episode recall identical node partially output purpose illustration type edge edge node type per per neither sure exist type contradiction r z z opposite contradict partial order x z contradiction arise proof generate among closed possibility prove drop belong z l ix z l ix l need close perform list lemma exist prove reverse indicate possible hypothesis exist lemma prove exist type closed hypothesis never present demand node scenario closed iff statement z z exist hypothesis l type prove hypothesis b hypothesis existence type z l close node analogous add I I e l l remark claim frequent episode framework event episode node node separate episode serial episode trivial parallel episode discover episode partial order specialize serial episode flexible specialized mining frequent alone sufficient propose partial episode effectiveness episode discover temporal pattern symbolic several domain www biology text mining etc stream event partially collection order episode event constitute occurrence episode call serial episode whose exceeds currently exist discover frequent serial episode stream available episode relate sequential pattern order collection sequence contrast frequent episode discovery look pattern repeat make quite different discover episode restrict attention pattern episode episode restriction order algorithm handle constraint episode occurrence occurrence specialized discover frequent serial episode episode method discovery certain partial maximal serial episode episode example order inherent frequent episode episode occur often episode frequent episode discovery episode serial alone insufficient measure episode order tackle evidence event occurrence require extensive organize sec frequent episode formalism episode describe track occurrence episode episode sec candidate generation sec conclude event tuple occurrence sequence episode tuple node type alphabet serial episode empty episode episode neither serial episode imply follow episode occurrence constitute valid occur integer subsequence fourth event occurrence event occurrence arrange accord order occurrence episode follow episode say g w must hold episode whose exceed frequency episode stream frequency episode occurrence frequency episode informally occurrence episode occurrence appear episode occurrence episode stream stream episode say cardinality occurrence frequency paper consider episode call episode say episode episode episode alphabet subset denote partial induce denote simple episode episode episode episode depend occurrence come episode represent indistinguishable occurrence ambiguity obtain alphabet order note order episode early example episode ax cr v cx cr v v v cx b er v episode cf associate relate episode episode give track episode order manner serial episode node cm auto edge swap loop node node swap loop illustrate episode track episode subset namely already accept I ready initially accept first continue ready accept encounter instead encounter rather thus either move recognize occurrence episode episode parent see initially element immediately subset episode track event event ready initial namely pair ie e tuple constitute could accept accepted list property state exactly element e state make matter state represent pair contain empty type accept reach thus definition parent accept e complete episode episode per e since added e conversely consider consider true event show started accept set e e accept eventually state accept event proof sake completeness argument property belong ie l increment frequent episode extract episode frequent serial style episode order generation candidate combine frequent episode candidate exploit frequent frequency frequency episode frequent episode detailed explanation candidate generation episode give size serial episode episode generalize episode partial order start state transition prescribe stream state soon stream final increment start track candidate episode data stream look appropriate transition algorithm count non occurrence pass count episode track stream temporal occurrence episode often application occurrence span difference episode occurrence span user window implement pattern separate count occurrence track span soon event appear occurrence allow consider start episode initialize first accept accept would second second accept move initialize occurrence final since increment episode episode begin track episode method reach whenever exist event I make stream counting like reach drop track amongst end together check span increment episode track reach use non occurrence episode constraint input episode set type episode array store episode notation iff array element store list possible state pair c tuple list tuple next event transition know next episode store explain list list event ensure transition list array piece store episode frequency episode track episode keep track transition initialize state accept start properly also encode event episode initialize work line initialize episode main loop stream list affect processing line k state set bit line recall active episode record initialize memory initialize process tuple tuple initialize line become current transition list line complete computation accept hence contain type ready accept compute list also list event state process list current old list make first state remove element appropriate list line reach constraint span accept increment remove episode start new complete pass loop algorithm handle type event share process unconditional state transition slight compare episode accept state set event time st definition event strategy stream move parse accept time add thing element start process initially inactive parse next instant perform transition event time instant essentially check removal follow increment parse type store lt false e j start zero currently process event associate current stream transition j I transition exist g increment k bag add start start bag central currently entry recall currently accepted event transition start state accept affect transition hence time entry take find add list information make transition maintain ready accept episode employ procedure level involve two candidate frequency count candidate candidate frequent episode generation exploit construct episode cf simple episode associate episode represent order array contain sort per episode principal generate node episode explain episode combine explain episode frequent obtain respective episode ba combine obtain drop ba ba combine obtain drop last node dl candidate episode formalize episode combine episode distinct event index combine episode episode pick order construct potential episode share drop maximum potential anti closure episode event type get partial episode combine er candidate valid candidate combine episode mention attempt valid partial order candidate verify check share dropping last closure r close closure check subset l style font pt b plus b leave right leave xshift scale construct drop line note dropping already frequent find frequent important partial whether frequent episode generation detail easy generate episode hence different episode exactly candidate pair vary depend combination pair case come form candidate ii x lx l lx also candidate combine x r lx lx x distinct argument every candidate order uniquely episode would candidate episode episode contain frequent episode suppose episode node episode particular frequent note drop last episode hence combine frequent combination episode valid generate generation whether maximal induction list candidate thus output valid candidate episode serial episode episode combination singleton combination candidate empty level would generate episode thus order serial episode partial explain partial every partial maximal lie easily specialize refer maximal serial episode episode property order retain potential candidate belong mine serial check generate class way satisfy maximal maximal bound user episode b parallel non episode maximal corresponding partial contain parallel episode serial episode easy belong increase order episode end class number maximal threshold exactly serial episode maximal serial episode episode episode maximal belong check use candidate order upper constraint make process efficient compare partial order wish count candidate episode node map lie chain associate interesting serial episode contain special episode keep episode episode total suppose map remain map far along special ambiguity possible episode redundancy non episode either accept choose per count device track state interestingly episode e even try counting even though convert state equivalent note count straight episode argue general require procedure episode episode tuple repeat interestingly proper element parent constructive disjoint two element episode deal node proper transition ensure counting algorithm episode elaborate candidate generation combine episode g v v vary iii suppose generate iff episode thing need ask type partial episode ultimately episode frequency alone episode episode every constitute occurrence episode episode mining restrict serial episode serial episode consider general exponentially episode da c thus inherent episode partial evidence episode frequency meaningful tackle episode frequent set episode say episode episode episode episode episode mining frequent episode size episode frequent episode order episode partial episode frequent specificity satisfactory frequent episode output suppose actually partial episode occurrence episode serial episode great specificity filter episode preference serial episode satisfactory depend count episode instead datum decide order fit datum would episode see roughly often follow well dependency episode episode addition nice partial demand episode event type either formalize give episode j occurrence occurrence partial episode measure try magnitude symmetric entropy term tie subset threshold episode threshold ii count episode maintain end candidate episode initialize count initialize zero store already increment reach increment relevant output size may reduce output efficient well wise efficiency unlike frequency threshold anti monotonicity main set episode subset case embed pattern pattern evidence embed since maximal embed often mining simulation evidence maximal almost maximal frequent sub episode threshold non level specific pattern pattern point happen threshold generate partial order episode vary generator episode episode occurrence episode stream involve episode stream stream string stream order datum consist generation three explain episode stream generate occurrence generate event need difference occurrence successive event geometric successive occurrence episode geometric denote event embed episode noise event stream occurrence successive geometric similarly type stream occurrence successive embed order stream merge stream way multiple instant stream merge episode stream final order stream improve efficiency mining embed order candidate frequent find frequent episode table ii frequency threshold iii threshold frequency level threshold time give th c post run embed three threshold embed report candidate remain frequent episode drastically table marginally overhead calculate improve considerably reduction candidate whether essentially embed along threshold frequency episode non max mix super c mix super c max max super indicate candidate episode column table episode frequent pattern various episode indicate non episode event embed order episode episode contain event two embed column two episode either category column necessarily embed say episode belong category maximal category table episode frequent table episode maximal maximal super episode contain threshold episode frequent one maximal episode use threshold episode report non maximal never eliminate frequency frequently inherent mining point evidence report actual partial group maximal category event episode occurrence episode verify pattern table frequent episode use post provide substantial improvement efficiency sized pattern run threshold episode embed second count mainly inherent partial pattern max level contribute huge frequent high huge candidate
comparison figure compute batch em depend implementation long estimate fair online would em hide markov fix appear case correspond recursive smoothing obviously explain rather whole use numerical online em recursive em batch individual entry performance estimate standard applying averaging sequence range comparable unitary also display post process average start center mild averaging recover suggest equivalent picture estimate effect index start variance early guess parameter figure thousand important optimally limiting provide demonstrate hmms rely recursive computation functional proposition require quantity course encourage theoretical analysis miss although idea analyze become point view many generally carlo computation report direction theorem side sided bound allow transition q variation obviously note x x probability lemma familiar normalization factor second determine side bound g backward function pseudo easily check indicator interior family finite first markov borel quantity latter converge prove check imply deal normalize log suppose drop law proceeding vanish corollary one simple lemma prove equal increase decrease lemma concentrate apply equal define light analyze prove exp algorithm remark call fix time series combine root methodology problem consist exploit purely hmms recursion algorithm resemble sufficiently convergence potential recursion quantity involve propose comparable hide markov maximization concept range practical impact year markov classical variable value rise procedure allow modelling situation ever em dedicate routine maximize preferred alternative ease hmms store hmms challenge trivial smoothing computation approximation log principle comprehensive recent review advanced require method em algorithm hmms take ingredient recursion allow smoothing functional algorithm specific general address purpose case independent proposal hmms possibly continuous key observation recursion smoothing scheme introduce currently provide algorithm coincide interpret em recursion infinite generalize argument provide large organize brief computation hmms propose online em devote previous numerical proof finally online estimation observe gaussian hide state observation parameterize convention pdf arbitrary respectively state initial pdf hmm start parameter consistently trajectory discussion density function characterize recursion belong sx non necessarily invertible natural parameterization maximum assumption ii maximum belong exponential family algorithm stick representation section detail hmms usual familiar avoid unnecessary log likelihood briefly ingredient recursively recursion idea study largely exploit discussion chapter addition usual q quantity obviously quantity allow interest decomposition update recursively available observation check claim proposition constitute carry analogy observation choose decrease stochastic approximation require performing update compute role guarantee behave degenerate hence properly sufficient first early intend dependent reduce simple recursion analyze approximation random perturbation key hmms maintain filter become acceptable proxy auxiliary although recursive auxiliary put em argument maintain filter backward constitute usual complete mathematical though recursive author particle filter markov acknowledge constrained instance fail principle algorithm analysis algorithm time important observation tends interestingly result limit auxiliary instrumental section word relate limited important mle adapt space inspection limit suitable converge limit normalize n various influence vanish ignore result combine family vanishing rewrite follow define hmms compact interior continuously interior fix stationary contrast interpretation em sequence hmms investigate convergence kullback leibler divergence property filter case use find identity conditioning find infinite future recursively indeed requirement estimation hmms show converge constant h theorem quantity obtain imply ii stable limit parameter highlight contrast past verify sake case take state conditional case consider matrix respective constraint transition mean statistic separate form nature state example slight incorporate h early modify term iff notation refer application step g n nature update component approximate covariance derivation straightforward scalar additive gaussian model several continuous channel comprise batch may maximization directly easily constraint avoid complete example observe chain k q ni ni ni ni trajectory q identification separation noise actual reflect systematically start initial illustrate consequence em variability estimate bold observation slow iteration obvious similar picture limit guarantee theorem range thousand plot large rather computational
facilitate object live metric oracle retrieve sort list distance respect relationship new question rank relationship object rank r introduce notion rigorous approximate triangle rank capture inequality relationship notion rank investigate randomized scheme exist near neighbor oracle average ask partial develop decompose likely get object tendency stay generalize build similarly locality hashing retrieve near query hash depend distortion criterion combinatorial rank distortion space less homogeneous asymmetric every capture neighbor variation extensively see survey space intrinsic net structure search application net author growth metric restrict growth guarantee randomly object neighbor underlie necessarily prior generalization rank search oracle study algorithm question database author combinatorial near neighbor define approximate analogous bound depend crucially combinatorial database notion triangle spirit net show complexity complexity retrieve near neighbor phase builds exponentially improve polynomial accept list framework access distance force number question ask particular infer rank also sample top believe search ask address show build decomposition diameter set tree intrinsic ask framework access exist hierarchical decomposition constant approximate neighbor query query time remarkable survey lsh instead hash table case several choose object near neighbor lower bind hash space distortion depend knowledge study first moreover hierarchical demonstrate efficient formally distance object directly access return rank set equal near simplify notation indicate unclear ambiguity note rank object object r question create rank add ask precisely ask object select question ask sort object characterization space form approximate first define relationship triangle imply rank object small triangle rank around rank symmetric column give distortion monotonically exist linear four triangle inequality imply first inequality nearest problem neighbor problem object hash sensitive probability search rank whether problematic sense tell close point candidate hide extent violate see triangle provide randomization lower randomize require neighbor exploit fact choose database densely ultimately sample observation close close sample conceptually randomize reduce consumption performance small retrieve near bit average search retrieve near query randomize develop indeed constant search answer oracle randomize question ask neighbor expect consequently within object database limitation might considerably inaccurate come point extend tree analogous result new hash application sensitive hash sensitive hashing depend rank capture rank pick sort object exploit far distortion special locality sensitive one consequence retrieve point question rank retrieve section assumption consequently rapidly exclude candidate object algorithm neighbor build succeed query neighbor w build top database bin close find r oracle comparison bin one chose hierarchy top answer question ask level question need question even sample close query low level near repeat database retrieve neighbor leave union ds ia nj ni ji ji union probability higher succeed h immediate easily modify neighbor hierarchy p interested hierarchy desire section configuration guarantee neighbor query expect question ask query database show star branch weight database object connect edge range object branch star edge connect neighbor object answer question ask need near neighbor graph short distance see randomized example direct find neighbor direct neighbor fewer expect sure retrieve ask find query know exclude know building decomposition separate root leaf close show neighbor lie centered object lie object r w narrow close appendix query retrieve neighbor question requires expect sort sample retrieve nearest happen scheme fact illustrate exclude object nearest vice versa width exclude ask characteristic word decompose separate know characterization build search recursively decompose database binary tree clearly illustrate rank notion diameter set proof diameter symmetric distance decrease diameter hence diameter equal diameter choosing take assume want appendix cut diameter reduce diameter constant general roughly good cut divide h interesting fall outlier likely bin bin away ever set node rank object end object notion median randomly select hash also separate computed time randomly sort object popularity next exploit randomly cut stay together sufficient search object rank rank distortion say single object input rank prove retrieve neighbor distortion distortion intuitively situation roughly constant r box distance address object database exist whether efficiently object need object assign meaningful numerical similarity raise interesting good efficient right characterization work capture one distortion relate present combinatorial bind search characterization sensitive hashing depend rank manner locality sensitive hashing search comparison bridge search situation technical lemma ball uniformly ball bin choose less let fall bin else see denote appropriately object query visualize distance htbp locate property tell rank set object less w close proof expect close less half identical property choosing make five true object level level object object let close property triangle section eq object object bin less tell close sample fall bin close level less inequality property summarize need question total level fail probability fall level upper question immediate store close requirement exceed bit property proof appendix external query find object ask question particular object near neighbor triple let two inside star contain case could ball even place end branch contain node xy htbp branch star edge line graph path connect database choice neighbor query weight assign direct direct neighbor weight path principle expect run solid star object know answer question position point fig query branch object call edge equally word near neighbor must neighbor equal question except question
svm parameter gaussian denote cardinality model hold set follow q finding scenario confidence prediction make unbiased label argument give non outline model strategy training randomly train training point label proceed split validation calculate validation proposition require specifie function would test use empirically determine would estimate uniform cumulative difference bounding difference therefore measurable vc apply estimation accord unknown generation minimum misclassification satisfy choose function true opposite bound proposition apply unable cv traditional svm model cv make vote acquire uci repository sample unknown list attribute gaussian negative sample vote carry list fold routine split fold testing procedure train validation training carry procedure validation exclude training sample cv strategy result standard validation selection fast cv exclude set exclude selection cv less percent cv inferior number approach select margin testing despite improvement close cv rate overall bad cv exclude rate except standard vote since number traditional possible show bind value subsequently small figure bind early monotonically considerable partly quite vc correlate encourage achieve validation generally gold strategy margin perform error htbp give novel encourage shift selection competitive relation present restrict svms measure choose base margin applicability future research near measure acknowledgement would acknowledge project ep project fp cs ac selection measure batch help leave selection bind comparable method theoretical success demonstrate choose analysis fit analysis popular practitioner loo concentrate vector svm loo number alternative literature instance explore model span scale application drug automate methodology classification svms selection use put employ predefine propose pac bayes measure performance classifier estimating svms use optimisation recently achieve maximum obtain select svm cv involve cv keep model cv applicable margin idea label measure observe unknown map sample svm svm optimisation slack primal vector denote dual optimisation nonlinear optimisation problem x lagrangian discuss validation whole fraction z measure validation functional large
enter distinct forecast distinct functional multiple function evaluation functional participant allow possibly distinct forecast issue relate huber huber bregman characterize probable arithmetic perspective apply conditional quantile traditional quantile bregman original form could employ generalize least distinction example census census measure use census estimate include squared mean percentage census impossible designing estimate aim se consistent distinct statistical functional census way point open nonnegative nonnegative part strictly consistent equivalently part fy dominating hold unless relative consistent hence sketch statement immediate argument general necessity prove let probability f exist x strictly remain necessity principle x pairwise partial yield score validity scoring function sketch yet part x fy fy fy fy nonnegative positive strictly prove necessity principle usual integration convex measure point functional relative class focus center absolutely compact appendix forecast scoring forecast rule take normal forecast robust forecast international journal stein loss berkeley mathematical ed university california note consistency guide evaluate quantile j banach norm ed north average law lead journal law refinement quantile median message toward fr mean journal la et de quantiles quantile quantiles journal distribution conference f performance volatility mathematical point nd competition implication international journal k competition study international journal forecasting newton j forecasting competition journal forecast national evolution sale forecasting management forecasting journal forecast j k usage application deterministic forecasting technology forecast science r pp r forecast verification weather journal american association asymmetric van p proper h ph thesis california berkeley k schemes journal public economics l error volatility forecast comparison volatility volatility correlation forecast time pp machine journal van design apply statistical economic analysis university journal journal building minimize transaction journal american approach focus statistical activity news c survey production possibility inform journal statistical functional west asymptotic scoring conjecture example universit single forecast continue science forecasting assess depend forecast observation error inference forecasting point scoring function forecaster functional score forecaster forecast rule loss forecaster receive functional scoring rule forecast link bayes scoring scoring expectation ratio quantile bregman quantile piecewise ratio expectation weight score consistent functional instance quantitative finance phrase rule median point forecast aspect human activity major uncertain forecast nature still reason report requirement communication type situation assess average thus criterion take realization list commonly score generally score orient forecast absolute discusse point forecast proxy se mm percentage mm table public table survey volume review forecasting group application area iv article forecasting paper contain predictive forecaster forecasting score monotone square surprisingly forecasting scoring function square popular particularly group absolute percentage survey conduct summarize al square business demand sale monotone transformation employ percentage scoring sum column exceed column simultaneous scoring article estimation study fp se mm international journal forecasting journal forecasting statistic mm apply journal american journal statistical iii journal business journal mm american mm weather journal weather mm se option consideration practice standard theoretical arguably theoretically principled year note verification measure effort concept principle forecast verification nothing change ask deterministic forecasting verification al states requirement still circle day ahead forecast blue focus seek forecast asset realization series issue forecast asset actual true forecast forecast along asset successive trading day little performance forecast score list low absolute percentage scoring perform yet forecast scoring se se mm value growth case report probability point prediction square specifically means ask ask provide distributional point may report mode apply function may similarly receive concern example help variable square minimize refer rule practice argue complementary way ex forecaster functional forecaster scoring permit mr issue point forecast scoring forecast absolute bayes median percentage scoring density optimal forecast median random whose call arise summarize discussion forecast rule distribution generate mr forecast forecaster predictive distribution study understand follow fractional exist readily see forecast small thus forecaster predictor score mm rule forecast se mm correspond mr mr bayes forecast mm bayes alternative request forecaster quantile scoring roughly quantity quantity distribution concentrate mapping ft consistent equality strictly consistent remainder notion comprehensive addition scoring finding forecast ratio expectation quantile subject weak regularity bregman subgradient apply quantile piecewise functional notably functional popularity application practice forecast forecast either expectation quantile scoring develop forecast theoretic whose comprise outcome probability domain equip constitute probability distribution observation decision maker represent cost maker act rule maker uncertain future represent loss bayes act bayes domain action simplicity common equip borel algebra furthermore score theoretic observation score pe forecast optimal forecast act interval case partial derivative subsequent impose score share multiply forecast pose predictor concern argue homogeneity desirable scoring domain bc x decision problem scoring probability distribution instance prediction b decision resemble al much work assume domain b favor forecast focus forecast distributional forecaster functional huber current point presentation definition score class scoring relative note opposed scoring line probability finite moment parametric property forecast decision dual connect forecast evaluate scoring give forecast state differently consistent identical function despite immediate define appear widely result scoring suggest score satisfie score relative proper distinction useful fail make forecast mu discuss score consistent framework assume expectation domain penalty arise forecaster forecaster loss belief encourage probabilistic scoring act domain score consistent scoring induce natural score functional proper general theoretic proper describe evaluation notion back whenever feasible definition consistent relative relative domain map consistent scoring strictly strictly consistent concern scoring function admit dominating domain weight w integral proportional strictly relative strictly weight score predictive probability density proportional density result forecast forecast function result score median functional recover correspond mae prop forecast permit table notably relative quantity represent freedom limit multiply carlo derive see appendix b optimal original scoring consistent functional functional become forecast positive squared percentage optimal point percentage scoring derive situation weight forecast routine show consistent eq weight scoring consistent characterize class scoring general functional al condition functional sum quantile question include converse generally characterization practical way describe characterize scoring function consistent useful functional include point identification available argument partial derivative example expectation derive square fourth identification mm mm forecast probability argument respect yy consistent principle subsequent principle example expectation quantile ratio expectation technical rely property refer know squared scoring functional probability moment expectation turn subsequent identify bregman result score compactly subgradient score mean functional measure bregman bregman representation score homogeneous arise multiplicative constant unique symmetric introduce rich family homogeneous bregman function namely homogeneous scoring restriction worth forecast bregman event compare mr mr forecast bregman forecaster mr bayes functional measurable f function satisfie compactly subgradient arise section relative consistent subgradient scoring respectively cumulative finance var evaluation forecast generally relative measure heart quantile score function consistent equivalence historical quantile functional relative class suppose satisfie class compactly quantile form generalize piecewise order transformation functional monotone mapping homogeneous mae log mae sd score mr green simulation score mr bayes introduce functional measure rule point piecewise score similarly rule asymmetric piecewise surprisingly quantile class scoring interesting combine key characteristic bregman family measure interval moment scoring satisfy compactly probability strictly finite expectation distribution convenient quantile huber popular finance vary elegant appealing sense consider relative distribution mixture continuous challenge use measure evaluation forecast lee consistent scoring remain unclear might forecasts measure mode state informally mode optimal forecast rigorous forecast rule scoring modal length explore differently know member lebesgue sense represented become available put scoring attain origin lebesgue continuous unimodal median coincide theorem convexity survival adequate people accept reasonably accurate death forecast apply et restrict attention forecast domain wind forecast volatility theoretic forecast target take discuss assume wise forecast bregman denote product sufficiently smooth score generalization representation
lie cube u contradict combine contingency integer wish estimate number constraint program begin expectation determine solution integer satisfy approximated see discussion deviation sum independent geometric characteristic assess refer characteristic time separately associate theorem parameter require absolute zero drop away hold matrix satisfy since hold possible non increase maximum r uniqueness achieve determined row maximize tu st r tu constrain lie add fix maximization equality follow concave fix equality strictly strictly strict concavity tu say paragraph unless hold turn maximization fix accomplish choice lie compact maximize maximal point contradict fixed choice row entry conclude occur extreme take away require occur nm mn quadratic dt jk jk jk jk dm form establishes allow independent relationship covariance diagonal note w ie ir ir ir reduce dimensionality reverse hold v concluding proposition prove q corollary bound taylor eq jk jk jk jk ga ex u u count direct construct progress small index index pass value remain variable I tt ct condition require prove analytic prove evaluate rest behave integer characteristic individual characteristic function value happen handle transformation constant prove q proof integration denote condition iv nm enough v v ir characteristic function constant validity sum cm sum equal calculation convenient dropping exponential c estimate contingency constant column exact approximation long accurate cm approximation joint correction produce integer row sum undirected degree begin consist q provided solve uniform degree eq lattice possible degree near produce total twice formula near regular graph graph whose asymptotic depend characteristic expectation approximation expectation similar contingency case binomial degree let expectation invert variance fourth formula estimate degree identical formula improve give note symmetric since degree graph maximize far assign degree get degree exactly correction constant table error vertex formula work near half number degree bad even consider degree entropy edge maximum variance p j lattice possible degree compute p l gauss accurate formula degree I approximate correct entropy satisfy integer satisfy satisfy entropy density first moment validity correct demonstrate contingency graph subject polytope volume ann mi usa mail address new ct nsf grant dms dms united count integer c distribute central apply sum suitable moment select propose maximum gaussian volume kt entropy discrete entropy distribution cube integer satisfy density form density may constraint uniformly integer suggest mean say expectation ba maxima j far entropy sum infinity variance sum sum approximate accurate correction produce proportional derive cauchy integral coefficient generate maximum widely specification clique cm extend integral asymptotic enumeration contingency table integer sum know sum maximum formulae vary formulae integer integer advance unify use example moment sum variable determine maximum de square statistic uniform
collect undesirable time concern heterogeneity difference category model likelihood bias relatively variance category na estimator need increase variability bias trade recommendation actual study motivated reduce particular focus estimate genome wide control disease status study adopt model significance methodology discussion specification assess performance specific utility association candidate three conclude log odd interest follow asymptotically snp code copy allele loss minor allele vs case essentially familiar significance conduct vs statistic standard calculate subsequently without adjust nan hypothesis critical rate simplify although conditional unless na I bias demonstrate genome wide linkage study correct q standard conceptual connect logistic control correspond correspond statistic correspond bayesian focus factor influence association disease population additive dominant allele snp power association simulation study practically range significance sample turn normal moreover conceptual cover association analyse quantitative conceptual coefficient show build conceptual normal publish association coefficient threshold unbiased mean conditionally development evaluation performance author mle correct reduce simulation tend large estimator treatment conditioning author unbiased population completeness family estimator conditional statistical n b dirac define equation completeness n estimator unbiased information genome diverse genome linkage common apparent snp positive performance bayesian paradigm belief significance mathematically discrete probability hyperparameter prior history shrinkage theoretical discusse spike frequentist procedure reflect belief imply favor could belief extreme outcome value small correspond snp beta evaluating propose robustness prior simulation indicate preserve shape inverse shaped density inference exist although variance precision subject hyperparameter inverse produce component represent however estimator snp difficult influence problem influence sample actual reflect disease trait snp know perhaps genetic date show long largely association detailed equation provide use characteristic n traditional chain technique mixture augmentation extremely alternatively carry inverse metropolis iteration discard first burn discard sample conceptually method take account uncertainty lack information test confident detect reflect robustness paradigm inferential say eq set model discovery belief specify utilize decrease q view normalize constant therefore two normalizing use bridge sampling propose iterative st compute begin easy ij obtain carry simulation examine performance bayesian compared develop second nine examine na I unconditional mle estimator likelihood recommend estimator prior high power discovery estimator negative truncate zero flip occur snp population snp associate disease allele allele allele decrease factorial design factor investigate reflect genome association rate typical snp throughput depend snps statistic population uniquely determine detail simulation simulation generate statistic nine genetic apparent relatively consider h inferior implication put put little I estimate behind bayesian bias well compare perform power bias big standard deviation root mean square half match pair base run factor influence bias qualitatively sample small significance stay show table show slight variance difference drastically association study snp keep comparable increase comparable range simulate circle bar long horizontal deviation sd root rmse conduct range generating summary size apply let detect snp log qualitatively genetic scale study practically significance na I discovery report twice association power size na I never sufficient snp na I sample center believe practically useful sampling examine level equal report apparent perform snp power association circle bar average sd rmse rate calculate na I simulate significant circle horizontal bar long horizontal require genome wide study outcome specifically association diabetes previously via genetic estimate report easily study original genetic na I five estimator literature design power knowledge snp rs snps five complete serve variation original ci interpretations ci different although sample way credible normal averaging estimator equation ci ci specifically ci quantiles conditional note propose compete estimate mle wang et candidate study control significant I rs independent considerably rs I association ci ci ci b pt follow quality allele test focus snps analyze snp propose use discovery actual control control discovery ci ci ci b estimate mle ci pt ci l follow snp roughly value apply report table extreme correction estimator little change publish less general produce longitudinal phenotype diabetes control trial identify conventional value association c vs snp perform pass effect pt ci ci ci follow follow robust reflect belief iteration odd addition conceptual normal
share probably rare position challenge enable simultaneously improvement sequencing machine include enable convolution additive dominant averaging e replace build add column proposition conjecture assumption variant rapid sequencing enable prohibitive improve trade pooling design individual relatively low individual pool design sense general simple computer enable rare allele especially high individual technique enhance genomic sequencing recover identity rare genome association successfully association phenotype numerous find various trait limited bias although find association trait far explain trait disease interest however study large infeasible recently currently sequence sequence throughput cost hardware generation sequence utilize read genomic order higher sequence high rapid sequencing address question infeasible sequence possibility genomic sequence individual extensive amount genetic ability enable rare population thus fill us discover rare predefine seek variation nucleotide minor large population certain sake discovery rare variant generation sequence million typically dna pre genomic sequencing hybrid dna rna region interest region throughput make task naive costly option hundred thousand since whole genome throughput much capacity naive inefficient pool feasible pool several individual together sequence pool allele population measurement allele allele traditional pool sequencing infer rare recover rare rare sequencing pool tackle identify individual design e mid biology streaming communication see comprehensive survey work try rare allele pe er design code enable allele construct provide signature enable allele observe contain signature offer resource recovery single rare allele detecting albeit al enable identification sequence accord ideally could mix use therefore suggest pool read obtain identity chinese accurate recovery allele pool keep approach recover individual variant testing yet study allele identification identity allele analyze deal cs enable testing thus identify snps cs paradigm reads effectively sense active development research daily basis cs many field pixel zero reconstruct term dot measurement reconstruct testing set allele interested indeed measurement dna pool individual take rare single whether consecutive theory theoretical robustness noise numerous technique benefit development fast improve suitable identify rare allele direction extensive aim benefit apply cs identify rare scenario sequence find benefit apply large improvement per organize rare reconstruction pool simulation efficiency approach offer conclusion provide cs description identify mathematical sequencing utilize standard cs reconstruct k solving equation measurement sense vector case scalar recover uniquely uniquely specifically solution find number zero entry somewhat stem contain information require isometry rip briefly state matrix almost matrix invertible vector able follow computationally relax close still solution problem reformulate follow problem convex realistic problem measurement norm therefore reformulate maximal level obtain add term cause available enable practical thousand choose wish reconstruct individual reference allele alternative allele represent allele allele count individual interested rare minor zero cs real restriction expect reconstruction enable fast namely sense matrix individual random individual sense sense measurement perform pool ensure affect individual equally dna read genome perform allele together cover number measurement represent measurement introduce various section measurement measurement present illustrate formulation people snp green allele assign pool probability example first pool dna individual pool pool sense incorporate constraint original rare range aim sequence apply generation clarity experimental factor add establish allele vector rare allele perfect pool likely read sequence read pool segment read allele probability vector allele eq q dna pool practice position cover read read generally read cover specific main variation bias content experimental adopt rare allele fluctuation sample c assume noise exist besides limited read scenario sequence read sequence reflect read sequencing may sample due basis read mis wrong genome base dna read different result three different allele allele read read produce reduce observe read vary sequence technology library quality control alignment algorithm realistic value know correct r unknown run one single incorporate framework pool factor far error resemble one propose model namely dp hard exactly amount amount result measurement actual obtain dp error entry mixture dp noise unknown unchanged effect modify actual rare allele oppose classic cs know dp region consideration give size snps study small region result treat isolate snps indeed snps read cover snp contiguous interpret target region read cover multiply length genomic read successfully sequence technology million sequence machine greatly influence perfect might desire varie read alignment million fix rather modern genome report simulation adapting need per snp denote merely section interpret read person pool approximately individual pool therefore average coverage pool visualize noise read error specific scenario instance rare identify level sampling noise number snp correspond zero separate read right panel dp actual error reconstructed obtain clearly visible infinite expect deviation size moderate error panel rough quantization vanish reconstruction aggregate information read level irrespective error reconstruct leave probably sample per sequencing lead cc dash expect frequency pool read dp pool dominant observe gradient trade fit specify maximal allow often desirable measurement adopt throughout although different entry integer post processing zero post large value ss equal describe cs combine strategy improve obtain dna dna additional identification mix together attribute sample pool dna pool dna specific apply still pool utilize increase effective number read decrease usage solve total read easily cs vary accord technology present trade run extensive simulation evaluate parameter range simulate instance apply order accuracy input reconstruct sequence reconstruct due coverage error term reconstruct yield false false show restrictive reconstruction reconstruction reconstruction occur problem read etc individual correctly rare allele false discovery discover allele measure reconstruction efficiently various simulation experimental simulation follow individual rare allele vector allele frequency allele low allele varied region length read length different coverage kept fix read estimate cs relevant error present different finally display behavior dramatically rare fig present number snps allele vertical display correspond demonstrate naive sample available merely evident panel decrease insufficient cause almost cc treat simultaneously mean certain parameter simulation yield reconstruction simply demonstrate right axis display panel region present scenario unit individual correspond case appear cs efficiency present simply I individual cs divide naive cs black line scenario snps score high number approach resource provide hand read per coverage question read successful small snps yet mostly noise coverage coverage individual code percentage reconstruction white line accuracy transition sharp overcome reconstruction criterion lower naive difference efficiency considerable naive allele cs simulate individual addition higher encounter practice take reconstruction fig cs way allele rare allele much naive one rare allele frequency rare allele noise specific reference appear dp figure compare performance read read person insufficient reduce coverage snps read read read significant performance see study dp pooling protocol overcome cs read dp solid read dp
stein shrink shrinkage estimator introduce empirical entropy loss class prior address invertible case minimize mse show advantage improve develop rao theorem idea mse matrix first haar obtain solution provably mse shrinkage try begin naive replace obtain remarkably expression refer limit oracle estimator intuitive compute provably dominate size estimation autoregressive show implement author substantially introduce represent simulation adaptive summarize conclusion letter letter transpose conjugate transpose frobenius positive definite independent identical distribute assume mse constraint specific shrinkage solution necessarily mse variance usually ill pose hand well reduce reasonable towards follow shrinkage matrix generalize restrict attention shrinkage shrinkage next unknown approximate optimal shrinkage begin derive oracle error follow subsection show optimal let equation gaussian evaluation expectation identity eq specify distribution depends prove show shrinkage rao estimator describe provably improve sufficient estimator rao rao expectation never rao covariance expectation satisfie q large asymptotically achieve optimality point indicate contradict reasonable parsimonious formula assumption oracle iterative procedure initialize initial iteratively refine covariance non negative yielding generate limit eq latter force meaningful plug simplify q easy shrinkage iterative plot estimator gradually towards htbp htbp htbp increment fractional long often internet traffic phenomena mse fig fig experiment evident perform closely small improve obvious figure converge increase decrease regard characteristic explain well correspond preferable covariance time mse mse mse always dominate occur shrinkage c coefficient next narrow band sensor complex signal interference vector assume linearly combine arrival obtain vector estimator improve method estimator yield value hermitian complex part covariance represent extend estimator element half mutually independent one angular db sensor implemented measure varie gain fig performance several even well improve great paper shrinkage covariance art virtue rao oracle estimate analytically provably dominate conduct estimator structure coefficient significant target scale theory identity target target theorem require treatment haar singular wishart state lemma mean singular decomposition comprise orthogonal joint jacobian denote diagonal substituting factorize similarly column treat separately haar matrix unitary independent complex fourth since eq take identically substitute element identically note change variable obtain eq jacobian similarly eq scale eq calculate go prove th column expand coefficient
usually bad case instance square instead eq work term essentially close close bias estimate crucial linear theorem non oracle section hold numerical constant consequence yield asymptotic leading imply optimality front mean actually observable check experiment prove select near close selection restrictive relaxed sufficient family estimator ingredient uniform probability event define prove uniform result refer proof instance ridge regression certainly additional deviation core least theorem would assume gaussian asymptotic oracle heavy tail clear scope comparison estimator close term negligible oracle asymptotic meaning implicitly assume problematic may grow difference since like statistical non selection therein inequality procedure mostly ridge assumption sure asymptotic optimality ridge whereas non exist bind weak compare e exist regression require assumption near regression case method computational price increase study comparison mild lead penalty deal put aside nevertheless complex dependent since explicitly appear mention special feature minimal may appear dimensionality jump occur theoretical risk prove interesting correct minimize proportional simulation illustrate take use spaced goal regularization learn number neighbor bandwidth size jump penalty variable get jump penalty leave amplitude jump particular due replication knowledge minimal consider find freedom integer jump second degree jump heuristic jump occur penalty outperform outperform multiply penalty factor useful minimal proved extent slope heuristic modify light penalty prove theorem assume nn extension example regression problem simultaneously covariance regularization parameter penalty successfully mild regression e base certain generalize probably modification rely case slope still still linear general appear naturally conjecture valid dimension contrary modify x repeatedly every every define main every actually consider separately ridge parameter prove useful elementary specific true symmetric convention define hold equal result q lemma schwarz every concentration main extend form gaussian vector ridge set come result standard proof constant new framework event proposition concentration real extend optimal good assumption satisfy case combine prove prove lower similar slightly constant consequence proposition prove separately apply ia orthogonal ib corollary let particular least n q proof choose become eq soon derive imply deduce event jump use controlling consider showing imply distinguish eq take compare choose hold soon inequality hold soon satisfied section yield take use n compare hence side since soon q comparing get eq hold hand soon choose lemma proposition hold true define replace proof start eq become algebraic bound use definition bound remainder every inequality eq imply handle implie since reasoning union theorem us hold get deduce noting remark use end inequality upper bind assumption continuous common well vector therefore corollary similarly every taking possibly zero term limit ss integrable eq q hence result first root well rational fraction every q either polynomial degree general orthonormal hence every actually prove since concentration similarly exist every realize let q every similarly eq follow apply every finite lemma proposition acknowledgment improve acknowledge reference detect european grant independent post cm cm cm cm project team cs paris france l cs paris france select several linear non include kernel ridge spline context plug prove simulation multiple significantly calibration validation smoothing spline kernel classification issue framework practitioner use cross drive procedure rarely choice square select among quadratic include choice spline neighbor study signal constant minimax rich harmonic function main present class drive selection satisfy section parameterized ridge spline one parameter inequality context heuristic curve improve procedure kernel near neighbor locally solve example estimator let assume observe design set quadratic norm linear rest vector family deterministic parameterized combination result apply mind input projection projection selection wants parameterize column matrix assume reproduce rkh spline estimator regularization unique equal parameterize case positive framework back b norm f pf detail identity lead smoothing combination depend group single parameterization parameterization particular way alternative oracle selection procedure particular upon value large drawback estimate drawback give directly drawback minimal study risk proportional although several paper projection hold intuitive behave completely rigorous conclusion bias well classical selection empirical opposite suggest sense behave distinguish huge increase penalization follow first analyze paper indeed suggest around jump select degree describe generalize previous assume projection
follow note initially begin deviation weaker large weak sufficient error rapidly careful dynamic proceed need lemma weak need say bad hoeffding bernstein permutation fix price period linear let objective least b ij nb satisfy similar appear next relate value offline ready long violate achieve bound fact condition competitive item unless vector demand demand demand pair complement vector consist least item achieve boolean string represent item illustrative string p cm vector c c c c complement demand pick bit string vector share boolean string demand demand coefficient input demand demand demand accept construct hold eq proof offline sum input demand w type item unit item remain e prove future research x proof bad bad p inequality lemma sum price inequality ease omit subscript primal bn bad permutation term define easy z h lemma p second define tn p price distinct take union prove proof similar p pp price learn optimal dual kkt x nb probability permutation h nb z nb next distinct value customer bid bid conditional bid value customer bid bid constant similar necessity end bid potential mistake instance w bid mistake call mistake size mistake total mistake claim potential therefore consist v common complete probability central constant probability equal lagrangian optimal pg pg pg return differ value expression expression distinct online wide attention management reveal come online management arrive period stay request decision bid capacity receive sequence request user intend usage utility objective appear match online management resource recent development reader problem formulate online linear integer discussion focus linear programming integer online linear constraint reveal corresponding coefficient function far immediate without observe precise offline restrictive constraint meet q linear objective maximize propose algorithm programming need make assumption adopt column coefficient arrive pick start permutation number priori adopt comprehensive review intermediate case bad uncertainty evaluate bad input offline hand priori input great extent choice critical suffer weak draw one drawing algorithm history price relax knowledge multiplicative error let model offline solution permutation arrive near program also multi decision use offline program decision choose satisfy use objective maximize entire special online programming usually refer maker face fashion maximize application arise context worker schedule permutation widely paper constant competitive sized near sized paper higher near computer edge request request offline decision integer discussion problem name online packing problem main yahoo etc ad attract past decade majority daily budget relevance bid display maker engine dimensional allocate corresponding offline special case programming vector except entry attract interest recently competitive comprehensive section paper competitive dual act pa work threshold initial price decision price initially step precisely state program way competitive ratio condition far demand large prove necessary counterpart online lp theorem show appear algorithm problem permutation competitive random first depend side model quite uncertainty implement dependence dependence strict sized however contrary input engine search category million reasonably large input furthermore result might interest finish section programming entry condition q science operation management community recently random model attract grow popularity adversarial still capture matching online packing result obtain category near dimensional prof competitive achieve look multi need technique apply randomized maintain programming schedule prove necessity dimension achievable clearly high programming online utilize decision author develop update price carefully achieve competitive check problem although idea nature require answer price significant general packing vary extension propose achieve side dependence remove also improve dependence improvement dynamic recently study competitive way root cubic competitive expense contrast cubic dependence table om opt besides allocation call adversarial generalize permutation situation input model develop achieve competitive dependence make comparable technique operation research price various management resource problem arrival availability poisson bid price investigate author bid asymptotically arrival idea discuss arrival make contribution scope many dynamic learn knowledge arrival instead reveal design answer update often quantify update trivial bind add difficulty apply concentration cover dynamic share idea trick trick unknown horizon price horizon careful design section ratio simpler regard necessity condition extension notation integer price state initialize p dual price vector decide allocation execute constraint attractive solve small program side guarantee trick subsection learning rely program observe unless ratio first optimal dual price sufficient column reveal online fashion algorithm substitute substitute constraint optimal value simplify discussion necessarily point add uniform p simultaneously pa effect perturbation program differ offline program q py dual p pa condition therefore value pa value optimal dual close offline however price p discussion attempt price p random input replacement construct feasible dual linear program precisely follow price p say bad bad every union price
affine subspace avoid sphere incorporate necessary algebraic improve problematic minimum local minima affine initialization explore number develop would semi set substantially stream algorithm minimum assume currently develop robustness instance identify careful initialization thank anonymous helpful comment chen constructive manuscript hybrid question thank benchmark motion segmentation support nsf grant partially nsf zhang school mathematics school california se mn edu mn zhang edu linear e cluster minimize implementation amount storage modeling subspace require evaluate http www edu model vector video lie affine face condition face give rise hybrid linear mixture suggest utilize generalize algebraic agglomerative spectral curvature use multiscale hand e separation subspace probably kf aim partition formally parameter try minimize function example following assign newly cluster significantly worse global recent subspace affine replace function replace goal point beneficial moderate stream accurate order address algorithm name see approximate strategy approximation implementation apply affine superior particular outli relatively intrinsic motion intrinsic dimension well minimum carefully hybrid segmentation video conclude brief possibility introduce storage discuss implementation algorithm convention subspace I identify approximate subspace correspond try partition subspace normalize element lie sphere use gradient energy need derivative orthogonal calculation near subspace summarize partition update step convergence assign storage algorithm find near subspace operation computing update cost cost typically usually increase become complex etc stop ratio previous computation datum online replace functional square angle often ki large find near little empirically greatly obtain initialization initialization subspace initialize guess pick tuple hand kf guess respectively kf kf kf restrict represent various dimension equal follow fix instance linear http edu randomly distribution cross cube subspace center center corresponding far corrupt uniformly outlier cube maximal instance misclassification table time various hybrid linear advantage outlier reduce kf conclude algorithm ambient intrinsic run table indicate usually apply kf would deviation misclassification minima available www coordinate extract frame accord object background video formally video frame sequence move camera background point j segmentation camera correspond move live four kf htbp percentage misclassifie database kf median kf kf kf htbp misclassifie three kf r median median kf kf kf compare connect improve segmentation kf implement subspace subspace consensus kf http www vision edu base rate misclassification kf segmentation result randomness kf time misclassification kf ambient kf initialization
standard argument identify similarly rule fix interpret stopping criterion control cg particular consider ill filter know iteration polynomial unlike additional technique establish consistency attain square regression learn reproduce stand minimization procedure linear solution nest exploit conjugate algorithm empirical estimation iteration step partial second condition provide pls technique construct predictor covariance dimensional representation fitting regression latent component principal component trick pls nonlinear dimensionality pls consistency e ridge component pls define sense depend linearly instead pls minimize nested subset define latent estimator pls study consistency pls target know pls early configuration scenario dimensionality grow regularization universal pls infinite derivation pls pls combination early stop equivalence version pls step population pls converge ii population pls ensure pls empirical condition estimator provide universal emphasize stop rule base joint convention perfect knowledge bar represent technique true mapping kx boundedness surely dense briefly interpretation inverse denote operator adjoint usual operator g reproducing coincide gx finally variable learning cast formally problem multiplication even formal motivation regularization come inverse order approximation belong right interpret perturb version wherein empirical counterpart note usual wherein map operator x operator space dimensional correspond perturb right predictor ill pose regularization ridge correspond regularization boost pls describe greedy iterative produce pls response project component th step conjugate cg normal detailed overview establish pls e use pls exact clarity remainder cg subspace range real polynomial degree step cg minimize e mapping equivalently represent vector extremely important scalar multiplication iteratively construct space pairwise equivalently uncorrelated algorithm convenient cg population version iteration replace operator component different geometry norm respect projection onto closure u asymptotic simplicity I infinitely decomposition eigenfunction version theoretically run population stop step modification projection closure conjugate population write projection onto first operator eigenfunction degree convenient property eigenfunction principal converge principal principal fact pls empirical token less biased boost correspond however finding suggest suitable control since ingredient stopping quantity version quantitie positive x justified banach banach satisfy compatibility eq bind monitoring note monitor initialization observable quantity u monitor procedure subscript estimate rule b u use stop step almost
functional study view statistical besides parallelism state utilize scad besides superior practice advantage penalty extra tune think formally little simplify nonconvex solution guarantee might think point computational scad visual arguably less follow briefly review tv point trivial hope reader follow development adapt denoise problem discuss minimization review regularization computational superiority denoise emphasize encounter paper tv model decade desirable smoothing carry replacement tv near weight avoid tuning amount variation partial tv derivative choose sufficiently basically artificial edge numerically unstable image difficult lead increase computation require mention introduction almost use different context identically distribute sometimes reason zero formulate encourage propose possesse lasso prove practice reasonable estimate rigorous see parallel processing desirable difference argument lead form tv respectively statistical study mention adaptive appearance address variable smoothly desirable possess oracle property behave coefficient scad processing get parameter superior linear penalty amount fig derivative even odd penalty penalty image formally write functional eq reader solution seem note encourage reader change expression continuous prefer nuisance compare tv consider minimizer compare scad penalty penalty defer appendix tv I simple pixel shrink penalty desire shrinkage tv already context difference big enough complicated functional evolution euler lagrange equation gradient potentially also tv scad taylor estimate scad getting actually minimize penalty weight involve extra inner evolutionary derive euler analogy bound stability extra tuning unnecessary scad produce monotonically iteration take thus comparable largely quality mse calculate knowledge technique carlo require denoise formulation noisy image consider image input prove divergence mapping regularization filter operation carlo add recover carlo pilot experiment empirically pixel four neighbor normal black white regularization compare whole deviation tv weight function good different intensity value choice see outperform show evolution mse regularization figure clearly scad significantly tv get histogram histogram tv tv bias colored pixel value generally shift colored intensity shift previously address scad besides histogram result result fig mse big well different wrong make sure burden even good scad mse fig former white square image structure large feature deviation regularization carlo sure briefly previously tv scad denoise sure accurately table situation mse require choice priori conclusion scad slightly complicated test performance library record scientific show fig transform piece wise visually standard add result scad image since visually difficult matlab format penalization functional denoise know scad originally statistical I pixel scad solve inherent spatially adaptive tv newly method stability achieve general denoise wavelet become popular exist pde extra recover also current proposition b make small derivative constrain easy quadratic form minimizer meanwhile imply partial add subtract partial q combine prove solution argument contradiction functional tv cccc tv e cccc tv image numerous improve different context motivate efficiently realize spatially theoretically penalty inherent variation
hour c chain c chain hyper well value around truth confirm estimation increase parameter reach orthonormal wavelet basis resolution image noisy image denoise noisy b reference image separable shift filter denoise mmse frame frame recover image wavelet frame depict purpose bayesian experiment hyper denoise ball interest deal wavelet literature assume frame generalize another propose framework situation explain sampling ball independently along appendix focus difficult norm pdf k unit sphere ball derive pl distribution ball ball radius straightforwardly ph paris est la france mail paris est fr university france mail fr ph bm point france mail fr sup des communication sup com en et communication mail tn example assumption study frame become necessary characterize frame coefficient difficult general synthesis observable introduce frame hyper markov subsequently posterior experiment propose accurate hyper problem image denoise impact bayesian frame generalize sparsity sense wavelet crucial operation processing include signal transform representation domain fouri good frequency localization expense spatial localization localization wavelet tool wavelet mention basis redundant become many decade sake clarity point frame understand sense advantage frame process use frame frame synthesis operator determination frame frame conceptually frame focus gaussian restrict attention concave function provide take current development carlo instance separation brain investigate impose reconstruction assess redundant address mcmc move accord deal denoise framework contrast overcomplete representation basis hyper estimation organize brief overview hierarchical representation introduce draw digital endow vector frame adjoint synthesis whereas redundant orthonormal tight identity signal write frame fr model impose belong error nothing measure adopt assume realization characterize parametric great image denoise actually denoise domain wavelet investigate seminal description signal relate estimate estimation perform inverting deduce hyper since present need coefficient representation pdf frame pdf close convex random hyper parameter frame assumption use lead follow prior shape coefficient modelling signal laplace recovery rewrite distribution define split group hyper vector multiscale frame frame resolution belong hierarchical complete improper kind summarize cover encounter reflect prior parameter posteriori square close study sampling generating distribute accord posterior generate sample unknown sample posterior provide sampler generate distribute precisely iteratively accord straightforward accord detailed method frame norm inefficient especially design distribution union orthonormal euclidean analysis adjoint f f kk ml x sampler generation coefficient accord f nn n mh support generation pdf define close acceptance mh preferable choose ball region associate propose n b n b adjust exploration enough sampler successively see strategy consume component dimensional follow parameterization truncate achieve use mh move note sampler I accept intuitive implement point restrictive consider frame orthonormal basis assumption hold propose hyper exploit algebraic sample impossible replace move mh globally accept reject acceptance efficiency strongly depend candidate stems fact yield algebraic frame frame decompose x realization value f sample h perform x u account ff f interesting generation technique proceed generate u ff explain simulate x reason draw ff easy pdf semi rule transform due q express remain yield finally simplify hyper sampler initialize u b ff u ff ratio accept accept result application carry frame basis filter wavelet j basis stand horizontal vertical resolution accordance model form uniform hyper distribution frame suppose principle reference value square hyper parameter belong monte
iv equal partial differentiable therefore assumption together characterization subdifferential hold addition hold proximal assume active I follow subset q resemble traditional tucker optimality since necessarily subsequence converge every iv iteration n proof indice linearly space gradient linear small continuity linearly large rewrite convergent subsequence subdifferential exist subsequence equation I family ij claim finish theorem ij pass use outer semi subdifferential sum moreover active index imply tucker type condition addition either r ij ij ij ij optimality hold penalization function aic penalty approach lack variable increase method type selector subset variable enumeration optimization bundle however purpose difficult situation induce certain em chen presence finite mixture realization example chen chen generalization smoothly deviation scad fan li propose spirit convergence mm purpose cluster theoretical scad chen penalty eq define label component complete case invertible choice chen jointly optimize variable consist successively alternatively respect differentiable non differentiable option turn conditional kullback like subset successively trivially iv imply ik real http www report bic criterion algorithm plain em obtain resp plot optimal start similar plain close fact notice plain chen expectation algorithm kullback stationarity cluster show nonsmooth tucker regression differentiable require reference locally admit directional directional subdifferential singleton subdifferential relate map value I crucial subdifferential outer say locally lipschitz optimality tucker particular main fact theorem section definition section france email fr methodology penalize provable convergence likelihood relaxed penalty smooth paper alternate penalize kullback proximal extension em nonsmooth stationary lie sparse em methodology extensively study generalization wu propose attention variable I many several approach contribution penalty contribution among alternative selector attempt penalization logistic chen present non use kullback interpretation em prove nonsmooth coordinate extension component version acceleration speed apply al cluster alternate nonsmooth tucker penalize kullback point penalty present implementation regression fan li far study chen ml observe random sample parametrize em maximization density alternate parametrize consist maximizer accept current iterate maximize concave point iterative write penalty control convergence relationship algorithm discover detail analogy motivate alternate generalization sense absolutely projection distance differentiable kl st coordinate vi kullback positive strong continuously differentiable definition real number convex iv assumption notice know divergence satisfied gaussian check make assumption iteration define maximizer em order prove technical assumption important theory likelihood important establish tucker often satisfy fact regular scad assumption need simplify analysis iterate lead always onto impose ensure grow assumption tucker behave simplification requirement basic proximal
note table word semi bold detail substitution many choose word bold candidate neither ccc anti sup semi sup oracle word language consideration alternative remain strategy perform anti sup sup comparison complete great note variety possible take last word comparison anti remain two candidate anti oracle set entirely relative way comparison sup sup set contain summarize validate trend sup quality result affect censor ratio speech article select evidence distinct speech processing depend upon detection phone rate consistent limitation neither candidate ever recognize candidate approach interest requirement notice finally recognition optimize development set leverage amongst generalize expert censor likelihood likely perspective much speech utilize acoustic main appealing direction future forward acoustic ccc error anti oracle sup semi oracle acknowledge speech team speech university plan co acknowledge wolfe manuscript manuscript conventional pattern selection accomplish minimize ground mean label development costly train error automatic amenable robust minimax censor limit validate experimentally candidate use utility suggest application sign category powerful speech speech engineering community likelihood long play prominent role aid system development log ratio turn ever area likelihood many processing compete serve regression likelihood observe speech assume know datum come acquisition limit scalability sample speech engineering likelihood evaluate compete serve truth yield sound algorithmic datum selection strategy speech standard metric serve benefit unlabele construct permit algorithm derive consider label incorrect assignment influence incorrect labeling limit well technical show optimality apply semi maximal induced labeling minimize relative compete significance select close kullback divergence notion fundamentally model predict possible comprise instance acoustic traditional devise fidelity effective unseen comprise classical optimize calculate held label truth manner validation fitting goal building training testing subsequently practical assume training draw speech engineering benefit ever great amount build amount training datum whose use semi paradigm application supervise limited augmentation self involve speech system desirable even certain instance new engineering approach acoustic acquire digital amount employ unlabeled approach understand system indicate bring likelihood error automatic mean supervise label unlabeled speech select compete performance section art speech recognition detection force conclude implication improve speech processing processing manner speech recognition acoustic likelihood typically model parameter simultaneously stage maximization amount match validation adapt aside purpose recall represent continuous function density evaluate acoustic pair training proceed absence direct knowledge speech likelihood seek overall use choose amongst compete compete approach leibler distribute empirical likelihood respect sample arithmetic average amongst multi clarity exposition admit outcome represent natural describe fit follow careful expectation work potentially likelihood possess technical assume give force appropriately standardize straightforward proceed appropriate statistic asymptotic deviation evaluation root fails statistic surely directional fixing normalize evaluate select model evaluate decide favor model conclude insufficient evidence conclude distinguish compete already wish leverage model class seek fit replace ratio ratio course label decode incur error section compete model procedure suffer misclassification error reasonable marginal tend course recover label encountered true principled adapt p misclassification recover nonparametric simply actual statistical enhanced robustness error automatically limit reasonable provably misclassification automatic consequence inexact procedure distribution respective contaminated determine exist contamination incorrect amongst whenever ratio monotone enough ensure contaminate disjoint comparison include possible comparison currently machine tailor select amongst compete prototype selection processing task amongst compete two speech system differ conventional audio crucial processing speech recognition synthesis form term comprise mapping er create however expensive inconsistent individual lack broad interest approach put must word maintain speech processing system create vocabulary must extended name usage rare significantly dynamically adjust generation thereby effective automatically amongst date focus modeling letter sound previous building large phone additional speech work variation concern compete scenario word current however end consider speech condition say subsequently supervise hence oppose conventional show ccccc ax ax ax ax ax er z n n contain word force acoustic assign viterbi g cast follow word comprise q decide likelihood force alignment conventional method production difficult consume external potentially need identify speech segment instance use candidate news item rich serve correspond example occur know weather record episode give word know many word segment choose example output alignment select candidate speech recognition candidate datum acoustic evaluate entire likelihood semi analogy sequence likelihood use decide ratio indicate section consist scale speech processing force output candidate hour correspond recognize speech total recognize speech evaluate decision trade curve phone speech experiment conduct state recognition retrieval supervise selection variety g retrieval synthesis variety word place english english consideration often require selection word acoustic interest verify system contain consideration acoustic candidate consider letter sound word letter sound produce subsequent sign log ratio correspond reflect priori equally likely enforce candidate selection vocabulary build recognition train datum hour detection train word news rt hour test employ system use lattice detection task detection index refer procedure letter word purpose make either supervise
estimation equivalent add constraint problem add constraint redundant claim equivalence lagrangian function add spherical constraint optimize sphere lagrangian eq use variational formulation eigenvalue symmetric notice parametrize linear lagrangian dual weak lagrange duality use provide primal weak duality duality bind precisely value relaxation section weak duality eigenvalue relaxation quite tight nesterov problem well function admit make prove fact exist say go denote subdifferential subdifferential affine adjoint whose subdifferential prove nothing define product e know set subdifferential perhaps duality prove denote closure hull true constrain technical property prove duality apply relaxation norm consequence specialize short subdifferential differentiable must prove nice relaxation exact combinatorial relaxation optimum one help answer two recover binary problem max graph nesterov allow precise view practical favorable good average eigenvalue natural approach sdp thus eigenvector solution simplify presentation establish nesterov obtain sdp problem equivalence relate matrix last q nonconvex sdp relaxation give program definite follow subdifferential formula minimizer hand candidate initially prove give relaxation preferred argument recall get use obtain well eigenvalue equal relaxation sdp relaxation use fact eigenvector great obtain sdp find work specify obtain practice eigenvector square subdifferential subgradient obtain bundle desire lie feature bundle sufficient opt eq eq subdifferential multiply cauchy sdp desire common sdp reader solution recover quite subgradient optimum answer question although part section exactly necessary condition strong max sense lebesgue eigenvalue fix diagonal frobenius deduce subscript decomposition non say eigenvector associate differentiable eigenvector associate subgradient n strategy whose expression greatest round coordinate nearest give higher feasible round near binary sum cut objective randomized imply cholesky factorization clear prove cut exactly eigenvalue explanation take section image devote problem simply consider column penalization require sense index like quantity property index south east west main associate eigenvalue eigenvalue optimization outside diagonal prove duality hold perturb problem perturb eq image denote denoise perturb binary denoise cost page minimize notice confirm relaxation least hand eigenvalue complicate quadratic constraint experiment report noisy additive independent identically apply influence smoothing display vs recover result encourage suggested word compare letter seem posteriori consist satisfactory experiment numerous confirm interval identify study approach square k ds signature th take equal user overall filter bank match filter whose give equal approach compare extension shift symbol issue former primal greatly increase overcome semidefinite duality prove eigenvalue equally point analysis prove correlation outside strong signature research theory frame interesting viewpoint finding negativity htb signature duality indeed situation heuristic indeed compare value primal verify monte carlo axis software sdp solver possibly cost time vs user dash style sdp curve plain eigenvalue complexity reader complexity routine solve bundle software eq study polynomial property part physics sketch order easily case zero formula deduce root interval two discuss main square relaxation solution problem address sufficient allow solution nesterov binary detection image denoise strong hold signature thm thm proposition subsection france email fr survey property eigenvalue program obtain lagrangian dual implicit compact relaxation definite programming tool handle image several engineering corrupt noise address square relax recognition size bundle quadratic superiority bundle semi one appear favor eigenvalue relaxation user prefer sdp primal recovered motivation recover geometric penalize least eq parameter priori term binary vector know construct programming relaxation semi definite programming sdp play role inside signal problem nice survey lagrange general bound
estimation belief bp red dash expression bias toward look observe solve bp observe isolated one obviously nontrivial isolated ml calculate count straightforwardly derive recursion permutation b return case phenomenon split nonzero temperature one temperature nontrivial bp local dominate positive solution good matching exist constructive weakly ml configuration without generality observe nontrivial derive j v bp constraints translate u equation temperature straightforward verify nontrivial perfect matching conjecture solution bp extend solution smoothly another plausible minimum lie doubly stochastic polytope bp generic gm ls term adapt algebra expression accordance subgraph bipartite e loop even formula degree formula determinant derive evaluating observe eq ls doubly belief derive utilize version prove z ls x van minimum doubly attain conjecture open finally surprisingly short elegant allow van conjecture call van simplify van theorem non matrix respect bp transformation transform side one naturally bound van namely match without generality positivity entry sake completeness review specialize independent square hadamard consider study detailed size temperature provably nontrivial bp van deterministic provable calculus realistic course mathematic support student visit advanced study grateful national nuclear security department energy national laboratory contract na via nsf collaborative communication reference proposition theorem claim computation equivalently likely function calculus representation bethe functional doubly belief also pass propagation type z expression state bethe alternative multiplicative calculating context physics intrinsic particle solve unlikely naturally look randomize algorithmic problem approximated polynomial relative complexity impractical realistic motivate find continue belief propagation heuristic absolute bp originally code artificial intelligence state loop evaluate partition maximum gm normally expect result surprising existence ml realize bp raise question understand performance heuristic capable handle gm calculus gm bp call subgraph series doubly matrix marginal perfect describe minimum bethe energy understand first result correction recover expression pf key gap estimate pf exact technical bethe energy bp conduct evaluate bp entire ls collapse term state ls lower state corollary van lower derive original lower discuss text sufficiently dominate iv discuss transform application hadamard discuss negative j parameterize via pm complete binary interpretation allow represent perfect perfect matching temperature degenerate gm un variational kullback leibler statistics functional find condition belief understand proxy probability unity achieve minimum approximation underlie gm bp gm relaxation gibbs paragraph gm bp states perfect accord belief belief belief associate variable substituting bound absolute minimum simplify express solely satisfy
cell set consider cell contingency frequency vector denote denote I configuration denote exist contingency common sufficient element negative total frequency degree move write move primitive primitive basis correspond specify unique minimal basis coincide move reduce element distance markov reduce usual call basis connect reduce zero table reduce zero table finitely difference element large move connect investigation connectivity one table basic fact basis primitive sign side entry sign add therefore strongly reduce large connect entry condition strong frequency vector distinct move form markov existence two basis imply discuss end condition table entry zero remain one least reduce effective connect several generalization cell move condition distance reduce entry suggest connectivity table pattern b distinct one combine follow induction primitive suffice reduction check move primitive recursively reduction proposition cell remove zero last discuss weak basis basis similar weak cell move equivalent distance reduce reduce move reduce index see hold investigate common contingency long question table contingency entry express difficulty set sum statistic extensively study practically evaluate develop e sample practice case easily show exact need connect every zero row sum show connect table row sum goodness model imply move show three independence htbp table complete q move independence degenerate define move complete classify I rr rr I I move move require move connect see square free move even seem difficult ti practical test limit connect implement zero include social statistic move quasi denote cell zero table social quasi implement sequential two way way structural provide independence way contingency complete description basis quasi loop section df support df exactly element array df loop basic df contingency show connect zero df loop connect every table quasi square diagonal move df result table incidence matrix square symbol row symbol square axis array square sum square give may minimal connect case first table move interaction need degree move form interaction line slice loop slice loop degree generality slice loop pattern move contradict move see move move prove move degree connectivity program check basic degree hence require connect chapter connect generate array representation view move move square basic move level correspond degree move connect table general find basis table basis contingency result contingency table entry zero zero particular contingency table enough proposition suggest basis table basis behave cell consider challenge author anonymous constructive true mm mm section section discuss connect entry correspond minimal markov basis since basis tend interest particular minimal connect table table minimal table one table basis methodology test exponential family sharing call markov basis markov tend scale researcher interested subset markov restriction count one case interpret logistic logit trial covariate one element frequency occurrence non occurrence record
select open specific spectral already grow improvement translate directly compression size offer improvement wolfe latter approach practitioner dependent offer liu rigorously date bound quantitative guarantee quantitative nystr approximation appropriately definite symmetric definite kernel spectral coordinate matrix sort non approximate kernel follow notion unitary invariance unitary invariance term matrix depend singular coordinate evaluate quality task ordering however cost costly method rely dimension dimension impose computational burden illustrate former category datum via nystr om modern preserve nystr om cardinality submatrix partition follow positive principal rectangular dimension eigenvector eigenvalue typically complexity reconstruction nystr om serve complete row cardinality norm lower small eigenvalue general complement correspondingly general definition selection cardinality minimize remain open whether threshold spectral force enumeration divide minimize liu zhang quality kernel typically wolfe sampling currently stochastic determinant detail nystr om adopt norm trace norm arbitrary denote singular semi definite decomposition frobenius notion function e trace amongst invariant norm adopt minimax unitary result end definite om complement per obtain trace nystr om error norm complement norm induce nystr om approximation express denote index select om accord complement always definite norm subset term latter zhang trace regression partition accordance om obtain project admit decomposition loss generality choose therefore diagonal nystr om square spanned column characterization provide comparison semi fix exponent distribution principal determinant computation amount course mass concentrated practitioner et al al trivially recover associated wolfe error uniform sampling nystr om tight average contrast principal eigenvalue incur fail place wolfe kernel subset via om completion require reconstruction nystr om suppose nystr rank admit determinant rank square imply determinant know q selection gaussian minimax dimensional upon correspond represent first covariance maximize entropy maximal relative upon extend follow maximization nystr om j result wolfe bounding case improve let om extension large eigenvalue tx x al possible note present combinatorial wolfe easily cover approximation computation light selection draw field computer manifold stream particularly aspect impact efficacy diverse segmentation et al spectral et appearance manifold lee spectral world require indeed aforementioned task share feature approximation select serve video database lee al video dataset often dynamical evolve line rotation extract low object recognition appearance manifold lee recognition pose lee ingredient video stream obtain definite number frame video stream prohibitive entail begin test efficacy couple procedure exact embed dimension diffusion al database lee head front motion position straight camera circular low front right normalize nystr om video selection overall average map average trial monte carlo wolfe determinant subset determinant choice sampling determinant range observe maximal analysis tend concentrate front yield locally region properly avoid subsequent collect video data author slowly front camera embed diffusion straight evaluate visual quality display recover uniformly curve centre show trial yield good sampling uniformly analysis practical implication subsample determinant sampling centre uniform map embed video movement speed pixel recover almost scheme lose curve fold material base project grant national health grant national science grant dms perform mathematical science high dimensional average error nystr om reconstruction map sampling consistently outperform kernel nystr om extension year spectral appropriately define extract dimensional currently practitioner kernel follow spectral analysis term nystr provide algorithmic performance optimize selection implication bound real world draw vision whereby low video year capability development decade new treat structure cluster laplacian different computation principal eigenvalue semi definite spectral long hold central matrix variety principal subspace discriminant determine separate data embedding kernel scale cube wolfe accordingly pose severe modern construct kernel partial analyse select summary burden enable practitioner apply aforementioned original solely subsequently upon adaptive manner begin review dimension formally clear conclude demonstrate statistical landscape indeed pca introduce enjoy cost process normalize transition step simply eigenvector eigenvector correspond unity eigenvector eigenvalue nonlinear embedding embed show overlap colour indicate panel respectively obtain map
extension mechanic work general definition contribution cluster anneal experimentally optimize well sa also sa implement carlo sampler sampler mcmc thus approximate st derive tractable sampler look interaction annealing find optima continue stronger close pick st quantum work globally suppose equal fig local interaction search assignment quantum metric interaction chance go close finds often cluster majority cluster assignment pick locate st one th indicator denote assignment denote indicator length assignment product matrix b point cluster assign second matrix assignment store use anneal like sa th aa c optimum optimum briefly anneal give energy search next function sa almost choose high hill function summarize inverse sa update energy many intractable sample focus mcmc draw denominator tractable quantum anneal fold form section schedule mechanic equation hamiltonian physics example matrix exponential e e equal calculate easy equal sample equation effect matrix one depend work try couple explain bad later sa mcmc method mcmc intractable evaluate unlike formula product product follow completely different similarity finite show exact pass anneal schedule dominate anneal schedule residual annealing continue effect different anneal temperature assignment increase inverse inverse anneal address propose observation pilot pilot observe st suboptimal assignments st assignment far optima necessarily well compare become become close strong begin strong interaction close search middle anneal step eq th sampling f difficulty sa choose anneal schedule schedule next choose anneal st reduce schedule sa mnist lda b sa sa fig three schedule st sa schedule show st sa schedule improve sa gaussians inverse wishart latent model sa posteriori corpus point apply reduce corpus word vocabulary word vocabulary corpus st st sa st keep st similar sa term anneal vary become st st fig plot st sa interested st achieve sa column mean result schedule schedule sa third st still work experimental consistent work lda lda optima show st achieve sa slow schedule st still sa accelerate study permutation augment minima techniques exchange carlo fit cluster schedule simulate sa randomly sa run sa actually sa time fast necessarily sa thus experimentally try develop algorithm network quantum look like genetic multiple interesting acknowledgement research physics new material grant super
alternative conventional probabilistic incorporate center realize drop interpretation replace amplitude quantum mechanic associate center find quantum state point operator gaussian wave trace represent minima quantum begin focus attention wave construction expectation coordinate dynamical state mass moving choose evolve evolution accord expectation wave towards potential relation minima trajectory clearly locate near local come move apart dynamic visually trace associate one successfully move seem replace conceptually implement gradient solve complicated difficulty solution simplify considerably allow translate form capture less advantage formula analytic evaluate thus multiplication simultaneously multi processor time produce linearly display introduce employ introduce minima also connect nearby degenerate minima reduce calculation final worth wave strategy handle set discuss work brief outline assume datum associate wave center coordinate scalar product matrix gaussian orthonormal basis solution desire stop obvious restrict feature derive analytic operator computation experience difficulty apply text book five belong reader example wish simplest capture essential reasonably reduction unitary occur diagonal entry consist call component think assign five full within principal apply consist pc compose three use five quantum color place quantum potential good job capture cluster roll near produce separation three first column guarantee normalize unity conventional project onto sphere study temporal henceforth show unit sphere quantum visually separation colored class however incorporate unsupervised begin specie red green fairly problematic middle plot quantum stop cluster occur quantum evolution enhance separate accomplish e point band difficult tight toward cluster arrive minima pattern stop evolution configuration evolution hand happen end matter quite evident agree color group together color blind full proceed lead insight dynamic point lie due assignment true proximity difference extreme phenotype absence discriminative important conceptual message classification message include geometry measure euclidean influence analysis may replace manner define dynamic point euclidean clearly geometric distance reduce investigate evolve semi close example intermediate point evolve thus interesting exist reason scientific large dealing point problem pc intensive simply lie map cube hilbert tuple way svd filter simple modification dataset eigenvalue define entropy define quantity remove filter technique remove raw datum pc represented follow apply evolution latter cluster blue filter blue separate cluster separate point identify stage iterate stage however begin svd base filter trying enhance red start blue distinguish blue remove higher important biological clustering reduce go six feature begin among eliminate responsible separation track robust repeat filter blue sort green fact figure remove accord svd happen remove comparison time svd entropy five filtering separate complete svd filter stage cluster accomplished evolution distinct merge evolution certainly say reality cell look cluster give fp stand correspondingly high summary dynamical explore start show dynamical visual state derive analytic calculation temporal evolution treat quite put system sure produce gaussians associate center evolve wave gain full wave expand dynamic notion particularly core manifold methodology potential minima svd value reduction dimensional handle difficulty computational point define associate related computing multiplying give operator clearly number feature potential small feature construct experience stage evolution occur structure see plot readily construct use reduce addition employ contribution nature especially evolution note contain advantage carry see reduction assign datum remarkably explain unnecessary allow set contain large number turn hilbert point naturally hilbert display employed datum wish usually learn e appropriate stage dimensional filter lie visually subsequent datum accomplish point appropriate construct include system intermediate give full fact old colored accord original possible set plus datum happen original fail properly feature filtering change sort identification existence identify cluster quantum potential hamiltonian already evolve guarantee cluster accord cluster conventional quantum wave dirac employ operator relation hermitian operator identity calculate order meaning prove easily construct gaussian center wave coherent gaussian gaussian matrix orthogonal shift derive appendix q I generalize derivation expand point identity speed series original include computation purpose need eq behind remain mechanic evolution hamiltonian shift computed hamiltonian computing truncate operator matrix set find away compute whose exponential operator simply eigenvalue one basis compute q put construct pt cluster stanford stanford usa school university whose determine use hamiltonian calculation wave around original allow exploration dynamical point convergence formalism decomposition ill define nonetheless give point looks sort sense together investigate
represent may generalization corollary bayes appear risk unbiased priori may flexible shrinkage unknown notational eq operator adjust naturally unbiased risk enable haar haar wavelet shrinkage neither haar direct application haar soft iy rather haar empirical via coefficient simple effective empirical estimating rule substitute bayesian matrix poisson intensity estimate risk unimodal estimator available imply ix coefficient variance solve numerically yield consider early simulation overall performance efficacy consider bayes numerically laplacian soft laplacian linear ss soft thresholding adjust shrinkage haar moment shrinkage rule relative fig mse inference min std min max std median std reconstruction test bit image frequently literature house interest level reconstruction report snr compete conjunction implementation baseline comprise haar wavelet priori amongst propose offer alternative term quality visual appropriately locally yield dark fig importance incorporate coefficient process variance completely resolve smoothed texture entirely lose exception typically estimator widely error db readily confirm compete remain competitive great adjust snr save intensity way near frequentist haar along low wavelet confirm offer appealing method literature indeed haar yield exact along substantial application amongst haar gain presence additive proposition remark material base national foundation grant publish th imaging science international conference obtain independently et international imaging information laboratory usa mail integrating imply variety world model poisson turn proportional underlie article arise haar measurement type difference near provide unbiased frequentist new haar unbiased risk complexity demonstrate efficacy shrinkage estimator univariate wavelet test image substantial class device subject loss resolution e quantization inherent degradation prominent variety digital communication image popularity diverse wavelet transform spectral scenario readily transform measurement range effect across transform instance accumulate element detector model poisson random density ii signal sensor grow linearly strength imply inefficient detector back summary seek invertible typically compressive nonlinearity familiar white estimate directly poisson leverage independence strength upon maximum approach dependency haar describe well repeat directly haar date wavelet describe domain mean canonical tail prior analytical practical variability variant stein parametric estimator transform discussion subspace axiom require family kk moreover scale wavelet form wavelet family together expand time q scaling term analogous transform haar take scale mirror turn recursive canonical type frequency digital processing decompose component wavelet band recursive decomposition hadamard haar requirement transform make attractive haar orthogonality computational simplicity haar wavelet transform axiom analysis serve admit efficient sequel coefficient aggregated scale notational finite subscript generic turning transform domain estimate stein sure differentiable risk unbiased transform nonlinear thresholding example poisson denoise straightforward case haar wavelet close form difference poisson count end unnormalized haar transform wavelet element observe haar empirical coefficient effectively poisson coefficient characterize generate fix difference poisson verification summation observation taylor q poisson clarity variate limit difference integer hand skewness skewness random variable proportional poisson rate proportion distribution proportional fig text skewness fig haar transform depend similarly haar wavelet let definition haar transform empirical consequence eq verify entry leverage haar wavelet intensity conclusion variability variability haar haar first toward achieve turn univariate mean frequentist generic scalar quantity haar coefficient haar coefficient latent coefficient directly estimation assumption plug estimator haar scale constitute poisson context signal ratio argument moreover admit recurrence probabilistic derivation begin derivative derivative admit comprise operator act linearity similarly derivative property slope curvature identity imply likelihood recursion admit recurrence fix eq calculation likelihood initialize combine equation linear equation consider underlie prior distribution area range generalize mixture attempt estimation univariate however determination parameter infeasible end approximate obtain closed shrinkage random bayes score approximation equation expectation schwarz latter term control condition bind equivalence analogous derivative go average high shrinkage rule prior expectation formulation amenable former tail go derivative generalize parameter gamma unimodal expression admit truncate laplace prior px px
robustness direct consequence assumption central application central definition law consistently one since proof acknowledgment thank english responsible mistake lemma axiom exercise package interval central team laboratory france france abstract describe package implement confidence central mean context test express robustness normality keyword parametric test interval central package test chi variance however use even fail chi hypothesis level sensitivity normality may implement sample variance well thing much study test direct remark valid ratio variance user tool different community basic e wish hypothesis limit call already find chi square testing normality contrary test g procedure implemented available package etc never paper unified sample mean variance derivation test package normality large alternative fisher purpose term present test framework test unify present give concept explanation chi finally general derive asymptotic variance package notably finally section devote discussion proof sample identically version denote compare gaussian framework resp resp mean resp resp quantity variable chi ratio statistic measure gaussian normality theoretical explain section q obtain differ variance gap framework govern indeed asymptotic assess large c widely confidence thank limit express term central theorem parameter limit law definition standard nd alternative illustrate measurement specie frame tt length width normality width specie frame specie mean specie var var mean width specie par ref test width specie alternative less percent interval inf sample width frame tt n width specie difference width specie e percent inf difference width specie frame tt specie width test species inf ratio difference frame number width specie e alternative weighted percent inf sample simulation reliability sample perform datum chi test versus alternative chi versus false fisher asymptotic type error worse suit asymptotic direct summarize introduction order illustrate output resp sample empirical propose seem fit chi apparent normality may prefer tt par l ref statistic variance percent inf variance decision inferior case kind test tt leave variance ratio percent apparent prefer frame number two hypothesis interval difference variance think might test various write
possibility latent consider approximate simulate converge sequence therein amount everywhere right hand side uniquely depend version simulation joint converge approximation stress artificial sample convergent estimator biased availability ratio computational first alternative require possibility representation bridge formalism approximate factor sample z follow converge eq gibbs distribution normal compute estimator compare variation approximation mle distribution bridge complete mle replicate simulation sampler comparable section address http www fr evaluate impact diabetes occurrence diabetes probit mr acknowledgements grateful discussion comment team improve exposition author point bridge sampling university support de la paris project big mr framework know hypothesis allow demonstrate fundamentally rely theoretic version density involve impose mathematically completely measure bridge approach versus bayesian core choice indeed marginal mathematically uniquely exist differ literature chen improvement numerical precision bayes relate complex method representation nuisance decompose plug obvious notation marginal reduce justification illustrate representation artificial computing hypothesis embed model since representation address aspect give rarely face challenge deep nature consider consider uniquely alternative hypothesis posterior though second section difficulty examine axiom state mathematical difficulty proceed identity exploit early probit measure rigorously probability q measurable property conditional distribution test advance completely manner reason theory satisfied conversely assumption prior contrary state mathematically artificial arbitrary agreement instead disagreement valid specific version density derivation depend last rely namely rigorous availability already test prior experiment thus modify completely rigorous representation hold density verify stress mathematically prior choice mean choice establish approximation bayes obviously
continuity moment v k co closed complement co co relative assume surely therefore moment computable joint computable topology right random case moment computable moment give work regardless location consider randomized number computable absolutely everywhere real denote generate open continuity compute measure extension uniformly relative k measure multiplication convergence c moment aa rational interval whose equal thus mixed ia de relative immediately lemma computable relative joint determine moreover respect ij aa k note continuity union intersection ij supremum computable almost surely surely relative expectation respect relative make complete proof explicitly construction algorithm alternatively sketch computable theory prove computable could densely interval continuity set metric approach complete probability boundary less boundary set abuse define c ik ik arbitrarily polynomial pointwise polynomial define n n cv cn real co final follow order product call ready computable exchangeable computable de give show computable proof ij v cv uniformly supremum continuity range cv preserve dimension open topology eq recall polynomial v k p p continuity p c c cv cv cv cv continuity v cv q v continuity continuity low note dominate convergence cv q real supremum c uniformly implication semantic programming language eliminate code automatically context background functional language code transformation computable early partial exchangeability recent application programming language choice operator language researcher number language operator semantic program theoretical science randomize checking area somewhat different language type inductive evidence think infer program datum implement programming al describe extend mit perform execution monte think walk execution describe language implement approximate sampling describe calculus expectation g tree naturally helpful work express nonparametric model produce program entropy beta bernoulli describe language de exchangeable computable furthermore representation automatically make computable probabilistic language extend scheme binary value return false move definition call bernoulli semantics associate evaluation expression binomial evaluation procedure behave denote dirac concentrate real equal probability always modify use via non local may implement thereby mutation manner associate probability expression could keep counter variable repeat translate mutation mutation perform transformation throughout program represent pass operation original counter accept return transformation particular remove require rest section concrete mathematical beta bernoulli describe recall directly bernoulli possible mutation require keep track value instead introduce operator track black ball compactly scheme description beta fix beta coin coin code code environment create argument environment coin coin argument return procedure coin coin coin coin repeat beta coin coin otherwise coin flip sample sequence ten return hide beta coin coin beta coin beta coin weight independent application coin ball ball return application coin exchangeable therefore draw know beta coin bernoulli informally think beta coin bernoulli process although coin important coin coin coin change coin non sequence execution computable sequence measure transform mutation return distribution general program generate exchangeable computable produce mutation procedure random randomness transform code sequence produce procedure return coin semantic mutation reason example overhead necessary exploit independence exchangeability probabilistic language execution sequence exchangeable chinese turn distribution sequence exchangeable process stick break produce certainly form exchangeable object combinatorial write analogous p restaurant dirichlet detail result encoding subset countable support discrete measure computable complicated breaking process depend arise evy method construct process de random extension suffice type exchangeability variable exchangeable permutation nearly year show array separate exchangeability conditionally adjacency exchangeable beta bernoulli g inherent parallelism computable de exchangeable could representation wide range include increasingly exchangeable block process explicit nsf dms fellowship conference abstract would thank extended present comment computer artificial intelligence laboratory institute technology usa exchangeable exchangeable language automatically modify local computable de theorem computable language mutation classical theorem sequence almost exchangeable sequence computable de distribution computable measure computable readily de term exchangeable along prove computable independent interest proof moment random moment computable suffice compute computation general survey representation computable real automate domain study semantic programming language survey statistical probability measure coincide generate computer numerically science machine recursion probabilistic language directly relevant computer exchangeable use communication access modify program indirect classical description conditionally thereby implementation classical finding moreover construct desire describe would perform local induce beyond language semantic desirable term de measure example machine computable assume theoretic formulation theory let nonnegative logic basic borel bx value index real conditionally b bx measure expectation side characterization perspective interpretation exchangeable arise latent implication de prove mixture repeatedly coin draw extend value exchangeable hausdorff give conditionally history development book comprehensive role measure borel dirac uniform almost surely sequence example dirac dirac marginally distribute compose marginal give x n verify kolmogorov marginal clearly process exchangeable sequential turn q repeat ball ball return additional variable independent x n finally de measure measure notion precise programming language ask computable exchangeable sequence computable de computable exchangeable sequence converse computable exchangeable exchangeable computable computable limit computable rate limit able reconstruct measure borel open denote rational value borel uniquely place moment variable I de marginals exchangeable open lattice open note place boundary I rational lower property imply e moment suffice characterization computable sequence order computable topological enumeration set topological discrete topology enumeration precisely consider computable representation topology effective enumeration straightforward let enumeration function km tn computable recover definition computable measure topological unit interval admissible borel measure equivalence computable program randomness name topological algebra finite intersection second countable space topological computable closure intersection enumeration derive set enumeration fashion enumeration enumeration space measure computable note computable uniformly index co open co e close singleton space strong open strong arbitrary computable measure concrete notion topology computable sequence computable topological characterize sequence real enumeration simple computable side precisely corollary one characterization embed measure play distribution joint computable product right ik ij one note standard characterization integral respect topology topology variable
far construct w hull first svm convex reduce construct formally enable optimality solution value large writing vector let horizontal word formally boundedness finally know define choose choice polytope ray illustrated prove indicate three em change go discrete make complexity guarantee exponentially none hull attain equality solution program many subset solution general result program ignore svm solve program track path bad path solution prescribe quality constant bad path project pa work anonymous comment suggestion discussion proposition france cl france variety machine entire develop method support path assume exponential instance svm parameter become tool biology vision regularization contain special regularization tradeoff term tradeoff generalization performance programming extensively algorithm whole varie function prove complexity svms program worst exponentially distinct occur regularization change exponentially valid number space eq describe vary always semi gram include regression multiple square angle lar also pursuit denoise compressed parametric program control predictive geometry moving point mean portfolio variate solve parameter majority prominent g svm probably example path parameter piece path turn alternatively number change svm distinct number show svm exponential number path linear avoid confusion construction report exponential exponentially many exist conceptually construction motivated finding instance imply probably parameterize restrict svms exponential geometric path relate homotopy long particular particular entire parameterize compute exact solution machine technique similar program solution path svm give lasso multi svms svm point separate regularization path interesting quadratic path mention usually require invertible always later point svm indeed already arbitrary matrix invertible matrix recently g instead optimization would cube support simplex cube objective parameter compute method maintain linear vertex exist program variable many solution vary contribution adapt overcome parameterization regularization class svm different secondly continuous path solution parameter carefully transform objective turn geometry svm svm parameterize eq point solution appear coefficient distance reduce point note discover equivalence note slightly commonly svm variant geometric distance regularization parameter geometric explain path svm convex role first construct two plane class regularization class svm path piecewise linear path roughly align circle align class vertex depict polytope formulation end walk nearly boundary face path inner claim formal main guide surprisingly lower contain class parameter dimensional plane crucial construction hull vertex moreover walk fraction depict figure tool slightly cube vertex visible cube intersect plane cube keep let fact way hull polytope interior suffice face full vertex hull imply strong every statement equivalent use actually cube variant cube inequality standard cube transformation cube program rule property polytope interior hold inequality intersection exactly vertex index set vertex denote vertex v differ expression thus property cube hence variant cube coordinate dimensional precisely vertex tell cube appear polytope project interior transformation map strictly satisfy call hull set polytope finitely inequality boundedness transform interesting polytope origin interior polytope vertex define simple consider dual cube dual cube therefore perturb version polytope see outline intersect already property polytope cube summing mean cube geometric duality one correspondence q accord lie slightly plane ideally cube sure walk intersect accord achieve except omit version define face define point behind transform enough onto key define fix sufficiently exponentially property unique cube ray em dl define choose cube vertex inequality onto vertical chosen show eq eq see also projection item outline begin remain statement eq hence prove second part write problem minimize quadratic equivalent optimality pair satisfied pair prove unique relaxed dropping tucker relaxed problem determine solution observe tucker complementary turn hence actually easy
notice aggregate good performance stage diagnosis combination aa bad even choose particular aggregate mix stage coefficient aa result amount allow carry follow calculate approach peak intensity carry violate unlikely calculate nan nan calculate describe apply calculate event triplet categorical aa triplet aa value calculate state diagnosis trial assign label e present include value include frequent choose combination peak algorithm choose peak peak manner good time period error index loss algorithm value good number error contain correspondingly l l l l practice often choose suitable level advance time peak combination aa prediction disease predict suffer less intensity another investigate stage diagnosis aggregate mix predict check peak give time diagnosis get value reject statistic never bad deal cancer assumption direction future consider disease patient put triplet like thank discussion fp grant method application diagnosis development machine complex grant ep learning management optical network apply method level conjunction mass collect period year gives convert strict make peak contain diagnosis previously set reliable cancer appear stage disease lead difficulty patient improve diagnosis reliable early stage investigate quality detection demonstrate contain diagnosis stage peak intensity form set case choose age storage condition triplet sample individual cancer information peak precise reliable diagnosis use rule process combine prediction decision rule algorithm triplet convert label weight label normalize reliable prediction vast majority thorough experiment perform peak information stage choose organize describe data work description stage diagnosis section predict step make suffer goal prediction combine expert close impose loss probability measure loss function probability concentrate game repeatedly expert protocol learner reality cumulative lot probably weight aggregate aggregate behind algorithm assign correspondence expert adapt outcome strong aggregate algorithm exclude case lack triplet control sample sample individual triplet diagnosis triplet calculate triplet outcome represent apply construct predictor patient peak expert cancer diagnosis purpose sort date measurement sample person triplet choose theoretically aggregate evolution expert minus triplet clearly line expert aa expert line high bad axis present triplet aggregate triplet aa expert group cluster graph separate mistake stage aa prediction moreover loss aggregate loss say fair expert strict flexible find expert convert triplet present way strict aa prediction convert calculate see aa
framework analysis inequality analyze family template problem recent particular simple mistake batch multi ii categorization norm interpret decide multi perceptron previously similarly task complexity issue bound simplify previous recently framework previously task furthermore previous learning discuss convexity growing study organized class categorization online pca lasso e significantly derivation improve adequate tool context learn technique e set generative specifie task correctly identify relevant contrast focus agnostic understanding sample identify discuss convexity smoothness strong attribute relatively recently machine derive online generalization batch duality convexity strong derive online along regret algorithms case directly describe related involve bound detailed survey application derive duality complexity rademacher duality characterization immediate corollary need strong strong second different literature banach seek understand concentration smooth say recently derive concentration matrix compose rest organize section general present strong inequality show rademacher draw attention strongly strongly convex strongly number corollary recent next systematically adequate prior I turn applicability approach categorization section naturally unified enable simplify understand background new believe presentation notion reader familiar object short f euclidean equip fx deal space nm increase arrange f f infinite restrict proper define smooth state strong convexity property close smooth close strongly strongly dual proper important setting always domain smooth finite everywhere differentiable p smoothness find reverse implication easily people direct prove generalization convex denote st nd smoothness online round player prediction rule mapping assess risk generalization w excess online convex optimization bind family strongly lipschitz r dual enjoy regret plug rearrange tw tu tw w receive let I I know lipschitz bound consist predictor strong argument corollary proof essentially original highlight importance fw n get side last become divide throughout combine contraction w fw satisfies expectation choice easy bind expectation recall strong norm fw mainly respect slightly mean function shall set slightly become fw family strongly counterpart define n q corollary dual weak easy calculus group absolutely suppose smooth condition norm convexity duality combine corollary bind return learning corollary easy derive nx exist online w nx form problem online subset dimension learn argument absolute hinge batch set online encode particular shall eq analogously section bound simplicity ignore class property well know ignore organized refer usual suppose distribution still inequality course question dense bound away g k guess small tackle apply vector column vector us column column close dense small preferable class course problem might dense sparse similarly sense imply sparsity zero dense group sparsity use dense employ difference instead consider consider demonstrate general methodology order bound solve prediction example define mix different task heterogeneous multi task learn recent natural regularizer task together similarity consideration common regularizer comparison class consideration lie linear norm regularizer comparison describe learner use predict j dl j j also w x regret implication bound represent assume well scenario happen reasonably predict value let group roughly evenly evenly column well adequate prior belief similarly maximal singular assume spread draw k rademacher bind strong rademacher argument k tw line proceeding proof theorem k tw group norm row corollary corresponding excess risk table multi categorization online round receive predict number prediction zero verify r zero note two fact let w let learn bound give c implication happen inferior well space previously suggest observe class share much share demonstrate expert expert predict label e represent expert ignore logarithmic term utilize derive perceptron discussion multi perceptron perceptron class categorization online mirror procedure conservative update ignore round mistake recall di dt receive ty ty mistake mistake bound conclude mistake make sequence bound briefly family kernel consider k unconstrained class class margin jk class j lasso mild kernel large kernel discuss upon logarithmic inferior bind rademacher however result generalization bound devote dedicated effort candidate proof derive strong duality helpful discussion key excellent reference function vector equip inner fx deal dual property dual conjugate conjugate f subdifferential fx vector l n product inner singular eigen value von equality orthogonal g gx permutation define start ng allow immediately conjugate absolutely symmetric singular eigenvalue entirely fan inequality instead norm lf n ng corollary absolutely convex norm ng another allow subdifferential ng prove case
clearly mathematical iteration dependent predict paper thresholding inspire message pass algorithm associate encode relevant measurement rule mp lp paper albeit mp thresholding estimate significantly easy exclude eqn depend weakly dependence careful analysis lead correction correction capture hand side eqn amp statistical reaction reconstruction iteration curve success amp se describe property properly version mp powerful reconstruction conduct extensive amp mp intensive detail complementary show model map accord se mse recursion although map lead threshold notice sparsity function concave indeed claim derivative omit remark due vanish combination concave concave sufficient explicit please information soft define optimal se optimal se analogous minimax constrained support nonlinearity amp se formalism mse hope design monotone amp inaccurate offer room nonlinearity support exploit eq offer essentially se phase transition experiment little threshold sensitive detailed support ensemble exhibit phase often large apply operator example fourier find case acknowledgement award dms nsf theorem compress certain accurately reconstruct exploit characteristic achieve reconstruct iterative alternative optimization algorithm substantially bad convex modification thresholding make iterative agree calculation formalism derive tradeoff agreement formalism sense refer body traditional make content sampling apply formula scheme use elegant promising recover signal expensive reconstruction scheme image acquire involve thousand million despite advance lp dramatically would achieve sense lp reconstruction dramatically fast iteratively threshold transpose finally iterative popular researcher year focus scheme scheme per attack application lp solver attack fall sparsity reconstruction iterative scheme base ta z include model numerical carlo amp lp sparsity limit phase mp reconstruct success failure lp partitions sparsity identical reconstruction region strong formalism accurately predict dynamical numerous formalism square variable change formalism predict amp recover mse sparsity ratio develop find coincide precisely ie fail amp within statistical precision fast perform programming base random simulation formalism remarkably success phase lp give boundary principle entry measurement normal canonical nonlinear linear reconstruct nonzero cast entry despite condition procedure perfectly recover take tradeoff sparsity limit control fraction domain phase reconstruction formally whose domain two region reconstruction tend zero exponentially succeed exponentially definition green analytical prediction geometry line datum amp random dash present estimate percentile percentile iteration obey reconstruction green ordering say sparse sparse curve behave surprisingly amount sparsity green really help quite agree respectively principal paper combinatorial geometry need modern problem image demand reconstruction thousand million practical system describe behavior convex run slowly minute hour require operation operator vector number really rather section inspire use extremely stop iteration scheme depend iteration paper j empirical sign vector drop depend mean mse answer orthogonal stop invertible correctly really sketch truly understand course immediately behave kind random suppose accurately model entry version sparse recover noisy heavily understand error sparsity consequently level accurate valuable digital communication interpretation channel interference interference coordinate weakly thresholding interference detect many zero remaining cause reconstruct weakly interact fraction significantly reduce interference actual behavior sample reconstruction track iteration fix less stable point unstable formal mse form independent random soft thresholding sparse u sort familiar soft supplement recursive system stay
sum degree polynomial independent hold multilinear polynomial use notion straightforwardly structural restriction degree critical al index multilinear polynomial regular multilinear px x x multilinear degree kp lx structural lemma variable qx qx qx qx use show use large apply get degree apply regular kp lemma suppose kp flip sign suitably constant definition lp combine markov bad l lx lx sx bind anti polynomial multilinear polynomial et sensitivity degree polynomial degree anti qx pz last anti apply eq setting tail early error claim work multilinear degree polynomial I ix x l ix note value fx fy fx eq sensitivity boolean hypercube bound sensitivity boolean immediately sensitivity boolean nf observe n eq sensitivity degree q ig ig induction cauchy schwarz z f ix I term expression average vanish notation x ix constant go acknowledgment thank early version appendix polynomial denote sequence cardinality derivative calculate eq derivative since depend without multivariate polynomial basis polynomial sx call coefficient polynomial work p claim recall exist since calculate assume loss bivariate degree expand suffice constant sx sx constant theorem corollary question cs edu give first nontrivial sensitivity sensitivity boolean equal total sensitivity combinatorial case sensitivity application new resolve result al due iii structural theorem structural interest template transform problem threshold hypercube polynomial let real say boolean play computational circuit quantum interesting property spectra sensitivity characterize average sensitivity conjecture nontrivial bound average sensitivity sensitivity average sensitivity sensitivity fundamental arise boolean roughly speak sensitivity randomly input sensitivity definition average application hardness circuit theory social quantum focus sensitivity boolean function agnostic hypercube also efficient et learning begin boolean sensitivity boolean boolean boolean hypercube perturbation measure change order analyze sensitivity noise similarly let univariate boolean gaussian define boolean degree gaussian boolean handle boolean boolean degree gaussian noise multilinear ellipsoid hold total sensitivity random hypercube sum influence sensitivity function clearly know sensitivity monotone tight sensitivity progress upper use sensitivity boolean give elementary argument show sensitivity degree elsewhere depend coordinate would important ingredient sensitivity bound structural restriction result paradigm play fundamental theory least restriction restrict regular influence bias motivated restriction degree reasonably generic one regular precisely reason author generator nearly outline obtain bound sensitivity move sensitivity lemma al sensitivity argument sensitivity regard theorem principle majority proof majority main anti concentration bound ball probability bound change either perturb slightly noise involve large second easily distribute opposed hypercube anti concentration boolean invariance invariance polynomial e sensitivity result structural biased former resort noise sensitivity latter merely bound extend issue exist variable restriction restrict polynomial polynomial biased resort bind merely sensitivity easily turn broad also polynomial anti invariance boolean sensitivity sensitivity boolean boolean however bound sensitivity challenge agnostic follow error gx mapping marginal cn sensitivity polynomial degree sensitivity replace multivariate existence function imply agnostic precise relationship sensitivity essentially follow bound noise sensitivity concept class appropriate use sensitivity hypercube concept class degree learnable respect concept learnable respect first hypercube degree learnable spherical gaussian sensitivity spherical distribution open implicit boolean trivial broad continuous believe obtain bound sensitivity
post burn sample trace knot convergence appear around show posterior modal single around parameter estimate finally continuity pose discussion posterior knot modelling level daily theory upon approximately follow generalised pareto dependent determine daily record international almost previously figure model shape point model crucial making prediction concern event reversible merge model mix inference curve specifically day variation temporal fluctuation expect scale simultaneously express coefficient adjacent year curve impose indexing clarity unable analytically integrate mcmc update generalised optimisation posteriori sampler effort factor prior specification spaced interval display plot pointwise level year year return conversely change tail fit pareto density variation year return correspond indicate early article focus approach allow via metropolis adopt sophisticated extend method depend interval knot accurate particularly instance map find example overlap interval interval reversible jump gibbs relax need coefficient detection converted point complex implementation jump involve split birth death move case analyse reversible sampler handle complex non standard found implement software efficiently material program please relevant david discussion research scheme dp de universit france school mathematics method regression unknown allow non provide sampling make material include computing keyword gibbs markov carlo splines article curve independent curve wish interval article powerful curve function represent location spline well curve knot coefficient exposition regression model spline introduce number subset select potential knot bayesian g carry require sampler curve fit straightforward easy knot reversible avoid square mcmc nevertheless define candidate knot limitation sort distinct knot recommend place sort cubic spline clearly problematic spaced treatment conjugate reversible jump run number article auxiliary variable introduce context knot knot lie expect curve knot give general modelling carry extend auxiliary gaussian auxiliary variable beneficial conclude discussion adopt indicator unknown absence range adopt denote product value give denote prior parameter prior relate prior conjugacy integrate correspond default work recommend range uninformative lead posterior chapter knot use knot allow quantity configuration equal probability gaussian easily posterior distribution sampler successive update involve two add swap propose involve swap two exchange move accept usual hasting uniform posterior metropolis sampler circumstance produce draw conditional commonly use use average configuration value mcmc carry discuss selection cubic sample add rescaled unit gaussian evaluate grid compare use mean difference method order mse sampler perform mainly attribute allow selection finding estimate difficult visit time quickly relatively short component randomly secondly quick since metropolis hasting jump regard mcmc long particularly find result suggest issue htb example map x ex ex ex l l ex consider denote nuisance methodology employ integrate step update correspond propose parameter precisely update update add swap otherwise chain update accept metropolis remain step facilitate note application computational mle estimate maximum posteriori posterior case recommend acceptance probability greatly poor choice delay accept reject decision acceptance beneficial move make distribution facilitate mode refer mixing become poisson sample fit consecutive limit avoid numerical sometimes calculation deviation run perform mix use update map differ mle plug smooth updating slightly expense include hasting exist alternative propose importance
section user adversary adversary user belief adversary play role probability define admissible posterior schwarz norm appendix follow since w form possibility user collect term people user rule kx decide set adversary belief adversary user world form procedure dirichlet distribution estimate dirichlet multinomial concrete observation multinomial sequence observation c function parameter dirichlet parameter use follow htb propose synthetic multinomial estimate via compare oracle differently biased model adversary second experiment access discretize model course user role correspond figure enjoy perfect concern distribution world use last similarly bias observation case percentile multiple per result since equip degree model framework run select user error record call appropriate baseline model world use bias bias approach examine oracle half single world bad biased model run though biased performance wise note ratio false positive negative world partially model htb datum prior adapt label upper attack examine essential naive dimensionality outperform partially possible approach sensitive case probability make tune achieve desire automatic useful comparison complex give promising lack adversarial marginal x prove statement sufficient obtain induce schwarz inequality organization detection project support economic grant depend prior variable tend irrespective empirical approach standard review technique deal statistical decision toy validate indicate outperform useful make scenario lack classical example maker whose accuracy measure decision base decision typically decision remain decision training decision rule detection spam normal relatively wish generate ask decide whether belong person set overall experience conceptual adversary distinction shall example instance user must separate specific adversary people use adversary attack decision making employ current observation upper good bad approach approach world model result validate detect building discuss framework introduce sec present main disadvantage label attack learn label attack train simple concern datum base nevertheless many detection predict expect application place individual attack severe cost alarm cost decision unknown expect rather body sensitive statistical decision sensitive assume availability label dataset attack cluster able detect attack disadvantage adversarial view main detection efficiency decrease extensively considerable call examine paper since seminal model consideration adversary essence frequently et adversarial adversary adversary adversary investigate reverse allow retrieve attack consider adversary lot adversary main attack without adversary current observation create bad conditioning adversary prior population user world essentially soft bad adversary construct relate nan appear explore sophisticated problem case maximally similar spirit contribution experimental model consider assume set observation ambiguity decide whether adversary complementary adversary user
near elimination bad ii quick convergence former per simple implement speedup modification brief exchange strategy formally algorithm algorithm monotonically multiplicative case continuous space design denote closure represent proportion assign design rounding usually convert assign model parameter interest response independent matrix eq equivalently design determinant unbiased widely criterion shall design alternatively e criterion motivate state weight crucial numerical approach refer well multiplicative choose let mapping highlight normalize al paper concern improve multiplicative principle exchange point exclude modification large step iteration maintain monotonic yu another yu conditional concerned theoretical ingredient direction define follow set maximize directional great ascent abuse shall closely exchange k maximizing perform optimal exchange carry denominator cauchy one numerator constant say rarely exchange method point ii resp minimize maximize optimal transfer nonzero choice shall key ingredient near multiplicative difficulty mass adjacent design together measure metric proportion put support would significantly mass add neighbor easy example eq single quantitative close whenever consider perform fractional intermediate output step nearest neighbor support exclude I intuitively appeal depend natural sometimes encode factor neighborhood interesting approach dynamically determine specifically much experience number minimized turn exclude shall composite mapping rather adopt two exchange serious stand alone outside support assign put previously define iteration neighbor fractional intermediate help keep iteration cost effective seem extend easily alternative consider monotonic convergence property increase establish see al yu monotonic convergence theoretical concern wu example immediate neighbor monotonic iii consequence rank iii effectiveness model exclude code request author line purpose conjugate cg quasi newton bfgs powerful dimensional effective consider system one count favor inspection iteration spend receive e algorithm reader start set randomly point intend ensure keep small design remove iteration relatively insensitive initial multiplicative always start exclude design priori linearization design grid evenly spaced parameter include analytic consider surface nonlinear stop criterion meet exceed large experiment compare omit similar evident improvement algorithm vary qualitative remain table median count multiplicative clear improve upon often tend fine exist seem insensitive concern newton via r function quasi popular bfgs conjugate test design iii substitution al derivative bfgs general purpose stop bfgs moderate gradient table bfgs quantitative affect nevertheless limit confident global conjugate cg bfgs cg bfgs cg bfgs design linear either multiplicative algorithm optimality yu vertex strategy close
theoretical mathematical model biology complicate become primary tool analysis large complex different probabilistic exist standard frequentist intractable assessment observe know integrate thereby treat nuisance likelihood abc calculation replace simulate desire agreement approximation material parameter whenever plausible candidate model closely shift onto posterior framework conceptual non bayesian practical consider show make expensive evaluation model develop abc selection formalism smc sampler abc employ whole bayesian formalism new model reaction dynamic real describe explain joint indicator likelihood posterior model scheme idea smc approach powerful reader material derivation well smc marginal basic consist candidate otherwise particles smc particle pm intermediate distribution gradually present smc indicator calculate return calculate tm b km kp I weight q set go particle previous denote perturbation allow perturbation replicate fix particle particle marginalization straightforwardly distance informative sense reaction rate informative preference particular inform find try arrive truncate adapt specify result special algorithm illustrate reaction abc gibbs select publish selection approach pathway stochastic reaction reaction occur protein synthetic measure use abc smc correct confidence tm n b tolerance schedule gibbs random become application biology bioinformatic gibbs form smc computational rejection collection iid ising eq sufficient respectively simulate parameter abc estimate computational speed smc compare n km tm kp exclude correctly analysis rejection abc approximately fold speed average abc smc occur infection infection address molecular spread infect distinguish spread inside infect community become infect infer separate question consider four characteristic hypothesis share function denote strongly appear four overlapping distribution show posterior posterior median run support mean table confirm marginal three share week model genetic difference heterogeneity infection among combine well suggest shape molecular well population parameter c abc smc agree conventional abc rejection turn previously perspective though crucial survival create site bind activate become act hypothesis originally get histogram population distribution schedule perturbation dd interval distance function root sum square ambiguity development mathematical action activate basic leave add appropriate developed leave original analysis evaluation course physical act equation propose abc delay without delay numerically equation add time noise identically population marginal bayes population accord evidence positive appear receive without delay allow nest methodology monte illustrate usefulness wide applicability smc even experimental unknown dynamical apply modelling
efficiency however effective prior knowledge light construct proposal proposal standard deviation proposal mcmc datum light symmetric simulation set simulation baseline count center concentrated infer event posterior center near relate simulation curve run property bayesian coverage interval summarize appear center calibrate interval overall result exercise simulate serve increase confidence validity apply discard burn parallel approximately cpu second hasting event event figure appearance characterize change appear minima evenly interval zero notable tendency median classical median likely p deviation symmetry relative event conduct count even split median equally fall zero calculate certain deviation extent reflect instrumental maxima curvature deviation support detect far possibly instrumental instrumental close nm result dominate phenomena effect count symmetric light expect small translate study curve shape differ signature study range clear object correspond finding time origin present herein minor examine symmetry parameterization flexible inference make quantity width half maxima extend apply angle incidence account uncertainty light offer construct location bit tractable multiple c acknowledge would like thank participant feedback particularly run computing group plot event omit anomalous omit due anomalous department computing university light curve ray center parameterization event characterization event key carlo carry novel curve light provide detect survey vast database series become increasingly examine traditionally examine skewness stack satisfactory ray light number particularly nonparametric skewness significantly parameterize skewness estimator false design publication claim detection diameter dynamical early population observation ground star present observational finding year group claim stack profile asymmetric however entirely satisfactory heavily transform take uncertainty wherein claim x instrumental al present evidence event event detector unfortunately beyond stack validity conclusion remain uncertain present ks year paper parameterization unimodal light profile describe wide naturally parameterization poisson event profile chain introduction approach proposal center event datum identify instrumental suggest describe apply discuss curve conclude assume intensity event occur sake discussion event extend trivially character intensity count sign deviation characterize intensity characterize pattern throughout index parameter characterize event baseline peak put constant make characterize location deviation still event proper balance parameterization give enough wide range event shape introduce event characterize addition first two deviation refer life approximately correction numerator event characterize symmetry approximate width maximum characterize rate characterize rapidly source second characterize rapidly intensity curvature event profile change intensity event see figure priori center interior contain light curve complete analysis improve become near end light curve restriction chain metropolis hasting symmetry excellent please quite result analytically tractable simulate metropolis draw metropolis hasting hold draw choose accept calculate holding etc new proposal analogous shape metropolis version signal center move analysis window ten point narrow noise capture result value light curve narrow curve initialization far proposal initialization first robust initialization brief duration smoothing smoothed light contrast simply overall contaminated initialization baseline assume symmetric acceptable
process combination point string function collect process state translation language string act sketch cone generate closure closure routine exercise check cone ideal computing however act might state mean parameterization rise hide process determined string length thing aware dimension obtain generator onto class unconstrained hold theoretic equivalent would ideal unconstrained string maximum degree increase obtain equation induce positive summing case entry row section accordance law string string matrix associate iff straightforwardly exploit unconstrained image usual string obvious lie finite mean induce translate theorem assertion necessary rise apply involve connection theory introduction see trace shall try general dimensional language trace trace therefore string application function dimension hold assertion application string function large stre kind relation trace relationship trace hide characterize chain correspond characterization state thm thm conjecture thm thm definition thm remark thm problem outline novel arise string function theorem outline model identifiability early view around value stay discrete accordance contribution example random process walks mostly quantum computer introduce string formally string function far provide helpful sec generate ideal algebraic obtain conjecture list hidden model detail sec obtain base precede trace relatively detailed proof finite letter concatenation operation direct string discrete view rise associated function discrete iff condition hold string call iff apply accordance terminology string function notation prediction case binary string string string also characterization source exist theorem prominent markov state alphabet emission probability initial define eq representation characterization string equivalent exist transpose follow actual dt dx matrix contrary resp row vector string function note govern condition generic need entire string practical resp span resp q resp row eq need theorem driving theorem finite minor column string resp string matrix size naive exponential runtime determine string basis row resp resp string imply let prove string parameterization case parametrization fact parameterization accord idea string rank eq done string matrix necessity claim give rise desire parametrization term elementary contain statement induction induction p span suffice induction q let two fully condition translate state column condition induce therein linearly dependent closure irreducible eq choice handle certain care rise unconstrained process string big give rise string ideal would consideration mild lift set generator generators string give great string define string member let unconstrained process step nh n nx n na generators generator generators ideal generator uniquely string least
disk pc addition inverse distance relatively uncertainty uncertainty dominant count uncertainty developed star star correlation negligible want describe star frame around system axis point rotation toward measure projection velocity individual star observation object project sphere coordinate term observe frame angular position angular frame depend coordinate transformation matrix epoch position coordinate north north respectively epoch deconvolution radial star use component remove restrict study velocity star uncertainty star star van reconstruction star gaussian variance restrict parameter rather method survey aa velocity one fit velocity good fit velocity hyperparameter km compare reconstruction star neighborhood base star penalize star available aa aa aa agree peak consistent move therefore well complicated velocity turn cluster probably cause dynamical bar way show merge plot actual high number give axis step implement programming language depend code reduce ensure full optimize increase describe maxima extremely merge select gaussian merge merge gaussian merge gaussians split split equally gaussian initialize random vector dm split merge gaussian gaussians hold affect gaussians sum gaussians convergence follow parameter great triplet split merge candidate question remain triplet quickly triplet define merge criterion observation equal probability gaussian merge therefore merge probability th gaussian kullback leibler distance local complete il il th density current leibler two nonzero finite local split determine problematic kullback model gaussian candidate merge merge pair rank decrease summarize step em specify correspond merge split sort candidate triplet sort list merge equation run affected gaussian begin triplet none merge list first stop go split merge constraint find maxima thank fr ed comment anonymous associate valuable support grant nsf grant perform w von expectation carry unique missing distribution even individual uncertainty project conjugate merge procedure avoid maxima infer velocity measurement infer find significant commonly many science case property different though uncertainty source well variation uncertainty significantly vary apparent interested really description uncertainty never know uncertainty property underlie distribution incomplete pose similar miss exist approach approach technique e zhang uncertainty recently attract none approach use account noiseless mixture generalize incomplete give model objective multiplying point obtain expectation algorithm optimize much way degree projection estimate incomplete current limit reduce show report merge search also incorporate merge issue datum purpose correctly velocity measurement component velocity measurement cover nevertheless good velocity describe application sample uncertainty mixture maximum likelihood seed goal fit true value datum matrix matrix infinite latter good infer velocity star know star exceed speed light exceed galaxy point projection greatly reduce place probability mix q unity observation ip gaussians parameter model choose explicit justified log eq q optimize optimizer reach maximum parameter must nonnegative add surface follow monotonically restriction mixture precise indicator extra handling datum would use prove update also likelihood observation consist likelihood respect model log show component current draw I denote expand eq covariance eq q thus ij ij straightforward full monotonically em wu maxima capability compute unknown encounter iteratively maximum equation become update might get monotonic increase avoid merge problem singular covariance produce rather different penalize maximum briefly regularization introduce conjugate mixture density normalize optimizing replace amplitude issue improper choice switch bayesian issue maximization split start algorithm variance reach gaussian gaussians merge alternative birth reversible mcmc variational approach move component suit detail merge complete specify set regularization assume basically function reliably could hand small believe get objective mention validation gaussian could well significant overlap overlap available test explore star km sigma plane proportional logarithm amplitude panel contour contain percent percent dark km move correspond often fully uncertainty reversible infinite infinite variational extend heterogeneous incomplete beyond extension straightforward method true value literature gibbs infer neighborhood angular
leaf allocate leaf categorical leaf response count z leaf expect count default leaf trivial leaf covariate allocate leaf lead representative particle include encode upon arrival sample version particle learn pl sequential carlo original pl pl mixture dynamic pl recursive due particle resample proportional wide question pl address make quick specific crucially resample introduce reduce dependence particle find filter remain track analytical greatly monte rao evaluation l marginalization move identify particle update low detail approximation depend set leaf section restriction plausible move accumulate update leave consist upon observe covariate follow residual resample new ty via stay move propose grow draw non u calculate posterior probability root sample leaf finish update particle approximation division incorporate global tree provide particle practice careful assess mention gain consist split rule sequentially discard except predict pl marginal available tree overcome datum aside model parameter condition large proper predictive distribution root factor pl constant pl marginal specification marginal likelihood ratio bf comparison leave leaf favor leave leave general covariate regressor bf assumption datum see consistent serve effective present series consider constant mean pl compare compete section focus describe application multinomial tree classification throughout dynamic tree parametrization inference parametrization four datum available mass acceleration simulate impact two row show filter fit pl row interval variation pl datum model appear adapt leaf lead high height opposite leaf evidence favor leaf estimate bayes present filter leave calculate first random ordering comparable preferable although evidence favor majority bf run increase agree visually leave particle interval cm filter log factor leaf mean compare tree regression bayesian constant make partially problem mcmc previously take broad partition relatively complicate response surface model practically leaf global additive tc gp forest hide respective size uniform root new candidate predict adaptation lead global summary finding leave relative fitting rapid interestingly routine flexibility gp augment candidate range hard standard addition repeat application focus response informative location efficient automatic covariate prediction error search draw I mt heuristic statistic optimum solely understand specify input gain heuristic alm choose maximize average upon maximum design alm heuristic alm extensive alm constant active inherently dynamic tree remainder illustrate alm scheme build particle alternative leaf functional evaluate candidate particle hence heuristic tree update alm seek maximize simple allocate mean equation leaf unconditional straightforward maximize give constant approximated candidate maximize alm leaf illustrate two heuristic column alm look much alm different exhibit additive ccc surface p cauchy rmse alm alm alm I tc tc alm alm alm design alm alm alm tc alm tc alm gp alm I gp design progress tree model leave heuristic hypercube statistic rapidly compare static start follow candidate round square predictive mean I repeat expensive dominate alm static fit mcmc amongst bad neither evenly gp alm offline learn tree clearly dominate second leaf I gp interest space tree categorical input high particle surface available mass library book response multinomial leaf dynamic gp outline fit gp latent p outcome tree particle gp max result bottom chain column figure surface fit decay cm class result soft classifier bottom black plot label mean surface illustrate problem top offer gp cluster status categorical encode aside note encoding incorporate categorical datum linear leaf adaptation exclude binary multinomial repetition sample soft classifier I gp correlation sophisticated leaf winner order gp particle feasible application dynamic max fold cm reformulate tree dynamic entirely class automatic specification avoid sampling tree point tree space posterior search essential model improper constant leave look find mechanism lead cm dynamic option surface accumulation learn efficient line filter state major partition division potential local newly capture along propose default specification integrate conditional illustrate nonparametric regression classification detail methodology motivate application commonly fraction partition cart school business edu laboratory university production article research partially sequential detail example experiment design characterization dynamic model grow practical effect algorithm example demonstrate propose generic formulation probability simple powerful situation priori hyper simple rectangle state induce restrictive formulation conditional simple tree depend approximate model updating sampling alone thereby specification alternative dynamic herein appeal e expense correlation specify realization easily finally perhaps problem serve engineering computer prior additive gp decompose build around sequential new datum intensive nature problem search calculation explicit expensive serial sequential gp fall design search due specification may area interest along response optima stationarity modeling scheme counterpart contrast fundamental consideration may flexible function set simple posterior suited class model specification automatic inference sequentially filter state prediction surface complicate review core outline characterization evolution define leaf prediction detail provide application alternative literature sequential experiment provide tree relationship relevant base approach partition regression cart force transformation use conceptual rectangular partition scheme recursive shall outline detail covariate correspond tree consist hierarchy subset split also terminal node new tree include sub tree subset diagram form recursive partitioning child parent contain equivalent node otherwise leaf internal leave parent child illustrate split set tree complete simple leaf covariate response regression tree allow assessment partition predictive band generative specifie place rule leaf depth implicit partition create contain ignore prior probability seminal develop partition tree stochastically propose incremental change swap move accept although provide chain move probability represent drastically batch inefficient characterization lead particle framework able difficulty mcmc trees model introduce recursive rule associate observe time define function newly covariate allow evolve small neighborhood specify section likelihood leaf marginal likelihood evolution detail describe multinomial outline partitioning statistic update combine resample account finally section evolution keep possible localize evolution equally move node include parent node uniform dimension
analyse independent dependence efficient approximation distribution study negligible piecewise competitive infer uniquely commonly allow change application posterior liu producing draw efficiently sc base assume note parameter segment give segment segment within start conjugacy side update piecewise polynomial segment ccc x x independently distribution geometric segment underlie refer segment note determine segment consistent dependence conjugate prior gaussian km km previous start observation end curve calculate interact multiple let step piecewise tm match moment number simulate give final segment simulate simulate model repeat segment conditional c pc pc tc segment simplify notation mp get need simulate smoothing algorithm give assume simulate mc simulate simulate smoothing position accurate obtain simulate simulate segment tractable piecewise polynomial give calculation become pdf inverse multivariate evaluate come independence simulate multivariate normal multivariate normal hold give multivariate segment filtering smoothing point linearly expense approximation particle resample liu chen discrete resample result bound constant investigate substantial obtain negligible evaluate model first look filter smoothing simulate accuracy curve underlie implement use filter rejection method resample filter observation variance segment segment smooth roughly second sample run value underlie could draw true quantile demonstrate deviation equivalent performance datum set equally spaced posterior plot dash dot dot line simulation black dash plot quantile case close extra quantile interval generally plot suggest approximation posterior look regression new fitting firstly implement parameter uninformative estimate hyper whereby default choice simulation study choose see repeat effect quantify set curve square coverage credible wavelet detail ccc c mse credible simulate mse estimate substantially two wavelet mse method coverage likely use bayes effect repeat default scale increase default increase mean error repeat last case repeat time lead table advantage study cubic expect modulus efficient increase underlie new coverage interval curve method still substantially wavelet curve ccc error coverage credible piecewise cubic simulated comparison various henceforth dms datum block signal denote dms reversible piecewise continuity approximation approximate segment magnitude analyse ccc snr dms dms set set give point observation mse dms compare method considerably estimate curve underlie dms set peak rapidly use polynomial cubic fit curve suggest dms method accurate wavelet investigate calculate error obtain use wavelet implement table integer power apply wavelet dms block detect curve curve due curve dms point
hoc task distinction sentence job create query applicable multiple document formulate term document relevant document correspond search query build concept behind ir represent query document probabilistic bag look rank make researcher build build kl divergence strictly assume distribution assign probability exactly smoothed final query focus sentence sentence eq system sentence document relevant query background information contain english appear relevant query word english level query background english general english capture english background document language query language specify exactly word rest sentence layer also degree lie dimensional simplex english variable language document corpus parameterize multinomial restriction continuous normalization term generative define document generate document select generate word relevance document query document wrong english depict know square circle relevance unobserved circle indicator degree contain document observe give expression datum accomplish prior sum value final word condition select intractable coupling integral give rise technique monte mcmc saddle approximation effective deal expectation propagation propagation roughly long propagation generalization belief propagation assume filter thesis superiority variational product integral give inclusion leave expression fix method ensure integral well approximation ep reliable ep omit experiment document document return search engine short conference data relevance require query typically break title sentence sentence concept keyword model train query relevant amount roughly median document relevant query remove seven manually extraction ask select document need overlap evaluation agreement annotation doubly annotate data inter agreement keep sentence precision recall criteria reciprocal calculate sentence order average reciprocal rank first precision relevant sentence human baseline four information retrieval rank fourth interpretation ir query rank sentence fourth cosine compute sentence document smooth collection retrieval blind feedback expansion first return relevance retrieve top expand sentence method interpolation parameter oracle optimistic relevance however access model evaluation corpus system ep hour query field evaluation description consider concept alone baseline position bit turn p maximize baseline either experimental show system thing standard without perform position model title relevance feedback might initially seem actually available several sentence however blind map kl kl kl input kl description title make metric add improve performance improve substantially model give collection violate access collection relevant deal collection irrelevant dotted kl star title red circle title indicate ir evaluation field engine linearly relevance interpolation obtain ir engine improve relevance six interpolation observe dot ir little difference obtain roughly dominate ir engine look perhaps surprising difference ir title believe title field contain nevertheless trend ir conference evaluation competition fortunately account redundancy document technique select sentence central well accord pyramid automatic element evaluation confidence significantly understand conference competition focus nearly mse additional compression component summary query system score competition linguistic evaluation worse likely sentence top automatic perform third never significantly bad describe generate focus bayesian focus state retrieval force relevance alone achieve score primary operate purely question arise outperform formalism relevance access
standardized compute bind pt x assume choice skewness equal skewness formally proper example positively skew symmetric skewness initial introduce skewness inverse positively due rarely symmetric algorithm search skewness skewness guarantee make impossible see return either upper positively skewed rarely shift transformation initial affect triple alg pass update input accurate level skewness k return triple great skewness student uniformly symmetric disadvantage probabilistic skewness remain must separate slightly second jointly underlie algorithm check reason optimistic true ii ignore branch leave considerably available kolmogorov input package vector specification skewness rv simulate though simplification would guarantee indeed general obtain although nontrivial expression alg sim finite input particular mle gaussian skew package whereas accuracy skew versus rv additional value return introduce reveal branch mle skew replication input rv relation set euclidean gaussian skew standard variable thus comparison imply estimate x impose cause loss finite sample symmetric distribution ml rmse unbiased whereas moment explain effect truly distribution estimate give confirm normality skewed gamma asset exhibit slightly skewness c ml skew present ignore mle mle rmse present rmse small reason though extremely skewed gamma skew mle skewness lie theoretically branch mle skew mle fail accurate heavily skewed practically rmse skew increase sample rmse ignore branch surprisingly size c c normal ml pt whereas extent skew inferior compare restrict extend class mle little normal precise location skew skewness absolute skew normal include bottom iteration right input degree freedom table increase away origin close bottom size find total number number bottom approximately time show rv degree freedom take input result return start process property fairly assumption quick heavily skewed demonstrate skewed transform triangle pt top normality test min mean skewness distribution appropriate well standardize auto risk body index dot several fairly skewness normality assume give yield positively skew datum reject w estimate significant evaluation triangle adequate lie half open lie local closeness density improper approximation health explain skewness support sense natural albeit financial introduce excess typically generalize auto volatility theoretical however usefulness distribution direction note resemble closely future news return interpretation stock input latent news reject consequence location table coefficient particular increase solely address negative daily leave transform data mle table kolmogorov student adequate unconditional vertical horizontal news scatter observe return outcome bad worth location powerful market show bad financial estimate potential asset fix period percentage expect confidence period statistically quantile distribution way empirical quantile comparative c pt quantile capability excess degree freedom high skewness datum skew tail capture news winner skewed skew one median true concentration return financial call return mle parametric model successful uncorrelated unit detail student available find standardized residual fit return package return mle standardize residual still give estimate study combine possibility skew suggest area research finding skewness asset skew lead suit asymmetric price financial return recover f rather skew control tail idea skewness specific get skewness heavy tail reveal important play role mathematic physics field introduce financial great flexibility respect rv wide transformation acknowledgment grateful give I cat de anonymous comment suggestion manuscript notation type point generalize parametric skew particular nonzero skewed variant counterpart maximum estimator show affect finance relevance particularly useful skewed package author publicly exploratory look gaussian asset tail skewed sense fairly generalize prominent generalization skew include skew rv density skew skewness lead skew cauchy skewness skew skewness seem put cart skewed variable start skewness propose novel naturally model observable rv input either chemical physical biological kind simply represent restriction analyze detail methodology instance stock asset asset percentage price skewness excess call left daily return percent series excess skewness excess kolmogorov skewness asymmetric xy skewed choice regression quantile estimation convert back skew world news w asset return perfectly empirical evidence student skew fundamental consider result bad market return good news positive return empirical news skew really thing happen really thing return get news get news drastically negative news distribute act skew rv way exploit skewed procedure skew skew transform result skew truth take skewness consideration ignore summary test kolmogorov ks pt st skewness define counterpart parameter sec show skewness affect estimate useful skewness set return particular output detailed quantile estimate essential appropriate plot evidence link square rv distribution figure simulation realize source statistic package notion translate terminology rv continuous rv parameter family rv value possesse continuity expect also rv factor positively exist rewrite exist typical scaling signal affect system suffice become useful transformation branch branch unique two real stable principal branch similar input extreme denote cause evidence student ignore root matter much branch obtain generate stand estimate branch alg w cdf likely ease yy u monotonicity split separate derive f scale analogously branch coincide xu scale rv equal analogously depend restrict c c importance value rv equal scale rv nonnegative rv inverse f f follow note rv f rv flexibility expression researcher easily create variant four coincide become skewed although particular sometimes aspect might return whereas solely generalize analyze concentrate inspection directly rv median transformation pass furthermore input median corollary interpretation general u analogously explicitly obtain quantile rv well insight gaussian moment particular
easily v consequence nd h converge surely lemma together imply combine surely respectively n ks nm x nd converge surely pt square integrable martingale equation x ng dx g say k furthermore martingale g conclude consequence dx x proposition check n combine dominated theorem constant g nj hx c nj j n ng eq inequality inequality v give acknowledgment grateful helpful discussion point reference helpful comment technical argument supplement section algorithm nsf dms asymptotic variance chain almost result chain literature weak result adaptive algorithm flexible framework sampler see interest mcmc markov hx kn role assess performance variance sampler markov chain markov precise constitute framework analyze mcmc well method notable mostly stationarity broadly contribute variance ergodic general example chain mixture variable geometric stability markov bandwidth almost coincide converge deterministic random derive bandwidth mcmc carlo describe method overlap batch consistency chain assumption ergodicity moment variance time model ordinary square condition converge estimation version call various hold martingale approximation adapt treat law number array differ sure take class section adaptive markov understand behavior result also supplementary logistic act measurable fy qx metropolis throughout impose ergodicity exist constant q ergodicity assumption moment probably redundant geometric ergodicity imply dr either drift large establish adaptive short proof state theorem assess g omit notational convenience nonnegative eq easier check positive metropolis independence similar metropolis langevin reflect fluctuation metropolis adaptation side surely good kernel often say value h justify follow hold section eq twice continuously kernel impose strong replace restriction continuously supplementary article instance fail choice satisfy markov chain transition kernel satisfie holds apply ball continuously imply vx xx assume thus small focus conclusion derive theorem vx hx h hold choose almost surely similarly deduce term give eq take choice issue take th autocorrelation choose high bias autocorrelation process decay issue one hand asymptotic square fluctuation assessment adaptive adaptation chain weakly metropolis subset typically adaptation h multiple clearly even confidence interval become asymptotic consequence valid run chain advantage adaptive monte assessment important adaptive adaptation mechanism define case rwm mala langevin illustrate markov follow I assume process chain take define introduce hold kernel choose bandwidth follow approach outline run discard burn sample path plot
density update procedure slightly density serve illustrate initial importance multivariate consist nine place centre second degree circle indicate mean proportional weight start shape become separate tail target importance target density normalise normalise ess first iteration simulation shown normalise rapidly importance approximately importance normalise increase importance normalise start indicate need density general need observe horizontal axis vertical axis second indicate dot every rd importance plot run plot thick contain outside circle start algorithm poor initial importance function narrow may part tail match importance inform guess possibly play small adapt may provide sample consideration density take large increase issue datum sect simulate compare consider encounter density normal change co unchanged density center uncorrelated jacobian shape two density interest tail target curvature slightly typically highlight difficulty cover guide choice mcmc absence except choose student distribution randomly away centre variable mean component degree adequate coverage albeit region distribution preferable variable importance dimension show iteration panel pilot importance density fitting require seven coverage issue relatively long unable sufficiently problem occur dimension curse prevent numerical updating less density adaptive fast use either independent walk proposal concern common gaussian stationary product target despite however effect recursive sample suitably condition empirical pilot run schedule quantity pilot value sensitive choice position counterpart find schedule every assess update simulation ensure burn period sect acceptance proposal update chain stop burn outline appear th follow final point point burn successive interval provide performance approach simulation interested mean also tail target indicator respectively result show deviation estimate calculate fold reduction compare close look reveal see quite reduce quite majority run panel difference explain visit positively second despite estimate overall variability cccc mean htp variability plot measure density skewed bottom nevertheless significantly point panel panel highlight variance evaluation simulate represent challenge importance take properly see significant alternative structure observe adapt true change proposal covariance precise comparison mcmc apply importance parameter constant dark energy parameter test ia release release analysis publicly public tt te bb theoretical return part compute associate covariance angular scale tt te bb angular compute pseudo te v te tt uncertainty ignore correction due impose large constant alone degeneracy acoustic peak spectrum therefore angular diameter distance energy weakly amplitude determine normalization peak dark matter density optical wolfe probe describe use curve fit frame band colour ia colour respectively parameter specific distance regard year release mass dispersion observable likelihood correlation angular theoretical dispersion scale galaxy rescale distribution galaxy histogram introduce parameter model multivariate uncorrelate include independent weak angular amount universe probe degenerate survey include latter determination weakly constrain hypercube exist implicit represent numerical formulae break exclude non allow rarely solution exclude rare description matter dark l normalization optical magnitude colour sect important function rely estimate hessian initial proposal use find fisher mixture consist student freedom turn bad shifted scale fisher shift shift result shift component stay near tail sample randomly typical case fisher element adequate use reliable explore parameter effective quickly reach although posterior satisfactory yield consistent mean efficiency sampling tb mixture close consecutive tb contour movement plot circle iteration mark circle panel position importance method order consume parallel obtain core readily acceptance normalise large final sample posterior calculation make computing mcmc issue mcmc algorithm avoid adaptive result reach acceptance rate well find excellent marginal superior follow inherent sample way mcmc chain visible second exhibit anomaly chain converge feature importance uncorrelated issue nearly unconstrained illustrate choose weak fisher stay small direction flat jump flat proposal acceptance rate modify initial increase sect explore parameter step adaptation modification low acceptance fine initial recover tb tb mcmc dash green dotted alone respectively interval less percent region agree mcmc carlo aim overcome achieve towards alternative lie massive parallel posterior essence form produce exploit regular sampling output previous combine approximation storage sample improve posterior may successive importance importance tail importance procedure usually involve matrix case alternatively iteration quickly poor ess normalise outline significantly inform iteration iteration counter potentially parameter absence importance available reasonably variance reasonably sect information example point covariance point approximation singular along interest component k reasonably successful examine place support reduce iteration difficult density main worth point ability increasingly useful availability multi cpu computer cluster computer software implement message parse interface sect computer reduce target significant use fold reduction mcmc time requirement total variability valuable adapt importance target combine across iteration absence desirable attribute reduce sample sect sect simple assessment potential associate diagnostic ability evidence naturally far explore acknowledge lambda lambda provide office computational support contract base mixture principle error function increase iteratively equivalently formulate maximization bayes posterior belong evaluate intermediate quantity concavity easily check em maximal requirement maximization solution whenever depend routine calculation denominator integral propose normalise weighted n empirical eqs sect density except equal shape factor tail polynomially decrease tail whose finite family mixture student mixture student sake formula student density multivariate easily derivation chi advantage straightforward opt call adaptive importance population considerably benefit assess performance problem actual type ia provide art markov mcmc type parameter hour cluster recent advance observational availability quality testing high thank technique monte produce markov chain burn regard sample approximately design mass region visit spend grid model mcmc particular metropolis thank package form hybrid hamiltonian interesting usage estimation spectrum reference advantage sampling approach suffer correct convergence presence greatly third need computation slow speed computer posterior public code precision second explore flat course require order magnitude efficiency apart improve likelihood code availability computer speed want big complex effort devote recently code interpolation network look improvement algorithm former provide pre step latter requirement availability computation derivative non markovian monte apply present improvement availability computer computer partly way lead speed improvement iterative mcmc running chain end big chain explore converge absence bias sample determine chain support expectation integral link q provide converge context right hand normalise importance ratio normalise converge approximation normalization
significance triangle c significance empirical even power alternative implement monte carlo monte critical level segregation ten skewed almost symmetric occurring estimate skewness get kernel skewed skewness get skewed segregation alternative observe get estimate symmetric symmetry occur skewed skewness large estimate solid dash power mc alternative imply power two alternative density nan dash estimate segregation ten skewed right occur skewed leave solid maximum carlo estimate association base monte estimate value leave middle estimate asymptotic alternative value asymptotic critical nr nr power critical association alternative circle significance triangle figure monte significance nr z appropriate approximation severe association power significance plot curve power consideration asymptotic efficacy local involve limit well test sequence neighborhood nan q pc equivalent pc pc satisfy pc pc testing region n pc need calculate asymptotic efficacy segregation consider test small suppose equation h h q derivative detailed yield thus second pc pc sr give segregation sr efficacy segregation segregation large test segregation small moderate skewness density moderate around convex behaviour efficacy section sufficiently whose r equation segregation get pc r numerator pc hold association alternatives r ar suggest small association appropriate skewness moderate alternative unlike involve require asymptotic variance see rr r sr degenerate sr plot asymptotic association alternative finite denote triangle triangle convex wish segregation alternative realization realization segregation association realization segregation association segregation association use relative base yield triangle corollary corollary respectively appendix similarly segregation triangle replace case h segregation association segregation e realization figure figure realization value segregation realization implement carlo empirical power table alternative alternative realization small empirical segregation use segregation small notice also increase estimate get alternative significance realization function right realization circle empirical significance triangle leave circle represent empirical significance triangle conditional unconditional size triangle poisson point derive arc triangle rw r j r adjust nr triangle size triangle optimal efficacy segregation association alternative asymptotic efficacy segregation association right realization vertical axis differently segregation association notice efficacy moderate segregation segregation sr efficacy segregation alternative conditional severe segregation ar sr efficacy association j r r severe association straightforward detail invariance asymptotic normality statistic proximity similar factor map literature slice proximity central proximity proximity advantage tractable dominating set geometry triangle mean high segregation association survey relevant independence triangle fix nan label complete randomness interest number triangle segregation alternative asymptotic efficacy suggest efficacy unbounded case arc moderate efficacy preferable preferable work project air force office scientific contract direct position employ parameterize proximity map relative arc summary provide alternative employ relative arc properly analytic study central segregation efficacy efficacy classification receive considerable year position cover give apply employ involve dominate set prototype find minimum dominating particular tractable proximity respectively triangle proximity advantage dominate tractable latter segregation arc properly rescale hypothesis segregation association alternative central efficacy relate segregation association test detail visualization describe dimensional nx xx use represent briefly proximity triangle interior form map segment edge partition fall fall adjacent opposite line euclidean orientation edge proximity notice tx rx set set occur direct graph base vertex arc since need define functional represent number arcs arc density henceforth proximity statistic brevity arcs variance simplify central statistic depend segregation involve tend away test segregation generally spatial randomness thus sample alternative segregation association let segregation occur near association association definition invariance follow triangle vertex parallel triangle segregation geometry triangle simplify subsequent theorem transform triangle v uniformity transformation boundary median parallel edge line joint intersection bound line content preserve uniformity preserve uniform assume triangle proximity nan rx occur geometric calculation limit establishe summarize q degenerate form increase sn sn r rx degenerate continuous observe skewness analytically much asymptotic variance r successively calculation value approximation small indicate figure severe skewness depict histogram replicate severe skewness extreme recurrence note normality hypothesis segregation segregation alternative covariance r normality obtain likewise segregation association h asymptotic universal asymptotic relative proximity appropriate segregation association use segregation standard since degenerate great mean alternative follow detailed segregation likewise association follow alternative segregation likewise indicate segregation alternative implement monte degenerate degenerate large empirical relative value empirical n h segregation alternative eight skewed skewness get symmetry occur estimate skew
post manual explanation calculate limit manual provide publicly make voting include device bring together expert follow computer program fair discrepancy ability strategy complete whether outcome expand sample algorithm development candidate select look college methodology principle resource routine limit outcome need go development two acknowledgment thank david article helpful suggestion political science suggest describe author field post thank develop weighted limiting size sample many I make mathematic valuable suggestion dedicate motivated derivation vote small margin report win small win candidate vote margin number could give cast report vote winner report vote close estimate minimum cause outcome margin sized unit could cause winner win vote unit could margin sufficient cause outcome eq reduce eq increase win winning unit vote candidate margin need incorrect outcome estimate detect actual win pair margin win plotted vote candidate excellent calculate win pair bound may chance circumstance win candidate produce win margin post vote initial express q denominator get cast candidate apparent two apparent initially receive vote error win vote initial cast way way vote situation vote margin occur candidate initially incorrectly incorrectly possible margin vote vote candidate really vote vote really candidate vote really vote pair vote candidate candidate vote margin error add scenario scenario occur margin improve four limit paper b c c c c c confidence bound small unit maximum occur confidence improve size conservative unit article advance improve post post extensive sampling various focusing discuss risk post exist unit winner tool four calculate show apply margin efficacy exist discuss mistake reduce adequate post sample three article two article discuss algorithm purpose detect incorrect act computer article define post check report manually count randomly vote checking record risk limit post minimum detect correct control worth million minimum outcome incorrect winner ahead post amount cause incorrect vote incorrect largely political recommend ad systematically recommend post national institute technology us development recommend voting voting discover ed modify consider computer provide calculation vote count vary rely small report vote count detect vote propose individual voting system design report count thus make sized develop detect outcome sample size win weight fair selection prefer ten sided generator publicly voting cast live voting voting device count batch unit automatic associate number maintain unit individual produce public report vote preserve privacy size manually count vote cast vote vote candidate security procedure substitution record effective manual therefore minimum count manual count ensure voting machine percentage publicly report inaccurate outcome outcome unit need detect outcome incorrect vote detect incorrect outcome post appropriate post purpose checking outcome achieve risk limit post desire sample unit minimum outcome voting machine accurately tolerance typically margin small every vote compare effectiveness cast margin calculate uniform estimate could cause outcome flat inaccurate high detecting could incorrect outcome red roughly vote detect correct regardless win size limit win margin decrease could eliminate automatically manual necessary ensure outcome bar ensure wide house vote roughly rate limiting way approach detect incorrect outcome category flat detect incorrect risk limit inaccurate separate limit risk incorrect post author continue detect risk limit post various limiting conduct california united limiting state ensure small incorrect outcome post sampling weight risk sample size depend outcome margin article calculate post detailed datum draw quick planning precise achieve desire minimum incorrect c cast vote win total vote win vote total vote vote vote candidate vote number vote nr vote win candidate winner margin vote win vote margin margin divide cast vote margin margin upper unit nj formula depend reverse total unit assume confidence initial unit select number size manually count might small cause vote count incorrect maximum thus margin look cause assume overall winner immediate minimum could cause incorrect outcome unit fewer incorrect large unit unit still bound would crucial consequence size candidate additional unit multiply could exist without see author margin cast inaccurate assumed level necessity additional one maximum article level ratio measure herein margin replacement course notice fact vote cast candidate expect target vote notice unit upper calculation method determine precise win pair calculation recommend method multiply time vote cast maximum derive vote apply place recommendation number cast incorrectly take normalized margin win pair risk limit large author continue recommend less margin insufficient use favor candidate expression margin vote win win impossible amount share margin available contribute incorrect case candidate vote account cause vote winner versa vote win count vote win vote vote win precisely cast assumption proportion vote switch winner vote vote cast vote could vote within margin size detect vote incorrect outcome limit win pair compare actual bound upper margin win candidate win pair sign difference margin manual win margin margin author agree occur pair unit winner individual upper margin winning reduce margin bind win cast expression q error winner vote vote full manual initial really vote winner vote cast margin vote number vote margin error thus incorrectly report winner percentage vote find actual minus margin winner b c bound win initial incorrectly record vote vote margin individual candidate contribute vote incorrect vote margin contribute win unit vote vote cast vote vote win winner maximum margin become winner win perhaps simple win pair formula include vote win candidate vote vote win one contribute could cause outcome contribute margin incorrect cm c c win accurate incorrect win detect incorrect win win candidate bound reason conservative margin small ratio vote margin margin win unit margin win win candidate vote winner vote vote vote win analyze limited win win margin win candidate margin affect pair overall unit win pair margin win consideration analyze error consider separately win pair unit analyze unit wide upper margin error win candidate pair win give cause incorrect limit presence sample sized probability large size could incorrect outcome calculation cast accurate detailed variation minimum initial cause order order bind win candidate margin win unit take margin follow pseudo create array r order cumulative margin vote count vote count etc compare margin minimum vote count incorrect calculate calculation vote win pair within calculate large sample prior random algebraic rough planning purpose initial detailed vote count vote estimate size need c c bound calculate est margin occur summarize three mechanism risk limit obtain rough overall total estimating suggest estimate quick vote cast cause incorrect margin error bind eq incorrect unit error cast large cast unit derivation equation simply margin cast ratio far close value detecting object sometimes detect unit unit eq eq substituting formula combine get substitute expression combine derivation sample size upper formula follow desire detecting cause incorrect notice cause incorrect outcome avg calculate c avg column risk sample various formula calculate need provide vote proportional probability margin post size necessary desire unit randomly select propose sized calculate probability probably good avoid initial win incorrect contribute margin error win develop ensure approach adequate sample margin total margin win unit describe error recommendation calculate certainly weight select unit bound win vote cast receive treatment avoid margin win candidate avoid assumption incorrectly report know win provide vote initial show order vote need margin win candidate increase detect error candidate vote vote unit winner initial win candidate f margin unit benefit winner multi suggest winner winner candidate win consistent amount contribute incorrect win candidate pair incorrect proportional bound look many pattern cause sampling develop et size unit improve possibly cause take reasonable winning without immediate unit size sum contribute outcome probability bind develop recommend win margin calculate overall replacement vote win sum margin win number cast initial vote win candidate vote report desire report outcome incorrect total vote win total sample another sum win win candidate vote unit calculate win multiply error time evident easily add plus total winning minus vote margin error win candidate summing times total vote win calculate win candidate initial derivation describe previous calculate calculate upper candidate htbp example probability size conservative sample size vote count win stop j find method unit vote approximately overcome select kb win pair ai ij ai win candidate pair round vote count incorrect outcome avg kb w r two c desire select unit supplement necessary size margin investigate necessity crucial would candidate pay separately confidence detect essence occur within unit include part manual fail sample separately programming system sample providing chance unit variation vote small error available sufficient incorrect especially require wide manually count risk
laplace measurement give h define control resolution tuning ill pose section notice deconvolution thresholding alternatively give choosing perform density wavelet confirm superiority base fouri decomposition nd instability interested small study reasonable estimator compare divide oracle independent average model selection repetition thresholding model selection expect uniform see also ill pose inverse direct introduce level quality combine remark reasonable satisfactory estimator solid jk em jk jk j jk jk inequality obtain j complete exist inequality constant c cc n c p q q remark although always various uniformly study behavior one j ps ps ps remain term dense decompose j r pt jk jk j em jk jk p v jk em jk j dense imply integer write pf j j r combine n n complete proof cm assumption lemma section aim usefulness wavelet deconvolution use wavelet computation rely fouri wavelet band compute avoid wavelet dyadic main drawback classical wavelet threshold performance logarithmic performance direct deconvolution deconvolution threshold inequality adaptive minimax secondary de universit acknowledgement acknowledge provide compute density identically iid representing size wavelet know advantage spatially use estimate function therefore receive attention last decade detailed subject wavelet usual coarse resolution frequency dyadic various instance sufficiently scheme wavelet limited wavelet transform approach fast transform exist wavelet band deconvolution moreover wavelet fast contaminate q iid represent additive fundamental importance communication theory e density additive relate indirect observation instance problem deconvolution wavelet thresholding wavelet fouri numerical scheme contribution threshold density threshold empirical wavelet threshold poisson estimation haar compute wavelet use threshold exploit variance wavelet compute compute threshold attain performance logarithmic depend quantity wavelet inequality gain popularity procedure performance recover inequality currently introduce wavelet problem white intensity poisson name background wavelet correspond direct explain performance compare oracle oracle study depend large propose estimator compare proof function compact support hold mainly mathematical outside center fall apply transformation wavelet respectively construct smooth scale wavelet space wavelet fu du fu u wavelet transform indeed estimator fast show wavelet band avoid scheme dyadic analogously c deconvolution e unbiased difficulty deconvolution quantify error ill fourier depend decay estimation ordinary fourier decay mean rate wavelet depend smoothness base wavelet wavelet form achieve optimal smoothness quadratic risk however nonlinear estimator estimator hard thresholding positive direct take level deconvolution deconvolution however threshold practice coefficient depend deconvolution let definition jk algebra deconvolution v make direct also write n calculation obtain dyadic point upper close oracle derive coefficient suggest threshold form tune universal conservative paper small moreover tuning depend finally propose threshold instead context nonparametric regression exploit wavelet oracle ideal estimator practice depend unknown shall assess quality quadratic equation retrieve classical risk oracle coefficients basis j addition obviously integrable supremum choice tune oracle define satisfie oracle performance moreover performance classical universal one set additive risk belong choose possible fundamental constant one level price pay term obtain derive rely wavelet basis haar deconvolution ordinary assumption j constant behave tend comment make choose additive great deconvolution degree ill deconvolution cut usually ill typical various literature see small pose inverse choice understand influence hyperparameter validate result characterize basis ball respective replace ensure subspace shall allow local smoothness typical old piecewise piece locally element despite possibility one piece function local let theorem deconvolution near presentation hyperparameter theorem condition assume minimax study converge usually pay deconvolution smooth deconvolution q rate respectively spread function vanish coefficient effect detail white show reference therein wavelet toolbox matlab fast wavelet density distribution em e pt test figure scale compute coefficient theorem hyperparameter control frequency cut threshold equation rate additive inequality take
previously location explicitly paper node mobile point process channel receive act obtain success accounting interference use tool analyze scheme analytical introduce inclusion spatial mathematical wireless service consist area call bs mobile cell boundary distance inter cell interference base increase expensive paradigm mobile boundary significant architecture effectively benefit communication two may significant scheduling bs receive information mobile cell act choose subset fashion interference simple interference geometry analyze provide asymptotic analyze complicated emphasis methodology rather communication specific although extension bs bs consider act locate circle bs power simulation distribute distribute code tight precise choose maximize diversity coefficient average node location incorporate start spatial emphasis introduce metric section probability connection bs section employ analyze scheme direct assume bss arrange square lattice deterministic bs bs poisson would see observe necessary near bs outside cell assumption ms bs serve assumption location mobile base cell probability h cell bold dot bss dot space consist frequency choose node hence increase center around one form non singular path loss treating interference imply transmission locate receiver bss additional mobile bs want receive never able connect set bs connect bs connect bss origin bs cell subset potential intend connect belong connect receiver potentially ms intermediate reference selection compare gain characterize high q diversity gain transmission transmission interference correspond limited even x curve scale receiver b bs bss interference bs origin hence position intensity intensity eq able connect follow connect section respective ms channel path receive channel node fair direct success inter interference precisely reason cell b slot fair condition one begin first contact observe contact rf interference cause cell though contact lie pdf condition event yield calculate asymptotic bs easy average potential scale f fr gr fr gr follow asymptotic basic expansion dash derive interference fr remark higher limit may happen scheme monte obtain theoretically include selection orthogonal choose channel alternatively channel use well distribute fashion previous success cell empty early shall unconditional multiply mathematically selection interference aim behaviour denote eq comparison transmission easy interference cause cell intensity obtain expansion mx let difficult calculate reason asymptotic since unconditional gain low interference conditioning require similarly observe upper basic exhibit q cell maximum interference limit direct transmission monte purpose bs threshold
centre unit truncate agree confidence double restrict realize capability kernel benchmark leave tail outcome confirm realize consistently suffer small realize precision neighbourhood quantile show matter concern neighbourhood vertical set involve iterate fast double range low produce essentially transformation generate double precision approach branching evaluate iterate suffer degradation production suitable approximation except relative relative precision change leave region construct quantile double b realize precision interesting feature reciprocal logarithm map sample map ode intermediate composition cdf probit rational difficult serve fast branch rational satisfie furthermore give divergence therefore back branch computation computed precision indistinguishable implement branch implement central central fall back vote modern gpu seven algorithm four code double dp double precision pure exponential hybrid understanding characteristic capability gpu iterate iterate b hybrid appendix clear provide double range monte argue preferred advantage relatively logarithm double precision note branch optimize tail behind hybrid robustness respect increase per briefly polynomial representation central avoid divide supremum finance normal base side direct give shall explore marginal couple simplify computation base tail elegant q translate evident q characterize base solve result boundary get correct clearly choose trivial make differential right initial condition method parameter identity model return asymptotic asymptotic match exponential integral give easily first identical remain straightforward ode step singular otherwise neighbourhood origin elsewhere risk depend tailed asset return allow traditional distribution ode transform couple tail complicate handle essentially quantile interest distribution translation solve relevant develop copula hypercube course rely explicitly characteristic elsewhere concern good quantile implementation formulae approach speed develop offer precision double gpu preserving enough monte carlo acknowledgment knowledge transfer thank theory gpu produce windows architecture members numerical group system grateful allow financial modelling understand helpful manner noise induce distribution monte form monte carlo may traditional fisher expansion distributional assess free regard student variance change employ allow place single rational wide range branching statement avoid quantile offer environment divergence comparison old argue mode fast precision offer quantile yet monte library function keyword carlo gamma finance mechanic quantile inverse probit distribution solution make sample distribution characterize density uniform base transform marginal generator effort leverage manner principle associate form see direct way develop form composite mapping transformation way skew base control introduction traditional gram simplify brief review insight quantile expansion already series explore normal student case explicitly become object offer performance environment branching algorithm approach costly branching avoid environment paper computer mathematic approximation learn reveal type norm construction term precision function differential characterize one probability another give focus transformation student traditional expansion brief introduction might part quantile distribution idea quantile precision implement change argue offer characteristic section make double computation introduce generate two double fast gpu preserve full range enough ode rule rational pearson allow analytical order quantile suppose algebra quantile map ode ode arrive equation two rather suggestion eq ode arrive ode eq positive line ode eq brevity encode exact composition cdf distribution relationship information term know expansion fisher consider notable hill consider differential interesting illustrate know asymptotic series explore develop explore purely student case freedom integer ode look solution behaviour rather different always purely matter large value far go wrong tail gaussian centre wish apply ode derivative centre treat solution student indicate whether convergent algebra tail assume ode change cdf determine determine simple property tail behaviour cdf deduce step calculation ode ordinary quantile find literature reasonably student degree freedom cf transform expand power incomplete every matter observe series constitute summation coefficient series assume valid domain turn assess precisely normal student cdf formula student beta simple form available page interesting daily shall turn central within treat tail tail goal excellent course efficiency arise numerical make issue transform student transform form useful explore elsewhere consider building quantile live distribution quantile distribution know probit interesting see reference general modelling precision rigorously achieve intermediate distribution gpu four explore issue give quantile mechanic diverse think appropriate consider difference precision seek minimize seek figure mind goal typically rational abundance second actual answer significantly detail mathematically model newton quantile relative matter error accept normal quantile option distribution interest side exponential degree chi square give student freedom pdf cdf quantile function quantile chi square random quantile also look quantile formulae side essentially base transform gamma normal tail double tail computer impact mathematical try cpu normally sophisticated management branching extent branch efficiently branch divergence group execute branch execute branch problematic evaluation evaluation change tail algorithm apply lower value effort essentially region final precision vote whole architecture multiply operation hardware create performance support cubic support polynomial rational device reduce factorization speed reduce polynomial specific trick degree exercise loss several iterate rational evaluated architecture quadratic comment cost work polynomial various mode expensive main quantile helpful inefficient note reciprocal root suitable scale filtering explore precision exist double consider modification iterate construction implementation employ arithmetic way essential precision region outside interval precision majority theoretical relative rather supremum norm region plot close machine growth concern early design precision precise room optimize speed goal approach exponential sample exponential sample detailed mapping region odd symmetry cdf solve however neighbourhood coefficient far enough different need retain equation point statement computer branch rational positive quantile region quantile iterate break computation thing break sensible break split slowly vary central fig much aim rational pick fig show character function precision relative rational target explore arithmetic quantile tail create small neighbourhood near employ desire fig polynomial respectively nest form produce v denominator completeness normal sample scale simplify unity evaluate rational pair employ entire range appendix next precision realize formula illustrate fact main tail fig precise tail carlo obtain request reality pass c see computation reveal scheme realize precision reality utilize
heavy detailed collection tuple consist string eq large class vc call inequality relate vc dimension maximal event frequency respect empirical distribution dirac measure inequality measure tuple q expectation refine empirical well come expense large let measurable polynomial vc concern universal source universal parameter euclidean nonempty interior reliably reconstruct regularity implicit depend condition approximate invoke exploit proof back bernstein yu laws stationary nonparametric strong finite move integer constant show absolutely continuous root polynomial circle complex plane decay open ball radius center weak source indeed p function lie open around attain evaluate taylor expansion regularity fisher normalize entropy recover easy check fact form satisfied independently density idea class distinct note consist q pass hypothesis draw independently classifier nz yield classification q relation vc cardinality sample imply finite hypothesis test allow weakly list regularity must result source every effective notation description encoder decoder active variational one say code finite memory dimensional compression finite memory code additional allow decoder identify asymptotically infimum immediate improve lagrangian code code fx expectation l c n shannon gray universal also sense terminology minimax universal behind notation suffice universal variable exist code prove throughout code operate decoder access bit later exist achieve come lagrangian optimum encode encoder convention infimum empty encoder look database find first encoder vector string string iii receive determine happen ball around estimate slightly furthermore almost lagrangian optimum comprise si encoder refer encoder parameter decoder decoder stage define code encoder n nc n assess gp contribution x random expectation term optimum motivate section decoder fidelity inequality assume unless use exponent divide block parameter although act use distribute marginal denote copy distance increasingly mix equality yu shall heavily hide database codebook construct absolutely everywhere stage codebook infinite decoder order store difficulty generate encoding generator encoder entry suffice describe define th density call key md estimator regardless construct encoder decoder pair parameter encoder look codebook find encoder hold every codebook performance bind need expectation zhang x everywhere almost eventually surely surely respect source codebook geometric parameter q x lemma almost codebook asymptotic bind n event triangle write nf n follow parameter follow sufficiently sphere imply accord invoke sec imply lemma side take expectation q n codebook realization codebook turn code boundedness distortion invoke finite codebook bit marginal together condition furthermore n dp boundedness lagrangian nx nx lagrangian expectation term due examine approximate expectation nz nz nn term encoder eventually expectation estimate obtain put everything together almost codebook np wish surely cf recalling imply follow inequality choose condition borel process class source remain order entropy source np algebra fix around degree polynomial six vc therefore autoregressive source autoregressive filter exist filter coefficient unit set root lie outside unit absolutely invoke process geometrically mix sufficiently condition asymptotic fisher information recall discussion conclude condition verify set form x r entry real variable bivariate process chain conditionally markov observable interest reference therein finite homogeneous chain sequence matrix ia ergodic irreducible exist unique initialize sufficiently far away past side alphabet specify density lebesgue channel source fix channel fix parametric though proceed verify meet ij n exponentially mix measurable fs random uniformly independently bivariate invariant mixing bound mix well condition condition examine fisher transition density invoke finally write satisfied mix universal scheme joint compression identification lagrangian redundancy variational estimate converging block quantify marginal generalize previous outline research paper priori would interest parameter hierarchical could technique structural g chapter adaptively distance play especially gray require select code base scheme toward compression would devise objective issue optimality interest say conceptually indicate link source exploit present universal coding therein treat statistical problem distribution close spirit minimum source complementary perspective code detail lagrange optimal exposition modification elsewhere alphabet alphabet induce measure marginal define np p n see gray eq infimum side eq set finite concentrate countable ci satisfies minimum expected see probability eq intuitive lagrange multipli encode point deterministic lagrangian block variable operate distribution py q np I c associate q np nx n x c prove give lagrangian close np q np np dp use triangle show distortion lagrange positive rate let encoder encoder pc p zero encoder decoder pc pc author anonymous useful suggestion joint rate compression performance source source mix smoothness universal joint compression lagrangian infinity source several parametric source minimum universal quantization source code complementary objective capture provide optimal precisely within class coding show modeling accomplish jointly regular alphabet source maximum likelihood stationary ergodic alphabet source amount construct address statistical modeling universal coding long knowledge statistic certainly helpful design apart hamming source error measure extract reliable rate distortion code distribution realization rate distortion emphasis compression instance chapter control observation controller modify discrete govern finite controller digital capacity bit
loss generality normal degenerate stand kernel hoeffding statistic symmetric kernel result h also eq order z z independent circular symmetry length arc circle point asymptotic variance z normal volume volume vertex spherical derivation variance correlation matrix formula symmetry derivation general geometric reasoning use correlation q note x expression spherical sake brevity omit reader david correspond concern omit integral express one though variable appendix label e nh h paragraph close strictly enough introduce find z inequality vanish tend well fact rs theorem v factor l nn h w w w w w complement event last expression suppose arbitrary adopt mean mean string alternate suppose r alternatively frequent throughout special rule refined rule along monotonicity et variant wherein use express pairwise justify proof six statement rt accordingly ensure algebraic interval algorithmic monotonicity choice lemma one phase second rule monotonicity throughout shall unless treat rather calculation perform software detail output make request rt ts format label stand phase dedicated lemma rt third rt prove increase perhaps even quadratic true yet proof eventually adapt r polynomial degree arbitrary upon recall visual q rational implication root shall simply similarly denote simply root concern us root respective rt argument accordance imply note general rule existence imply note finally hence continuity notation repeat rule next root show sign next whereas root imply next similarly imply lastly ts appendix ts adopt imply refined rule special case rule next root refine imply r g l establish program evaluate refine also rule similarly existence rule continuous root two root imply imply imply existence root rule note continuity lastly rule continuity rule remark precede hence recall reasoning proof stand regardless interpretation ts appendix follow adopt notation repeat imply case rule imply refine general existence imply imply yield z z f f g r g f g r r see lemma accordance rule single imply imply distinct root similarly root show general rule four interval find existence well via rule hence continuity imply rule continuity limit rs argument lemma repeat rule limit existence imply show rule rule imply exist special case show continuity find imply rule lastly imply place place begin six corollary replacing recall even desire proof sl r sign monotonicity b refine rule thm thm thm thm pearson common correlation setting nonparametric show normal efficiency monotonically coefficient increase monotonicity ta r another immediate corollary proof monotonicity pattern pearson correlation bivariate reduce correlation asymptotic could form show strictly stay additionally quadratic shown corollary theorem statistical certain sequence condition properly parameter neighborhood continuously differentiable express asymptotic ess bound mean increase increase
threshold set computational frequentist sparse network address yet give log likelihood marginal know density tractable value involve p likelihood variational maximize maximization minimization kl divergence approximate analytically believe kl depend nevertheless criterion rely algorithm bind criterion call estimate vertex practice result carry throughout experiment indeed contrary algorithm analyze core recall sbm criterion detection data sbm retrieve concentrate criterion experiment generative otherwise complex structure model connectivity matrix probability proportion class apply network various note l q simulated different initialization estimation henceforth keep tendency context behave consistently true whereas context graph make community class methodology represent chemical compound part vice section prior beta il repeat initialization indeed initialization class axis axis correspond learn frequentist dot representation bayesian maximum posteriori frequentist eight six probability connectivity compound responsible clique single clique sub clique class since structure retrieve topology class merge block sbm full bayesian informative approximate posterior non sbm focus density relevant number illustrate capacity sbm retrieve network look computation seem sbm likelihood accept acquire cluster vertex connection paper sbm sbm work variational expectation criterion estimate component criterion sbm study tend case propose criterion variational em random variational em bayes em date date field biology social science edge object internet lot attention develop group model mix vertex mostly cluster depend propose asymptotic approach map quickly position cluster conversely position simultaneously well consume r particularly suitable bipartite numerous encode group inter network author cite paper detail description mixing look efficient parameter belong otherwise introduce non paper focus sbm originally develop science network vertex belong describe intra inter proportion make structure account sbm characterize network locally dense extent many exist many sbm difficulty model variable due propose approach introduce posterior software package give posteriori handle vertex em sbm strategy lack bayesian criterion aic intractable tackle use criterion integrate easily compute sbm originally separate case fail interesting structure network sbm class deal emphasize sbm call integrate view detailed overview sbm tractable asymptotic obtain informative model implement work web http undirecte relation symmetric sbm vertex vertex vector x iid edge suppose sbm describe general allow concentrate without loop q z graph account loop sbm block membership stochastic block capture partial class sbm single identifiability identifiable except two sbm bayesian consider otherwise see constrain element sbm frequentist informative model distribution mix informative jeffreys distribution dimensional simplex fix informative jeffreys direct long variable hyperparameter distribution algorithm sbm full rely section asymptotic approximation term quickly become tackle well em stage estimation describe value old old
natural test vector generate sampling overlap face vector training search solver experiment dictionary weak order respectively threshold negative classifier order classifier w ix opt h ix jointly expect order value frame ix synthetic datum display comprehensive test minor modification find drop round eqn employ ensure refer investigation scaling suggest reduction classifier chose determine validation stop minimal result dimensional quantum instance variable result obtain coincide determined training overlap represent fit infer minimum quantum simulator initial hamiltonian quantum hamiltonian gap ground hamiltonian notational seminal need gap collection typical note extract minimum special resort consist unfortunately number variable currently attempt simulator minimum estimating derivative state relate gap ds correspond interested assume derivative polynomial exponential whether synthetic set encourage quantum wave quadratic unconstrained loss version traditionally learn preliminary usually attractive application dataset depict eqn employ adaboost outer square dimension large objective action unfortunately expense minimize small minimize training classify replace value purpose ever becoming need hardness objective could search conduct analysis resource lead small classify bit versa classify correctly objective contain need variable number weak classifier parameter process outside spirit thus far formulation quadratic amount variable live stay wave processor format elegant form sum introduce encourage weak b look like special context dependent exhibit execution importantly example incorporate principle allow incorporate priori principle example train detector impose image nearby symmetry continuity weight optimize formally thank zhang discussion boost david quantum initial g wave system com discrete classifier thresholded sum classifier motivation cast format amenable yield superior heuristic solver solver advantage communication training candidate weak weak learner use classifier exceed handle effectively piecewise optimization numerical loss add superior version boost carlo quantum detector strong choosing simultaneously set choose regularization complex regularization encourage strong build weak classifier maintain training accomplish solve follow number boost optimization bit depth represent small deal binary consist blue training color datum light color parallel hyperplane classifier situation employ four negative area adaboost subsequent round contain negative become severe greedy configuration adaboost see handle adaboost exponential employ quadratic show lead bind classifier vc dimension classifier bind compact achieve weak come look weak e merely demand reduction switching associate incorrectly eliminate expense vc lower equal adaboost classifier use rich need illustrate practice weak regime determine regularization strength perform trade increase gain baseline system namely employ adaboost rather exponential essential function employ perspective quadratic increase I possible contain global large classifier fulfil weak handle global look art solver wave train often sift moreover dictionary learner dependent mean cardinality typical classifier thousand rather strong solver problem hope reasonable quality inner loop algorithm handle learner need construct classifier small weighted opt hx unweighted yield small inner
share issue far later time contrast situation happen co next add statistic say likely formally circle support relationship dot statistic emphasis plug formulation still trade compete complementary retain flexibility model plug actor phenomenon would ability box acknowledgment participant comment fu cs evolution extension graph model well evolve network communication field social population communication relationship actor structure behavior global increase demand tool subject modeling depth network single flexible include classic extension model cluster model role relevance capture signature connectivity pattern statistic social probability intractable represent model graphical regard clique potential representation vary possibly include type asymmetric actor relation actor actor evolution sequential observation wish network community trend evolution rise time model dynamic example event represent actor link local neighborhood dynamic view exploration model beyond work quality series previously though recently explore issue flexibility parametrization model specify compare elegant indicator section refer capable evolution flexibility general furthermore formalism exist past decade readily temporal maximum likelihood fit capture signature dynamic hypothesis application begin way simplify evolve representation single make put property give generalize admit representation specify k understand clique specify accomplish simplicity presentation detail special form function possess framework weight adjacency matrix govern tie tendency control tendency link result tendency simple however actor possibility deal type task model network sensible normalize often mcmc study modification expectation network gibbs conditional unconstrained optimization newton direction relate family tailor model initialize convergence sample I I affect accurate iteration need however early stage sufficiently precision resource remain general procedure given computationally perform newton examine loss pmf z initialize range initialize density represent average convergence distance exact newton rather sample much approximation perform return almost identical indistinguishable concern express degeneracy issue arise model explore space mass complete several expect degenerate distribution intuitively degenerate slight variation become degenerate namely degeneracy generating degeneracy distribution converge aforementioned require fail question whether issue also affect temporal extension factor edge problem initial phenomenon example entry minimize empty maximize empty entropy quantity thus reasonably get intuitive entry entry plot example option fix yield plot plot bernoulli briefly graph calculate class analytically show class accord since edge exchangeable purely situation equivalence purely distinct significant calculation entropy computationally tractable small magnitude generalize discussion satisfy eq note number eq entropy imply entropy upper entry take factor expect marginal aforementioned hypothesis see generality pay broad scientific hypothesis hypothesis write significance represent serve plug potential compute united actor proposal resolution serve proposal possibly record create slide consecutive window direct relation window toward evenly spaced slide window proposal proposal window proposal series first test inherently formalize previously membership political otherwise potential represent alternative hypothesis write ratio compute estimator ratio mle value hypothesis mle alternative composite value bit seem tractable analytic unconstrained optimization population sequence nan mle calculate calculate empirical likelihood genetic decrease brevity introduction approximate form likelihood derivative analytically particular likelihood optimization directly without however might likelihood possibility likelihood divide value ratio dividing alternative even encode hypothesis well write specify alternative densely transition clique densely structure additionally actor allow random subset population know label evolve accurately infer actor label party modify party multinomial know party randomly leave posterior unknown infer assume label sequence fully observe since likelihood straightforward model infer party algorithm use observable unknown
select compatible assume compatible termination prescribe algorithm denote exist constant diameter confidence may constant either c usually hoeffde inequality bad rate region diameter ensure crucially depend construction regularity continuity approach assumption duration phase expectation regret obtain constant duration exploration terminate reach diameter maximal terminate balance terminate policy regret may instead bound terminate region rather assume actual formalize alternative construct exist constant policy bound decrease instance access interesting illustration different optimal long compute play role stochastically identical channel consider single channel bandwidth channel receive decide observe channel policy bring light define secondary user channel consists observe correspond policy represent straightforward policy infinity aggregate observation horizon confidence fully consists sense resp resp visit markov ensure confidence region regret algorithm assumption secondly half center large finally reward lipschitz exploration observe since channel lowest free positively positively long policy exactly compute secondary value negative knowing region similar verify rectangle whose include distance less illustrate run replication process horizon take equal empirical distribution may observe represent otherwise indeed actual policy exploration phase correspond inside exploration long policy capture exploration phase important exploration close resp really resp later central limit transition order hold observe transition difficult however long particularly corner indeed channel imply transition strongly positively decide reinforcement algorithm applicable channel balance monitoring phase pre specify furthermore guarantee single stochastically channel indeed exploration stochastically rapidly potential wireless communication concern able general stochastically channel adapt principle phase parameter possibility either region sum duration exploration violate f additional fact true denote use assumption minimize nc distance soon policy equal therefore c appendix prove event c c remark last hold nc nn conclude paris france fr consider task channel primary independent secondary statistical system problem cast markov decision aim exploitation requirement provide finite horizon regret performance channel stochastically identical channel cognitive focus effort make use portion band primary secondary cognitive user secondary user carefully identify resource primary access potential wireless previously channel secondary search channel availability evolve markovian long channel planning class partially decision assume primary traffic secondary user statistical traffic must secondary user select close reinforcement learning carry policy reach issue consider asymptotic rule exploration consider user simple source reinforcement none allocation contribution propose strategy adaptively exploration phase determine come form horizon sake clarity abstract parametric remark correspond transition availability channel rely parameter value planning problem case allocation detailed use stochastically identical channel consider article organize follow access detail stochastically consist channel channel primary channel either secondary channel primary sense channel secondary channel maximize transmission introduce channel channel channel resp channel additionally denote channel channel channel gain available unobserved channel reward depend observe may channel receive channel penalty channel receive channel exploit dimensional internal ti ti state enable secondary channel sense internal recursion denote last ki ti equation may channel interpret introduce planning bandit achievable number channel become important nevertheless recent near call reduced cost consist separate channel interestingly determine planning value explicit expression index sense learn act optimally learn high chance channel learn coincide cost question secondary apply well exploitation adaptively monitor function condition restrictive case mdps case like channel also q f unknown probability choose
modify denote run monte simulation fashion follow run simulation probability w p computed exact sec equal probability fact element threshold increase c one select maximum map model signal mutually mutually I train compute close thus step algorithm feed back appendix solution summarize discussion perfectly outside causal simulate camera static model acquisition orthonormal video denote norm mask contain randomly row entry take cs ls cs minimization table big store need solver include ii multiplication big like dft time solver use modification code interior sampling mask bit storage fast multiplication link image fig set table even size reconstruction measurement approximately result cs cs mod cs mod cs instant showing greatly work either tv former parallel modified question recursive noisy stability currently open dependence recursive time approximately anonymous contain pixel spatial magnitude nonzero minimize pixel subject designing homotopy cs sequentially measurement application contradiction nonzero full rank happen constraint since solution define iteration apply j follow apply c disjoint equation simplify absolutely convergent define two j r rt belong thus give summation u become dirac delta markov statement laplace estimate solution causal time normalize constant since I support institute ph university college electrical currently transaction interest current reconstruction sequential large tracking receive electrical engineering department china ph electrical research focus interest include theorem corollary definition edu part appear support nsf material purpose request projection although knowledge spatial support instant weak compressive error compare support important extension signal show noiseless measurement although part example mr discrete wavelet basis know reconstruct spatial support previous application dynamic camera video compression image refer energy transform number notice change sense cs conditions reconstruction cs residual ls problem know replace observation l residual greatly reconstruction signal require sparsity cs able reconstruction fewer noiseless measurement need cs reconstruction need modify cs relaxation exact reconstruction cs estimate weak cs support change slowly measurement true practice develop extension regularize modify reconstruction shorter appear work probabilistic include sequence use except reconstruction even demonstrate reconstruct difference approach reconstruct cs gradually become sparsity result achieve whenever cs actually point anonymous bp anonymous multiscale cs compression improve organize modify sec condition exact modify cs cs modify cs sequence sec sec count element norm contain belong notation complement w operation denote restrict isometry small cardinality orthogonality respectively notation vector part unknown thus denote denote denote size know denote property rip rip describe introduction unknown coefficient certain threshold tell intensity mostly index nonzero detail series instant occur case nonzero newly reconstruction current similarly case small may element word like empty know I threshold cs give result sec complete argument give disjoint reconstruct minimizer equivalent compare version true relaxation sufficient slightly strong next subsection whose unique condition cs simplify u k reconstruction consider reconstruct usually consider measurement give large ensure hand hold third term third term value small example differentiable multipli lagrange subgradient using replace satisfy subsection construct disjoint prove apply lemma iteratively construct satisfies condition theorem disjoint proof subsection outline prove theorem proof rs j fashion define argue satisfie entire appendix previous subsection motivated least minimizer need hold equal minimizer inequality less xy ax jj x full thus minimizer even thus set disjoint rhs rhs since ta definite denote matrix denote decrease take ks thus hold height eps nm nu small simulation exact modified cs sec compare cs cs express serve decide need meaningful cs corollary cs two obvious weak cs reconstruction either compare scalar e condition clearly alternatively probability binary reconstruction random h reconstruction random fig three different choice see allow measurement reconstruction cs maximum allow reason significant sufficient cs modify cs generate gaussian entry rip rip affect performance way follow time generate random generate uniformly element call output cs divide various value result cs cs hand cs work cs work reliably require say modify cs estimate b modify anonymous value reconstruction occur pursuit cs handle modify se c c e cs e cs
seed leverage implementation appendix metropolis sampling scheme seven scheme rr rr rr ac min ir ir median ir ir median ir rwm rwm likelihood leverage outlier allow outlier run small highest computed independent metropolis bridge cc particle pt also describe adaptive sampler processor furth table metropolis adaptive hasting parallelization rr rr rr rr ac min median ir ir median ir ir median ir rwm mn consider poisson gamma mcmc binomial model mean shape success marginal example poisson walk figure possible series priori poisson present monte replication seed simulation implementation walk metropolis seven hasting rr rr rr rate max median ir median ir ir ir rwm mn rwm mn show distribution summary result present mean standard particle binomial run use sampler eight processor nearly adaptive l rr rr min ir ir ir median ir ir rwm consider dynamic priori j poisson pattern model representation differ also explanatory multiplicative analysis capture change ht presents monte study seed adaptive walk metropolis seven hasting rr rr rr rr min median max median median ir median ir ir rwm c mn rwm mn eight use five show intervention consistent report intervention ht cc particle filter intervention intervention level intervention intervention run second iteration metropolis hasting run eight processor implementation summarize median ir ir rwm auxiliary particle rwm table contain trend ht cc cc research partially arc discovery dp datum resample sir fix notational suppose particle take filter density distribute filter dirac delta sample estimate denominator rao form efficient weight discrete univariate multinomial bootstrap sample associate method proceed time replace multinomial resample bootstrap strategy hold auxiliary density py z proof appendix paper walk proposal multivariate density covariance iteration laplace scalar sampler locally walk proposal multivariate adaptive simplifie refine take random tailed leave proposal adaptive parameter scheme stage throughout two term heavy third fourth tail preliminary run walk mean normal normal covariance begin component schedule depend ratio stage construct tail local mode vector explicitly bridge iterate q adaptive proposal positive reasonable py simulation alternative eq tailed importance ratio code matlab code write file use file resample step algorithm library cluster compute intel gb run processor give implementation simulation simulation processor particle iteration sampler metropolis hasting sampler iteration metropolis hasting draw eight processor update completion block standard particle detail sampling simulation processor particle filter metropolis hasting sampler perform particle hasting simulation sampler metropolis eight update occur particle eight process sampler corollary south edu se economics university south edu economics university ac uk feasible model adaptive hasting approximated filter base construct adaptive independent metropolis hasting proposal attractive parallel processor marginal obtain efficient bridge feasible exact state adaptive sampling evaluate analytically transition justified work likelihood uniform metropolis base time construct efficient adaptive proposal evaluate q likelihood compute mcmc parameter integrate kalman general auxiliary sample review carlo integral computationally standard particle approximate become particle tends infinity standard particle particle likelihood particle filter suppose wish hasting give initial proposal otherwise depend regularity detail iterate regularity iterate draw application available auxiliary particle provide unbiased particle filter conditional iterate use auxiliary augment u adaptive hasting scheme follow space suppose ii proposal metropolis scheme converge sense set integrable respect tail binary binomial similar ensure outline theorem theorem particle parallel processor available apply estimate processor filter use single processor make second apply hasting sampling
position generalize test memory link small integer total sum see define case decrease tm relie expression backward refer normalization algorithm correspond order interest introduce subsequent introduce two order either eqs iterative functions eqs function right normalization since step subsequent z boundary approximation constant eqs benefit tm another couple wish tm tm mc random sampling compute know technique set sample indeed happen become perform initial purpose approximation marginal e chain value tm gaussian random possible independently draw derivation iterate normalization constant expression replace return final tm introduces factorize introduce expression drawback variable connect arbitrary successive relationship probability interaction close neighbor consider variance expression couple write appendix behavior approximate impossible tm approximation average number perform start study channel discuss channel reduce eq probability maximized distribution occur expression event general finally expression channel consider write parameter notation error truncate x tt hand define gaussian impractical numerically take I use previous write expectation eq expression separate accordingly consider fact function expression distribution study subsection behavior section tm discard vary eight figure couple fig fig value memory thus tm vary dependent couple average error vary h bottom one top mc tm mc tm tm mc use tm mc mc tm mc expect trend tm notably tm bit decode perform tm par specifically tm g perform well tm tm mc also tm tm tm mc similarly outperform tm though link explain mc use poorly tm mc fig per sample different exactly combination tm tm case four combination tm mc tm tm tm function two tm mc tm combination algorithm relatively though g outperform grow study total length p f tm mc probability correctly decode total length big rapidly decrease explain graph fig long tm mc sample tm show thing tm mc slightly tm tm latter behave variation fig never vanish use notable instead error various incorporation inspire test sequence system obtain template range test genome gs machine separate sequencing model additional approximation estimate gs introduce deviation base compose alphabet dna read repetition sequence see consideration thing non incorporation rate positive read length datum estimate value small slight issue communication present mc plot correctly would extract tm mc gauss within total gs capable correctly read template basis template gs least indeed value since encounter decode tm mc concern indeed correctly chain length also decode sequence great error decode chain tm mc emphasize direct algorithm matrix tm tm gauss tm final tm g gauss perform memory small tm memory large besides yet test variation need quite future preliminary assume x gaussian variance value eq normalization keep recover iteration keep close reconstruct one marginal marginal need latter procedure calculate compute expression write need situation decompose conditional probability expectation eqs follow necessary distribution overcome regarded distance minimize wish constraint multiplier furthermore eqs constraint hand q right equal prevent appearance static choose place wish thank proposition conjecture infer markov physics language dimensional external accomplish become intractable several field transfer realistic model dna markov model application range speech sequence whereby state chain conditionally formulae fundamental algorithmic hmm infer thought state symbol reduce physics boltzmann dimensional temperature sequence act analogy marginal sequence state precisely multiply time become memory underlie multiple state standard hmm augmentation exponential lead severe length propose basic intuition length get transfer mean concrete infer dna carry trace position scale thus plain impractical dna motivate complexity describe analytical result collect positive integer e observe non recursive depend observation memory also side effectively sequence describe include except exactly noisy regard state precede high model posteriori boundary condition probability write construct use symbol decode map decode would computation entail summation grow section four limitation give subsection take success construct integer bernoulli correspond generate directly use decay rapidly still distribution distribution c enable decoding case otherwise derivation equal decide subscript original synthesis review concrete history one sequence dna chemical test become standard low efficiency sequencing
requirement practical believe allow assume truth concern corollary remark selection different concept relate eigenvalue slightly allow hence coherence restrict isometry keyword phrase compatibility lasso eigenvalue isometry examine relation oracle among selector property relation fairly study noiseless space dictionary consider fixed inequality oracle eigenvalue present condition tailor difference overview compare literature explicit display enable indicate implication section rigorously deal compatibility condition many compatibility approximation compatibility compatibility improvement study additional implication depend discuss invoke gram product form play small positive singular gram matrix compatibility name compatible definition compatibility condition restrict compatibility successively strong version size invoke simplicity define set q necessarily eigenvalue complement restrict eigenvalue call eigenvalue introduce restrict adaptive restrict restrict depend gram section compatibility condition ss l mainly situation study definition orthogonality orthogonality constant moreover isometry isometry eigenvalue eigenvalue restrict isometry weak isometry restrict isometry rip isometry rip modify coefficient condition condition strong definition condition meet spirit tight coherence compatibility imply derive later compatibility go lemma assertion lasso note inequality hold inequality implication lemma assertion statistical case condition essentially weak oracle noiseless isometry uniformity rip rip clear restrict isometry constant demand rip oracle inequality selector show weak isometry property restrict restrict n restrict condition compatibility easy calibrate prove compatibility might pay oracle regression end give put combining give essentially summarize sufficient compatibility somewhat elementary projection moreover adaptive word regression restrict oracle refer condition restrict definition hold imply conjugate quantity eq replace might eq take apply lemma give eq lemma vector use orthogonality constant q anti define moreover eq hence subset eq eq small positive since q kkt suppose weak know arbitrary must arbitrary weak condition equivalently restrict corollary q rip enough constant condition picture programming minimizer programming condition exact say restrictive compatibility prediction also multiply eq condition q moreover span uniform eq mu verification lasso type therefore look rather trivial non restrict population restrictive compatibility gram relevance replace design variable often population even design compatibility supremum generally may eigenvalue eigenvalue corollary r mu q result show find well eigenvalue consider column column denote population row one eigenvalue extend tail empirical section implication eigenvalue matrix vector restrict hold condition meet hence toeplitz sense spectral unique toeplitz small bind block drop minimal satisfy eigenvalue restrictive small matrix behave example compatibility calculate first satisfying small easily moreover uniform hold compatibility follow get behave noisy noisy design abuse notation define behave case suppose distribute noisy take define hence thus eq may case compatibility involve compatibility eigenvalue situation kkt noisy matrix repetition define generalization anti space span kkt
learn expression various agglomerative regularization hierarchical regularizer induce structured effect bias covariate strength coefficient regularization lasso snps association strength treat select separately lasso penalty union support capture penalty let define overlap response covariate snps properly calibrate consistent group organization although previously use penalty take advantage structural response due weight different group arbitrarily lead contrast systematic scheme apply balanced penalization coefficient lasso lasso weight adopt loss structured induce group computational typical time number node advantage graph fusion introduce fusion penalty fuse weak association false could regularization selection relate response different select response agglomerative widely preprocesse classification subtree height regression extract cluster gene form average regularization incorporate present datum detect signal successfully remainder paper organize brief discussion previous estimation section introduce experimental collect trait snp minor response center column intercept column th number covariate effective identify element matrix tuning control provide mechanism enforce literature group adopt multivariate nonzero share community estimate coefficient eq denote coefficient response covariate norm encourage coefficient encourage th covariate would vary different response realistic expression level snps snps gene pathway snps relax add penalty regression within zero share limitation regularize regression grouping structure response multiple considerably add advantage correlation hierarchical structured sparsity norm microarray measure thousand gene correlate sample imply share common pathway variation snps module relate act module derive run agglomerative expression natural extension incorporate output hierarchical cluster genetic variation influence module build lasso leverage statistical expression gene primarily mapping generally applicable multivariate dependency object speech valuable enhance relationship vertex illustrate node response subtree root internal bottom tight response root weak subtree knowledge resource gene database hierarchical agglomerative cluster associated subtree root represent correlate normalize tree expand overlap group overlap tree follow group whose member response variable subtree root regression tree bound weight near leaf common covariate amount penalization group internal relevant separately response child root tree term equation net slight difference elastic weighted nonzero response covariate heavily equivalent multi response share response parent penalty penalty tree single node leaf root node operation norm penalty norm encourage joint balance norm lasso penalty reduce penalty contour surface penalty show pt cc penalty ensure overall amount group article belong multiple different internal appear always weight scheme guarantee variation lasso overlap group previously overlap arbitrarily result unbalanced show weighting scheme inconsistent construct tree response tree recursively root leaf th covariate w w g introduce structure branch forest tree tree leaf individual main nonsmooth coordinate algorithm apply nonsmooth group overlap penalty prevent closed optimization tree formulate cone interior approach gene general sparsity induce penalty function share handle group fuse special overlap method introduce approximation nonsmooth smooth fista accelerate objective accelerate descent rewrite split two correspond internal leaf overlap weighted weight response contain overlap auxiliary covariate j v j j c matrix row column note tree penalty dual optimize overcome challenge smoothness nonsmooth g ty smooth gradient continuous v penalty fista adopt lasso give proximal penalty obtain close thresholding w give role determine compute back tracking iteration store tree lasso complexity quadratic cubic demonstrate compare regression assume response parameter model range error validation give low prediction evaluate criterion sensitivity specificity specificity sensitivity rate square base scenario generate correspond predefine group response share covariate cluster figure three avoid expression study divide set b hierarchical response avoid clutter row response column covariate method lasso regularize multi nonzero lasso strength positive coefficient across combine get response vertical hierarchical visually clear positive response true relevant covariate use average simulated datum systematically performance generate receiver operate regression coefficient average nonzero figure outperform regularize regression especially knowledge correlation prediction error average simulated figure show figure include error low error addition true hierarchical agglomerative along internal node since obtain manner represent discard root thresholded tree even available benefit account response error experiment range value cluster gene snps expression miss know module hierarchical module correlate agglomerative internal goal variation gene response incorporate able activity co express pt ptc ptc correlation agglomerative cluster coefficient panel accord agglomerative estimate row represent gene choose validation lasso extremely reveal snp weak unable detect signal strength gene correlate expression regularize task across tend vertical expression perform pt go category nonzero coefficient group module go interested identify snps gene module encourage hierarchical reveal element go gene nonzero snp absolute value estimate figure biological biological snps go regardless threshold select association estimate generally find lack figure finding provide snps influence gene focus affect gene biological molecular bp biological process molecular cc bp mf activity bp mf activity bp mf mf activity activity pt family stimulus response bp translation cc mf activity mechanism cc generation cc mf mf list group snp lasso strength category genomic location
assume follow symmetric select secondary shall decide channel bid bid single channel channel cognitive cognitive utility finally bid stay allocation channel allocation equal bid winner pay presentation assume mechanism write observe current rate implement observe shot entry well typically strategy history far participant knowledge player assess accumulate observed cost payment access history channel accumulate utility accumulate cost action utility intuitively beneficial stay incur accumulation thus avoid optimize even centralize manner space individually bid straightforwardly dominate strategy bid price consist game differ channel detail subsequent single drop stay reduce first thresholding identically cdf cdf state low evaluation difficult accumulate reward individually associate payment end simple approximate side maintain private update tc obtain associate payment choose stay payment winner payment amount payment payment amount estimate move old summarize output ta bm I som mt mt mt p mt access problem regret explore exhibit set action proportional play action two playing denote payoff bid view average channel k p channel prove access action decide though control channel I sufficiently investigate randomly place fix away center set transmission bandwidth assume length hz propose compare htb understand propose figure snapshot change monitoring converge quickly furthermore always bid decide bid past active pay monitoring decrease effect simulation user htb monitoring cost well see entry increase utility decrease gap selective likely high high show adopt resource allocation channel repeatedly entry htb htb next average bid bid generally agree high bid average bid also stay treat bid slot cost win htb vary respectively decrease dominate performance utility gain channel experiment performance convergence user ga regret users channel ga channel instead channel ga upper bind practical bad bid perform problem channel adopt access history always show greedy plan convergence may inaccurate acknowledgment air force office scientific foundation grant access cognitive model subject entry cost cost incur primary activity secondary user successful channel regard activity game propose outcome balance recent study despite actual spectrum period cr exploit spectral secondary wireless device whenever challenge research cognitive spectral efficiency tradeoff previous aspect spectrum access sense term throughput share instantaneous channel study sense propose improve detect aspect spectrum also receive attention allow maximize access account penalty therein sense cr network affect author access find correlate propose change arrival arise desirable access availability decrease transmission channel transmission spectrum explore transmission across heterogeneous common control channel information operator access allocate area intelligence outcome contribution paper spectrum access problem cognitive single rest paper organize terminology mechanism give cognitive channel reasonably estimate channel information primary
thus homology coefficient homology degree later topology form differential analogue adjoint definite operator theorem space identify kernel harmonic compute definite harmonic almost orthogonal class former representative space alternate due preserve proposition formula alternate define relate definition inclusion induce variant simplify relevant fact hilbert adjoint furthermore condition q former unique representative latter condition prove eq orthogonal hold imply v f pf pg pf w hilbert adjoint close include completeness q open suffice establish imply act trivially subspace map complement complement indeed leave zero leave side argument difficulty close sufficient first suppose bound banach finite without mapping close already finish lemma homology trivial check gx x proof guarantee k follow theorem theorem alternate trivial geometry borel measure non also throughout goal section alternate thing check bound proposition imply borel application theorem prove chain subspace constant assume function immediately alternate suffice think regularity pde continuous unique since bound identical proposition easily section chain construct whole trivial group theory operator throughout product observe closed diameter appear remark sense probability take alternate theory space respect f nonempty essentially note depend subsequent analysis q want emphasize dependence harmonic function eq component h furthermore equivalence inclusion consider map large rich condition metric hold immediate course formula harmonic minimize f ff see define section equal compact scale harmonic slice b xx x maximum component harmonic continuity fact slice locally harmonic form assumption unable form poisson regularity give show need regularity solution let lebesgue atom real number limit outside poisson regularity borel function measurable right suffice function continuous characteristic function measurable dominate continuity form partly theory put back symmetric extension define compact hilbert schmidt adjoint difficult section reproduce kernel poisson normalization kernel operator correspond operator eigenvalue complete eigenfunction reproduce next paragraph q reproduce finite limit theorem harmonic compact connect orient riemannian induce volume form ball convex geodesic lie ball point simplex face totally geodesic dimensional face geodesic geodesic segment construction vertex dimensional complex consider important visible determine tend homology orientation orientation orientation sign volume orient geodesic degeneracy open volume simplex vary continuously harmonic harmonic generate generate top constant curvature orient curvature harmonic scaling uniqueness degenerate x therefore orientation orientation geodesic segment curvature isometry onto onto easy define tt side establish let generality orientation orientation face orientation orientation therefore equal since case face depend face simplify opposite orientation right side equal geodesic term case orient surface totally generally geodesic triangle define case show orient manifold tuple iteratively combination assign orient evaluate generator space development relate homology theory metric metric section complex x complex necessarily complex open empty course complex chain check dim trivial intermediate complex rational coordinate intermediate case restriction b dx limit h xx construction see thus defined modification definition scale equip extent via continuous theory question extent map inclusion induce compact metric space equip borel topological reason see function space denote denote harmonic denote closed space analogous inner direct decomposition analytical analogous regularity equation regularity poisson answer problem continuous question show recall decomposition regularity show complete namely regularity harmonic function harmonic imply inclusion function induce regularity every degree representative suffice establish imply different class riemannian coefficient introduce similar sufficiently scale complex give detailed argument rather neighborhood diagonal cause difficulty compact metric alternate value discuss precede function manifold space ph h dimensionality involve fact collect rectangular space one check thus couple space operator respectively map linear map contraction follow fact algebra prove dimensionality leave complex augment chain map induce chain row augment homology need row de fact column lemma linear eq column follow complex also subspace corollary banach topological topology induce borel corresponding follow act horizontal explicitly tuple tuple chain horizontal map vertical bottom assign vertical map riemannian everywhere complex respectively cover complex complex condition borel open smooth riemannian induce natural inclusion map function arbitrary value suffice indeed vanish cover ic th measurable hc h th prove partition unity choose continuous riemannian manifold contraction replace ff restriction bottom augment argument row induce finite manifold ball complex take column trivially fix check define define contraction fix contraction slice x x empty finitely open borel measure hypothesis riemannian dimensional ball hold let covering since correspond show chain contraction column borel positive playing term chain contraction replace take give notion b rx rp x r proving assertion suppose imply x x iw u kb radius hypothesis open ball radius close closure let qx b b iw p thus continuity riemannian manifold riemannian metric strongly geodesic geometry hold hold proposition course second intersection radius proposition claim strongly minimize geodesic geodesic equal assume without geodesic sphere curvature imply sphere inside convexity claim single interior interior w hold denote exist establish single take subsequence take subsequence easy contradiction interior subsequence denote eq another subsequence large interior point interior contradiction proof formula arc point curvature geodesic comparison check sphere convex imply proposition convenience metric space metric converge respectively disjoint require require cluster diameter banach space canonical diagonal contain belong add tuple belong intersection center borel zero consider ignore tuple point characteristic scale look banach projection two time kernel complex vanishing sequence kf kf long etc obtain come six permutation degenerate simplex contribute without determined vanishe vanish value addition constant without constant observe function coincide way closure close ball coincide volume except ball ball point short inside metric write suitable similarly pair point diagonal neighborhood quite explicitly complex namely neighborhood projection map neighborhood compatible projection onto reverse map sided inverse function tuple cauchy bound projection consequently induce zero norm zero hausdorff inspire space infinite dimensional homology homology separable metric scale associate x dx point homology homology complex exploration compact metric homology miss infinite homology homology promise result homology derive fail attempt homology difficulty perturb equality homology high homology homology cycle equivalent high homology group scale homology length equal exist simplex substitute equivalence rely infinite dimensional homology group countable easy embed show sensitivity infinite consideration necessary homology dimensional homology infinite discussion diagram rgb diagram b degenerate cycle act homology last highlight must r shown suppose include eliminate boundary term new eliminate impossible return eliminate homology suppose need eliminate elimination either case boundary generator homology homology homology tailor nature homology change degenerate homology group homology reduce decrease enhanced version scale homology homology first case let homology satisfy value original exist cycle prove close
know unify confirm differentiable invertible matrix nonnegative nonnegative definite criterion hand side restriction concavity usually concave criterion illustration monotonicity assume iteration q word well denominator criterion simplicity formulae hold eq monotonic read consider monotonic read take ii monotonic criterion note point suggest desirable recommendation another henceforth vector zero denote identity design intercept nevertheless measure concerned coincide considerable numerical accumulate argument p use conjecture consequence g transform problem minimize jointly van ar yu although mathematic intuitively interpretation lead monotonicity monotonic establish regression key formulate specifically denote square obvious explain variance unbiased rank assumption square linear estimator sense definite matrix minimize estimator reduce upon immediately subproblem ii minimize formulate joint minimize equality problem upon natural minimize part closed form point maxima boundary therefore optimal present yu yu comment ensure highlight basic start assign weight point assigning tend valuable say check see satisfied coordinate nonzero equality check technical concern simplify positive maintain limit condition mention satisfied fail often condition illustrate converge remark iteration x z z tw rare practical think reason wide em monotonically although slow upon g monotonicity hold mathematical contribution way space possibility construct desirable monotonic logistic compute design prior guess space criterion weight design iteration locally verify simple serve monotonicity apply design space concavity long monotonic evident plot theorem monotonicity calculation monotonic evidence insight resolve alternative seek expected notation moment would section plan extension future work like david van grateful comment multiplicative introduce computing design conjecture confirm illustration approximate theory general focus finite space probability entry proportional closure nonnegative eq extend case allow use although positive typical criterion th criterion often combination naturally may special optimality seek variance unbiased coordinate blue interpretation argument apply guess chernoff optimality ignore prior dependence li henceforth assume dependence extension design early wu iterate
reconstruct motivated deduce small number application estimation wireless communication usually extra facilitate instance structural prescribed toeplitz etc complete completion maximize realization matter efficiently give arise context localization complete partially euclidean motion fundamental computer vision reconstruct analyze structure find matrix presence failure account difficulty complete customer complicate track customer preference system year maintain preference column correspond specify preference record entry consider preference item completion maximize force information specify reconstruction ill pose may criterion shall theoretic aspect rank wish reconstruct simplicity suppose initially available allow reconstruct natural ask entry manner theoretical problem network localization aforementione distance guarantee successful turn number pairwise distance reconstruct network perform solve semidefinite sdp get determine freedom singular decomposition svd orthonormal degree specify observe thus degree th freedom specify degree freedom specify freedom need entry infinitely many exactly whether reconstruct crucial reconstruction may zero reconstruct depend motivate reconstruction entry tradeoff entry reconstruct propose idea compressed optimization define notion coherence measure similar notion informally case reconstruct entry set singular whenever randomly formulate polynomial limit practice specialized solve sdp associate least subsequent improvement sampling runtime bound reconstruction certain coherence entry uniformly iterate probability refined analysis show input away reconstruction yet complexity first introduce notion speak sub intuitively stability large crucial present many small subset column amenable turn notion separable theory moreover analytic nature coherence nevertheless notion compare pursuit fashion note strategy rank namely strategy assume involve produce polynomial discussion et exact high particular low reconstruct furthermore runtime sampling complexity yield moreover essentially extra attribute phenomenon discussion derive coherence define study construction fact pursuit analyze sampling complexity possible direction section ability depend paper stable rank rank matrix unchanged removal nk nk row rank column may shall column stable sequel stability nice generalize notion stability I give let construction equal rank one whose equal easy verify stable two distinct singular coherence raise coherence formalize statement recall let orthogonal basis let orthonormal view span column coherence column establish arbitrary matrix let column iff collection complete ready negative suffice follow non measure linearly probability determinant turn statement haar conclusion reconstruction low matrix introduction reconstruction since priori information guess currently entry uniform strategy certainly reconstruction illustration form hope reconstruct entry randomly entry bound eq uses reconstruct large entry entry wise may critical localize matrix reconstruct first row general largely motivate follow reconstruction knowledge column examine draw column span eq proceed column linearly let entry express basis column terminate matrix illustrate flow rank row identify sub reconstruction entry low rank pursuit matrix proceed let remark speed lie index remove facilitate section analyze goal let terminate reconstruction simple theorem compute algorithm randomization complete terminate suffice b execute throughout towards end divide epoch begin end column iteration th basis column epoch execute parameter execute course execute r quantity distinct examine theorem corollary significantly orthogonal model least terminate reconstruction entry computational complexity initialization operation pursuit newly span index achieve via gram schmidt set orthonormal th orthonormal vector span index suppose select proceed select new column add basis pursuit since conclude pursuit bound row gram carry step use operation need reconstruct column follow total step summarize suppose let total perform suitably add counter time still however idea develop knowledge matrix reader bind compare sdp polynomial approximate polynomial reader issue early assume rank however priori raise question modify mention end last section keep pre specify pursuit proceed sufficiently probably find input exactly formalize propose initialize correspond recovered basis next column hence otherwise proceed b drawn belong increment span index belong new index find examine entry express th column combination basis coefficient turn rf least rf terminate exact total since total entry rank generic reconstruct note reconstruction much flexible need reconstruction rf algorithm find proceed step proceed facilitate proof divide epoch st epoch define exactly iteration st epoch hold rf find proceed rf proceeding probability inductive observe terminate
generate frequency word word database merge highly correlate coverage rectangle rectangle rectangle bf c bf level rectangle rectangle rectangle bf bf level word exclude right long ssc hierarchy acquire concrete age whereas continue visualize figure across entire connect across within triangle coordinate mark coordinate coordinate bf plot mark coordinate coordinate coordinate coordinate mark bf hierarchy write frequency stay factor analysis within exclude correlate hence effect analysis hierarchy across dictionary show c hierarchy hoc see test show triangle plot plot mark bf plot coordinate mark mark mark right bf right external initial age acquisition level level hierarchy whole abstract less writing far refine outer rest space level increase bottom level source whether effect distance hierarchy induce entire connect within alone turn successive level induce hierarchy beyond dictionary turn frequency continue frequency likewise continue decrease age acquisition level outer small acquire early corpus frequently ht corner fill edge style arc cm cm arc arc cm cm arc arc arc cm arc cm cm cm cm cm cm cm cm cm cm cm cm cm category abstract concrete distinguish thing thing category category reflect amenable ever category define something category constraint abstract category increasingly category abstract mathematic constraint though still law increasingly mean word turn counterpart cognitive des universit du universit du p et en analyse des dictionary need one reduce turn early concrete rest turn strongly core distance age write category feedback semantic symbol category thing trait state thing right thing specie trial category almost name category definition category acquire know define already symbol induction allow answer reach definition reduce dictionary word age variance correlate age remove residual abstract cause ground reach feedback vertex np hope able special case meanwhile dictionaries international english english begin analyze play important word acquire concrete rest tend acquisition abstract rest whole make comparison length distance strongly connect age acquisition object reader theory mathematic couple call graph edge graph vertex vertex nan integer starting cycle subgraph ii use couple word define dictionary graph loop toy color dark dark light good dark red dark color red light style edge style bad dark good good light light red bad color dark edge dark light dark edge bad good light light edge dark red core minimum contain good dark direct acyclic cover cycle find np unlikely find hope exploit around difficulty report effort extract operator define easily acyclic must stop step also include hence linguistic cognitive strongly subsection relation vertex path therefore construct exist fact graph acyclic acyclic induce vertex particular minimum belong dictionary turn refine division vertex eq categorization function hierarchy connect vertex q acyclic
n increment identical original way estimate approximate likelihood maximize approximation integration expectation break class version coincide improve rarely iteration robust indeed redundant reliable batch online gibbs joint variable sequence simulation simulate node j context version old complete il il simulated vertex online describe consequently label adapt original likelihood node notice network large associate expect log online equation gibbs modify already exist estimate propose use update convenient bernoulli poisson visit improved likelihood approximate variational newly new optimize leibl probability choose multinomial natural conditional expectation variational factorize hidden suppose entropy independent variable additive aim update step online new node necessary know lagrangian maximize solve give obtain thank successive statistic g update hidden term online use estimation give propose effect framework real growing assess implementation http software http project web package public online visually explore take advantage reveal first simulation simulate connection model free control modular structure enyi module intra module strong module module illustrate kind allow generate size graph simulate times r enyi criteria comparison reflect estimator estimate belonging partition lie partition adjust rand variational propose sequel comparison initialization start integrate consecutive step estimator mse online online precision batch behave well limitation burden impact online variational average precision rmse pt partition estimation reveal index partition illustration rand expect increase c pt c execution speed web web individual discuss page respectively represent edge like web focus study form community concerned community opinion existence page explore actually web comparison community detection consist compare political agreement aim find module intra tend another political link manual naturally module explain module value modularity community compare link community apply variational manual partition necessarily module modularity optimal definition complementary give global detect dense structure community rather cluster confirm political highlight role political com com com com central confirm political already political connectivity pattern affinity medium thought toward pt mean probability c c c c c c c c c c c c c determine characteristic conservative site political determinant com com know core intra clique position like website spread hierarchy intermediate interestingly c constitute core conservative compare community show expect model highlight political small way site explain tendency ignore internet interestingly core tight observation web interpretation short geodesic path vertex core political look similar random set method constitute potential amount estimation even online precise may size exist simulation study remain method political political classical module highlight strategy online flexible could membership online adapt context assume might intensity traffic mining describe article commonly publish author network n l n z z l n z z n n n n n page simulate political political trade speed estimating become essential include grid focus political new facebook show political far medium political political one day snapshot link manually classify citation strategy study distinction make model summarize connection consider spread among connectivity unknown cluster strategy mixture belong alphabet propose variable et structure computational strategy diverse constitute core put challenge development speed execution network extent bayesian strategy heuristic satisfactory statistical strategy limitation assess network time online alternative batch grow application study many algorithm model traffic difficulty graph conditionally factorize base simulation approximation describe principal modeling data framework estimate parameter variational design finally us algorithm extraction company day http com www manual classification automatic seed site web link visit still belong web create node page intra account political identify political corpus check validity seed confirm site http fr sg set vertex model connection suppose node among belong proportion formula loop introduce software edge supposed conditionally suppose belong vector normalize function consequently conditional graph exponential mainly nod membership whereas membership blockmodel play multiple role stand group whereas consider per node could dirac zero difference optimization strategy bayesian choose author allow integration structure contrary approach rely strategy multimodal approach lead switch converge local framework order political describe figure first provide category class inter connection intra
design minimize k interpret nuisance deviation e order particular variability local attempt understanding behave nuisance essentially roll believe mostly taylor expansion motivation set form lagrange multiplier compute inversion h top simulation realization random field know diagram corner scale near corner interpolation bottom bias job recover function region bias originally generate generate isotropic mat ern spatially smoothness proved convolution thesis let definite q positive definite equal g positive since combination limit definite positive definite g pd kind definite claim conjecture smoothly away work sampling trade bias reduce exposition technique estimate positive stationary field local na I likelihood stationary neighborhood difficult parameter control distant present estimating smoothness local mat ern random field observe sampling play role apply unfortunately stationarity real datum visible challenge spatial stationarity sort field stationarity statistical field lack due add assumption field local stationarity example enough scale dependency field approximate random definition stationarity literature enter discussion estimate observe realization dense possibly goal mention decompose log sum weight spatial covariate independence field neighborhood stationary random range undesirable present exposition local likelihood paper devote construct local likelihood distant weighted apply convenient local one prior fractional local ern realization observation present advantageous domain clear real distinction versus model local approximation random index argument return call determine stationary informally h l law remainder function concrete field fix denote could model tradeoff variability estimate increase local likelihood balance compete term smoothly single realization field spatial location observation parameter define first full likelihood incremental change likelihood field decomposition typically additive bandwidth typically start estimate function observe gaussian mat ern concrete estimation close solution need notation neighborhood response eq maximum weighted mean mat ern middle increment evenly space true hard thresholding I estimate line diagram I right hand simulate evenly mat ern use parameterization page unknown even variance change throughout observation region difficult compare however look increment middle diagram visible plot dash also section minimize global truth yield consuming make inverse formula may order study spatial parameter situation expansion high prior risk goal attempt understand aa tt thresholding I estimate local investigate estimation thresholding heat map percent threshold use hard thresholding choose leibler heat correspond column mat ern location evenly notice random field smoother possible explanation smooth neighborhood attain neighborhood high bias mention right diagram reach almost figure bias kernel hard mat random even location polynomial last paragraph likelihood vary depend estimate comprise realization datum highly leave problematic construct present present numerical claim justification heuristic bandwidth say one bandwidth variability first variability true try come realization field small bandwidth spatial variation random realization interpretation example might simulate realization stationary due field selector similar variation replace recognize statistic spatial stationary behavior quantity fit simulation expect stationary green dotted mat ern location plot criterion profile maximize dot blue green exception section evenly spaced sampling location interval mean zero stationary mat ern leave dash estimate green dotted smoothness true observation location kullback leibler right profile maximize green diagram standardize exception instead dash green dotted standardize good bandwidth mode drive criterion unclear moment two regardless think mode mode result truth fitting observation letting gets take smoothness observe realization suppose locally vary spatially smoothness spatially smoothness
calculate stage c make efficacy simulate datum student reach ap bp bp bp bf bp bf ap bf ap bf ap student simulate situation progress student event half half pose convenience hard distribution mark student pass distinction path illustration purpose combine factor edge merge point reach prior homogeneity strength exploit discover qualitative fashion homogeneity theory currently rich even category dynamic chain number case parameter big map class clearly paper student another arise similar search one describe case order reformulate great finding ki unit paths independent gamma root node leaf lemma theorem axiom theorem condition exercise proposition summary class advantage propagation naturally asymmetric event sample homogeneity technique event model finite discrete graphical however long scenario depend variable zhang create context however place class seek context belief lead encode common single graph end tree represent unit extend possible atom event space encode leaf exactly atomic argue expressive framework accurately belief particularly explain contain redundancy describe unable express deal combine topology independence statement possible follow run successful component allocate module student allocate module pass fail automatic module proceed fail student proceed module fail distinction round component fail depict ap bf bf r ad ap bf bf take edge edge pass node specify unfold student natural situation situation figure consider second pass component mark pass hypothesis leaf situation partition hypothesis consist stage p atom tree probability stage encode tree combine student record want take model posteriori common et specifically structure definition develop section analogous prior homogeneity condition algorithm use discover explanatory student finish event discussion overview direct v st possible event induce conditional reach v mutually independent event calculate primitive together primitive colour bernoulli pass module indicator variable subsequent module primitive multiplying primitive v starting define v e v child subtree represent random one eliminate two say despite conditioning situation concept stage write partition situation colour leave must colour situation similar stage eq path sequence map identical denote obvious variable position subsequent fine partition chain event tree position colour edge colour construct probability graph position use mixed vertex undirecte w w v construct closely analogous fashion conjugacy lead closed candidate make space demonstrate conjugate proceed stage edge vector q student stage assume analogous random separate component model local posteriori write posterior marginal likelihood function calculated theorem give prior score try posteriori intuitive c problem describe section tree partition tree stage situation except node edge search score high stage mcmc agglomerative local begin henceforth seek yield combined simplify algorithm assume stage form absolutely differ stage unchanged logarithm equation two proposal show calculation form two practical give value task way informative exploratory advantage potential cluster priori accord exploit modular state completely modular simple stage bayes obvious search exploit assumption likelihood cd marginal I set assumption stage equivalent vector c reduce set trivial prior cluster stage surprising global previous paragraph second form prior assume priori rate root leaf path independent entirely root leaf path assess uncertainty way dirichlet prior space kn integer I theorem leaf path equivalent least prior edge construct tree leaf path node extend possible describe atomic event variable event tree event leaf lemma trivial dirichlet show stage prior inductive identically compose identical
obstacle filtering sample arrive maintain gaussian kalman fix approximated tractable mixture approximation scheme apply nonlinear introduce representation linear call model behavior many boost machine unify explicit prior knowledge implicitly embed knowledge encode learning algorithm aspect require come bayesian inference posterior update instead issue incorporate update supervise sense scheme representation rule guarantee supervise statistical literature apply theoretical attack problem predicting depend therefore posterior parameterize updating equivalent sufficient augment incorporate statistic show fig leave state eq event rule update completely key observation capture general flexible posterior require approach approximation ensemble replace stochastic traditional explicitly problem explicit complexity incorporate learn advantage sophisticated supervise functional moreover derive rule hand sufficiently powerful believe approach model mis specification commonly occur change probabilistic dynamic goal remove refer dm map pre characterize vector essential learn middle panel evolution prediction panel time predict jt approach implicit evolution operator various natural part architecture essentially system solve kind graph understand circle make prediction predict conventional quantity integrate integrate evolve intermediate ambiguity problem middle panel use though source one solved integrate learn choice obtain rule without want computationally specialized provide straightforward multiple deal multiple correctness everything else difficult thing multiple choose take information near initialization improve train change future observation negative stochastic mixing apply improvement alone consistent gain execute imply system improve describe imagine done evolve state break state state transform depend example extra generally show dm limit algorithm limitation non agnostic dm exact agnostic dm correct technique insight agnostic analysis constraint dm vector generate nontrivial dm function understand dynamic show markov third set hide express state state invertible hide markov induce imply specification impossible invertible limited dms still nontrivial invertible long dependency accord accord observation state converge exponentially fast bayes chernoff concept look give class state short induce invertible efficiently different dynamic identical short known effect imply significant limitation efficient next recover consistency supervise consistency dms solve perfectly projection hold perfectly define c inductive imply exist consequently prove inductive case result dataset involve high structured come graphic nine leave average horizon model condition nonlinear use right panel state hmm predictor vector zero introduce use logistic dataset initialization previous gradient backpropagation rate robust conditioning step sequence angle plus body orientation invariant contain split datum sequence datum sequence dimensionality leave compare error autoregressive model nonlinear use compare show range autoregressive operate space linear prediction step fit prediction five step parameter autoregressive grow input operator quite term probably perform compare linear long range prediction panel perform obvious model unable cope unable outperform hmm long considerably video sequence
penalty b initialization observe start respect maximize algorithm imputation brief comment initialization estimating center instead iterate row ignore convergence computation matrix mle miss set imputation yet r way multivariate formula kronecker value covariance medium sized expensive inverting amount miss relatively inversion cost calculation sample bayesian imputation prohibitive thus cost multivariate imputation imputation regularize covariance iterate take expectation cm respect intensive iterate step iterate maximum starting often approximation solution seek start recall marginal variate one among column penalize imputation instead marginal start two missing applying row set complete apply em algorithm good start em expectation take imputation part however correction unnecessary goal miss imputation initial imputation miss miss mle discuss calculation whether good miss imputation detail final computed inverting avoid exploit property variate namely k j j l index row column respectively vector column I k k l j l j theorem row calculate take computation matrix conditional interest row splitting manner avoid invert kronecker conditional element value alternate expectation mi r j repeat alternate show invert kronecker reduce around complement matrix operation dimensional variate computationally model observe imputation example autoregressive certain report one slightly imputation could give missing estimate converge stationary point apply also apply beyond step denote log figure marginal indeed higher observe also iterate step comparable feasible dimensional datum data set variate normal begin star comparison follow imputation regularize variety situation exist assess one simulation suggest rating netflix movie rating distribute assess performance commonly imputation method svd reduce validation parameter effect pairwise correlation close missing compare cross validation array step take mean quantile close microarray value use penalty expensive compete imputation value cross choose indicate array independent often microarray mse error make investigate issue assess pattern miss array display absolute imputation quantile assess two method utility imputation microarray compare exist netflix rating framework suit movie customer movie customer movie customer netflix simply customer rate interest may customer set movie remove begin thus exhibit due miss value link covariance use rating find netflix movie assess currently number rating around rating customer subset movie movie customer one rating leave rating compare dense movie rating movie customer value compare absolute dense pattern customer rank movie entry leave miss svd discuss error netflix leave potentially thousand correlate customer greatly method predictive fact entire netflix rmse thus conjecture rmse use entire set small imputation penalty notably subset rmse average potentially great imputation large percentage miss model choose cross indicate movie power cross choose indicate well missing result follow svd right large absolute penalty lead error lead imputation netflix ensemble would individual formulate parametric advance miss set row step imputation approach one column vice roughly cost miss respectively application value imputation single extend imputation repeat imputation form foundation also address kronecker covariance covariance flexibility recall distribution restrict variate parameter say location within often reasonable either familiar flexibility numerous advantage potential mathematical practical interest covariance essential add restriction observe introduction make application many classification mining thank helpful improvement numerical covariance discussion property alternate expectation bayesian step gibbs penalty proposition challenge arrange column present modification separate place penalty inverse row maximum covariance imputation exploit allow imputation dimensional netflix imputation technique outperform great degree become common miss netflix movie around customer rate percentage predict rating movie recommend movie customer movie however relationship customer rate rating movie neighbor method movie customer rating movie customer movie rating linear rating fail sophisticated customer imputation netflix call movie customer column treat row interest base variate movie incorporate long matrix customer movie customer rating movie model interaction customer practice largely rarely real matrix variate restriction employ decomposition graphical foundation computationally expectation em imputation contribution single reasonable computational user regularize type imputation netflix section conclude section result enable give miss feature log regularize absolute square penalize graphical use likelihood penalty undirected graph analytical singular svd nonzero model alternative underlie imputation penalize likelihood covariance via except addition method fit penalize enable call discuss integral present previously model row column variate interpretable distribution regularize variate singular graphical variate normal restriction mean variate mean variate mostly sample formulation appropriate estimate marginal improve interpretability restrict variate n np mean model compose row column imply element column belong point random jj effect covariance share matrix variate error random product illustrate marginally jx jx distribution restrict variate statement two row ij n jj j j thus general multivariate completeness p np formulation variate mean give desirable term marginal section reformulate covariance estimate imputation seek two separate penalty inverse column penalize absolute square element penalty note penalty equivalent place kronecker great flexibility covariance penalty strategy counterpart place concentration graphical especially penalty nonzero conditional row correlate link form conversely imply regularize row graphical covariance hence mean column th row center additive result thus covariance difficult penalize concave coordinate coordinate wise block reach global due maximization monotonically log maximization accomplish set list penalty change penalty coefficient term parameter penalty eigenvalue coordinate maximization iterative maximum penalty global penalty penalty eigenvalue analytical optimal svd td pt ip penalty singular reveal equation eigenvalue quadratic give eigenvalue singular covariance globally say penalty covariance penalty decomposition commonly employ write reduce
datum constrained scheme perform presence element operational criterion design learning involve converse theorem formalism theoretic distribution assume infimum joint incur particular framework cover function indicator pg f function measurable problem paper basis
mostly probability often scientific job assign modelling though influence sampling fact entire rather look posterior thing discuss principle eliminate inference must belief carefully use evidence pd pd
term equation use triangular nonnegative u h ph p use sufficiently indeed definition lebesgue prove main theorem asymptotically vanish extend central markov accept additional assumption maximal exist induce markov start moreover measurable generality proof resp start let go infinity
fix actually extend classical extend take arbitrary continuity continuous dense lx l lx n lx lx lx lx lx lx lx lx lx lx successive therefore uniformly conclude end proof strong law number conclude finish since martingale write l lemma lf hold martingale eq f lf fundamental nd np k l combine dominate virtue follow strong nd l
go allow case original infimum convexity von various upper low maximize define joint array analytical try value joint put interval force decision root appear nature I maximize follow inequality I expect study follow distribution gap hierarchy analyze minimax introduce help expression derive inner adversary conditional deviation next functional concavity already concavity section show description interpret provide yet divergence begin simplify notice immediate application similarly interpret jensen albeit precise
resp vice overcomplete original coefficient pair global minimum representation entry whenever entry standard numerical opposite develop replacement admits side directional appendix indexing row index kn appendix provide sharp explicit decomposition kx k indexing nonzero entry index resp keep index resp also th equip notation state condition consider notation matter local restrict e strict open whether overcomplete dictionary side additional relating collection next discuss belong convex polytope project hypercube cf column resp difference column reference orthonormal necessary simply read polytope figure live bernoulli htbp
hence prove furthermore chance gamma distribute result assume k true evaluate establishe theorem gain k k equivalence event range summation implication mutually though factorization mutually force evaluation fact surprising dependent balance replicate form vector density mixture table structure q replicate take normalize constant stand x z z z z x z n n z leave side denominator factor control inspection suggest polynomial indeed degree reduce construction multipli assertion table indeed contribute high equal indicate contribution term zero zero equal reduce consideration able focus bivariate focus degree instance coefficient equal require
implementation expect artificial need need compare four class recognition face google identify content characterize sift descriptor object categorization detection face background google
relevant query result roc area percentile basically measure language rank result query average chance cca regular cca cca loading canonical approximately cca able good cca result section achieve cca loading cca sparse narrow effective dimensionality reduction technique short summary result cca application involve meaningful retrieve database semantic probabilistic acoustic signal tag word content decrease computational away noisy detail scope song represent song semantic tag coefficient song allow small span semantic highly audio sparse cca generate contain horizontal axis train area receiver operate vertical independent initially clearly cca dc remove noisy figure system base vocabulary heuristic amongst subject collect improvement initially summary selection cca improve retrieval computer effectively remove agreement special special primarily interested advantage formulation become clear sparse introduce relaxation gives specifically give regularization parameter perform write simplify solve version sparsity constraint constraint result convex convex optimum occur unlike discussion clear significantly
ne ne nd simulation reach need nash percentage ne table cm p cm p ne time ne games linear co co see lead nash equilibrium contrary version frequent nash games ne players ne version well social strategy update new previous strategy converge ne course function symmetry cost cost realize update learning individually note value display case hold state expect inter arrival time estimate arrival yield nash nash quite reach
small big may reasonable often obvious calculation term know directly markov chain see corollary easy autocorrelation fact yield loss generality obviously q bind side note state bind cover setting indicate set need apply perfect deterministic without burn burn corollary chebyshev
def define lack allow usual convergence give van condition space rate quantile surface unknown replace uniform partial quantile compact quantile hold metric typical rr logarithmic recovering binary presence issue identification criterion since partial quantile quantile space perfectly quantile point index consequence lemma quantile moreover characterize partial index make quantile index partial quantile assume quantile quantile point turn aim characterize quantity mild notation asymptotic quantile quantile hold assume continuously gaussian limit set gaussian limit generalization quantile relevant study monotonicity quantile counterpart characterize underlying establish independence measure quantile dispersion quantile detailed process develop quantile index let quantity quantile comparison contaminate result whose direct function probability comparison depend outli probability point nonetheless influential associate influence notion outside scope partial univariate rely partial reference well classic determine class probability identity index characterize determine determine order describe partial quantile characterize proper example determine acyclic follow necessary characterize otherwise partial order univariate order monotonicity partial surface partial quantile proposition deal assume monotonicity case monotone true quantile quantile partial monotone estimate quantile monotonicity show monotonicity condition quantile monotonicity quantile conditional
assumption together differential fraction indicate notation initially individual ignore choose interpretation still contact infect mix mass action linear worth pointing since track two differential equation show monotonically monotonically initially increase decrease start little happen eventually get infected fraction major occur first minor occur small separate different figure configuration right fundamental importance number refer individual epidemic epidemic balance divide equation epidemic recover balance determine epidemic infect interpret infect must infected infection pressure cause infect fraction infect useful related epidemic previous epidemic major fraction community mix homogeneous mixing accept model may example epidemic day care seem uncertainty infect possible chance epidemic motivate epidemic reason motivate stochastic epidemic disease equip standard error study epidemic disease general homogeneous uniformly mix recover time contact constant contact mutually contact infection individual contact
expensive calculate calculation algorithm polynomial practice trivial look step alternative evolve time way death happen happen birth mark happen next obeys dominate note property property monotonicity locally step dominate involve calculation process multiscale require configuration element construct thus refine
sphere radius derivative smooth curve say follow say resp smooth manifold matrix square view column view row especially corollary group simplify eq look spherical least observe singular decomposition rewrite obtain decomposition formula spherical act linearly one simplify inclusion drop simplify transform rearrange function smooth open geodesic cut subset short one cut complement tc ty since geodesic geodesic geodesic many intrinsic lead appear dimensional riemannian smooth structure hessian calculus
ground truth human berkeley parametrize map image quantify segmentation relative objective parametrize predict neighbor affinity affinity find connected component measure segmentation adjust rand affinity key edge graph opposite sign machine graph focus incorrectly split merge segment learn segmentation performance affinity previously number misclassifie train affinity partition cut optimize rand close however minimize rand classifier
sample calculation iteration two retain regret bernoulli benchmark ucb secondly bayesian simple optimistic htb select accumulate e suffer high determine perform cumulative average run compare ucb ucb bayesian baseline approach cumulative serial upper bound node expansion evident improve due algorithm serial expansion seem slightly serial consistently expansion
otherwise polytope often transpose denote denote identity covariance call function mind constant polytope regular require regular principle converge highly non main applicable proper constant proper hypercube spherical multilinear hypercube multilinear lipschitz function constant q pac sensitivity translate define noise define independently denote learn focus agnostic label example agnostic concept fit draw opt correspond agnostic see product multilinear estimating fouri concentration one function know work let submodular output example due polynomial polynomial output opt principle invariance regular polytope divide principle extension devote prove random closeness anti imply closeness surface bind ok result surface state invariance involve make hybrid introduction clarity next
five severe security resource might fixing exploit complete computer resource customer management attack database observe attack learn tune receive result extend model clause clause previous level attack might clause hypergraph model consist clause attack reward budget attack rule list set proposition proposition generalize directly valid row become clause matter obvious substitution become clause security game practice security objective show mathematically equivalent attack per attack edge interpretation attack root hold appropriate modification simple budget qualitatively informally design security account various attack patch benefit security economic shot security game identify
objective leave q quantity composite write put fix certain impose eq composite bias dms rank attain bias decomposition modify introduce equally weight bias local weighting definition easily extend basically gradient descent htbp locally post step generate user inspection size dm orthogonal find j gradient multiple solution solver produce solver optimization core use inspire solve handle transform via slack result constrain optimization problem trust use subproblem conjugate cg accurate handle hessian approximation solve region follow newton symmetric sr quasi newton approximation hessian use suggest reject step option bfgs option implement limited variant sr simultaneously test performance provide technical detail illustration purpose base profile response propose choose reflect choose choose vector zero elsewhere two column contrast htbp iteration htbp vector everything else convergence optimal contrast determine attain shift unconstraine htbp two shift
j e lemma n n nd fact condition conclude follow positive c nj nj nn nd first note event pa nd f j nj nj nj marginal let joint score snr fp pe snr tp fp pe remark correlation fan dimensionality true regression highly nonlinear correlation marginal nonparametric nonparametric member several screening general nonparametric mild independence screening extent quantify drive iterative nonparametric moderate dimension perform keyword model regression screening
regression assumption tail mild hoeffding hold coherence property u particular inequality mention early need domain eq mapping constraint property condition roughly speak concern one function indicate constraint well portion need condition constraint mild small comment constraint aa let nonempty counting measure extend rest give I see
dimensionality paper problem localize localize anchor intrinsic manifold code code anchor point map induce q moreover code theory condition shift origin coordinate system shift invariance requirement become represent code concept code localize illustrate linearization linearization code leave side linear coefficient right first second localize quality convenience introduce locality code localization tuning quality depend complexity coordinate depend manifold dimensionality manifold dimensionality call intrinsic constant
ex university national ex com ex national south david com technology introduce principled scalable learn direct approximation notion general reinforcement unclear motivate open provide first approximation monte agent context encourage propose future ex reinforcement confidence tree observable prediction tree reinforcement popular influential paradigm experience reinforcement agent environment introduce evaluate reinforcement agent inspire exist environment cycle history agent time environment achieve computable observation reward action compute machine mm horizon sequential universal horizon pick environment eq consider step ahead pick expect reward kolmogorov rigorously agent accurate environment act optimally description agent computable algorithmic bayesian making environment ai way principle view machine principle optimal behaviour complexity issue agent theory plan mixture reward past experience part many generalised generalised weighting idea contribution paper paper notation environment accumulate include familiar reward describe bayesian present carlo operation context generalise set put idea form related limitation highlight area investigation terminology agent experience true underlie environment string denote denote empty concatenation string space denote joint observation agent call drop way mean describe underlie may agent environment typically learn approximation environment agent agent previous definition environment wide variety environment reinforcement mdps familiar notion much seek horizon instantaneous reward
helpful measure offer optimize drive recursion assume euclidean measurable number function initial transition conditionally hold old illustrate metropolis langevin mala euclidean mala effective metropolis hasting proposal langevin diffusion symmetric definite brownian mean mala propose accept probability define mala practice computation run mala mcmc definite c mala proposal let family nonempty convex obviously choice initialization acceptance x x transpose q n define rhs expression recursion
require statistic student asymptotic bound closeness consideration test concern deviation expansion follow simplification uniform pearson statistic obtain variety value list try minimize however appendix choose let eq statistic q generalize though rather matrix us eq k let couple find closed ball smoothness condition let inf immediately finite degeneracy applicable yy degeneracy recall hyperplane appear central expansion normal proof corollary corollary assertion cf modify proof recall one prove remark note valid distinction definition schwarz yield remainder proof provide ease use extension obtain er absolutely continuous exponential introduce r definition concern assertion comparative truncation especially regard er transformation q also follow definition must recall restriction third two establish second case independence argument mt dt cf last recall definition easily identity next cf q chen absolute er random vector necessarily moreover replace define display use thus recall follow view replace establish
topology review mrf belief propagation pairwise undirected edge associate make direct denote graph compatibility normalization probability distribution undirected graphical generality compatibility function compatibility operation bethe bethe partition
theorem consider obviously imply fulfil thus problem generate see easily matter nx I side denote define recursively easy induction bf bf bf far pass equivalent sequential everywhere almost generate plan rest equivalent almost everywhere test everywhere everywhere fulfil test locally strict strict inequality proof
subset path addition triple disjoint definition decomposable exist decomposable subgraph decomposable example clique sequence perfect name residual graph clique residual property decomposable graph factorize eq occur forest graph cycle several tree acyclic edge forest forest bic minus ht spanning assume model homogeneous heterogeneous decomposable path occur path non adjacent pass show page represent circle dot
automatically chemical domain functional characteristic property limitation section classification start bayes specialized binary predict label unlabele model theory bayes bayes derivative formally differentiable order support class neighboring pair bayes highlight positive sign explanation form continuously change orientation understand remark become fit region issue estimator appropriate classification explanation learn explanation gradient dimensional vector sign decrease increase entry amount ascent test negative need predict model learn gaussian use interested classify drug discovery area process e gp natural datum worth insight predict capability complex definition directly start gradient
immediately note bound gaussian function distribution hypercube aspect perhaps symmetry original sensitivity sensitivity surface equivalent
mcmc include monte harmonic functional equality survey efficiently significance probit performance dataset keyword model choice factor functional probit contribution fundamental establish difficulty value bayes difficulty contribution area obviously present factor procedure stand hypothesis crucial framework give hypothesis compatible odd prior odd namely paradigm g comparison
variance prove normality strategy calculate assumption differentiable assumption order investigate ode complement open drive bandwidth extension link asymptotic differential equation current associate without measurement
protein datum result approximately positively data protein protein procedure correlation relational remain majority remove link total extra missing value normalize vector euclidean initially fit pt choice matrix rescale norm protein link mle computation measure rank protein pair gene page categorization retrieval receive concern link follow protein measure algorithm precision calculate area auc proportion vice versa belong multiple sense measure measure go categorization pair analogous agree believe categorization system organization protein lead category protein interaction correspond auc compare different variant metric retrieval cosine near score measure euclidean query given initially setup obtain fit use generate give analogous pl ij pl integrate parameter neither take account interaction theoretically require
identify trend score weak score e variance tail already table probability success curve figure curve strong agreement formal sensitive visual display score ensemble odd figure present odd eight panel difference ensemble problem along horizontal combine score consistent conduct test fourier hadamard panel dot red color panel hadamard dyadic blue dot fourier f hadamard rademacher rate panel e blue dot linear fit score match color hadamard dyadic dot mark display figure trend visually evident model drift otherwise truly difference probability score exact agreement success term refine tend early line fit within close plausible inspire score large expect find score reject weak model dependent record scale verify width drop verify success rate probit ensemble figure ensemble offer relatively poor ensemble size curve success shift shift character evident appendix row hadamard air present classical cyclic show notably special forward conduct extensive observe transition observe compete yet derivation phase dimensional informally approximately ensemble instance range empirical ensemble band phase band consistent prove behaviour ensemble independent diagram tend tool counterpart fit statistically significant trend problem fit trend finite dimensional class ensemble task geometry ensemble quantify drop verify probit logit logit well probability
size finite induce thus reliability ordering discuss reliability statistical compute hypothesis great bootstrappe computational confidence random sample create bootstrapping selection molecular
rank model develop mirror whose lemma define aggregate estimator subsample denote base subsample collection k consider function second subsample cardinality proceed eq goal collection mirror average recursive weight denote subsample aggregated density give norm restrict accordingly restrict l l mirror average restrict euclidean remark inspection one lebesgue restrict norm mild assumption inspection valid tend origin factor densitie integral ball value main learning label difficulty moderately dimension tail e classification purpose excess misclassification plug
change line sample fig trend need range panel statistically confirm environment affect frame cluster observer galaxy formation galaxy light particular optical effort red sequence herein bootstrappe cluster extent agree simulation future work illustrative purpose list make context literature five place comparison summarize galaxy impose color actual result relation correct red iv slope negative negative increase red slope fact rest frame difference come red iteratively sigma blue cloud present automatically initial fit account slope cut magnitude ideally fit line unfortunately indicator ii slope environment basic agreement
see core different sharing parameter allow delay core preferable consume synchronization core read write atomic synchronization memory architecture key multi date gradient computation retrieve copy message exceed bandwidth variant considerably less term template solely purpose occur g dimensional decompose partial value partial update combine partial cause stage already quite vector delay much processor communication channel access entirely delay adversary aware together time delayed rate political treat iid pay modification since exceed game analysis bound minimizer convert randomize
aspect draw uniformly commonly incoherence netflix figure decomposition say incoherent satisfie movie rating run netflix cumulative norm leave permutation notation define comparison generate independently plot straight plot among randomly movie rating matrix say movie user hence challenge reconstructed factorize diagonal singular start numerical understand provide rr max calculus establish prove thank discussion helpful comment work fellowship award grant dms pt minus pt plus pt corollary conjecture singular
drive extraction time experiment drive force regularity fail slow signal analyze expand present paper review well call sec drive force experiment demonstrate discuss originally review approach objective slowly multidimensional raw input indicate temporal proportional large place preprocesse input component also formalize generate output variation slowly measure derivative trivial meet calculus constrain basis contain yield expand reason become
incorrect chance outcome chance wrong bit rather small sample size keep total size would stop full count keep comparison error batch vote batch vote comparison control chance full hand count incorrect outcome error chance fail incorrect wrong chance chance
encode string substitute theorem straightforwardly two string code determine string mutual four quantitative difference provide admissible list distance account dominant admissible absolute want express string think string information bit universal distance list maximally maximally normalize universal list information say version metric alternative normalize divide either
discover factor loading spurious see variable effectively spurious gene without factor evolutionary search false loading show gene htbp result synthetic datum gene factor interpret hierarchy column matrix prominent tree column correspond prominent factor locate appropriate breast factor discover interested agglomerative usually contrast factor degradation likelihood result
merely trading consider say threshold find estimate restrict regularization parameter standardize solution threshold coordinate use assumption gaussian condition either analytic standardized significantly see appendix central function notice dt dt dt exponential family namely interior see chain central analytic interior generate namely particular hold analytic standardized moment assertion notice argument generate also desire abuse
aside reveal completely determine fast decide speed number concave converge toward look converge uniformity uniformity integer limit distribution law kolmogorov discrepancy uniformity rule kolmogorov uniformity month third country area square population million display area confirm rule absolute country fast
prove desire calculate f f term theorem combine estimate completely expression r tu eqs tu l x acknowledgement discussion support nsf generalization prove inequality chernoff p analogously proof independently reveal z j remark denote indicator tm ii max second x j x I max definition sum average adjacency denote later discrepancy property probability partition row j ib ib ib value group summation uv e definition leave bounded discrepancy show summation
back particle reconstruction reconstruction obtain reconstruction perturb initially calculation otherwise large reconstruction filter little perturbation individual run estimate filter function produce g independence observation detect pick trajectory particle increment value early trajectory single depend iii computation find correct value run c c run run filter design
much normal reliable datum normal nonparametric article concern density construct estimator reliable generating process function density reliably density relatively skew symmetric available disadvantage model bivariate copula normal flexibility density separately gaussian copula nonparametric copula copula attractive copula originally addition transform standard transform likely suppose joint hope capture copula allow one extra coefficient capture dependence contain remain multivariate overcome encounter large grow one survey univariate aspect choice bandwidth second normal example discussion univariate analysis
observation geometrically demonstrate establish geometric ergodicity uniformly ergodic uniformly also geometrically ergodic irreducible analogue albeit additional markov reversible geometrically geometrically geometrically ergodic geometrically relate result apply reversible yy application attention replace markov form usual gs update kernel easy qualitative geometrically exploit sampler practically model easy put say geometrically connect scan suppose yy clear conclusion condition instead gibbs easy claim gibbs sampler practically statistical geometrically reference ergodicity random scan sampler prove ergodic substitute I hasting hybrid composition
penalize estimation subsection smooth propose subsection covariance reference covariance certain zero since covariance sparse underlie graphical several dependency conditional identifying model well available give formulate control apply gene np hard g see relax follow penalize recently lin propose covariance suitably also selection et heuristic newton coordinate partially al lin effectively paper reformulate subsection notation subsection show q yield eq concavity xt xt xt xt xt two invertible upon side let obtain
error xu wu differ author advantage dependence exploit simple improvement inferior design work advantage available organized review boundary introduce hc boundary toeplitz low discuss dependence proof theorem lemmas hc unit sparse formulae quantity represent strength decrease increase increase ease would connect relationship adjust treat entry proportion much location without magnitude throughout sec variable uncorrelated identity variation equal upper result strength test two hypothesis uncorrelated case also detail identity record closely space test whether closely least clinical outcome et profile gene another expression quantitative trait genetic marker chen observable gene associate clinical outcome contribute clinical boundary partition plane region successfully nan error infinity drawback need detailed hc uncorrelated tackle aforementione uncorrelated apply th value sort high statistic version hc hypothesis infinity fall interior region type associate satisfy hc detection theorem use
ij adjacency bipartite true thesis hypothesis n conjecture claim stanford edu stanford electrical stanford stanford usa rank reconstruct subset collaborative netflix complexity combination optimal circumstance statistic process significant low rank implementation practical sparse discover unless reconstruct first analyze convex introduce tight carry iterative scheme appear optimization complexity complex operation theoretic reconstruct performance crucially
maximum sum order I sample interval plot nan hypothesis figure show limit plot vertical clearly fs support strongly present state know notice scaling choose well theory resolution one
surrogate predicting reduce nonconvex misclassification replace apply mean among nh prediction surrogate appropriate nh beneficial since interpretation weight node interesting nh cope missing imputation surrogate split fit replace value technique estimator calculate prediction member question whether node group simple turn say node membership sufficient node membership sense value usually interaction node membership observation still member involve variable th observation member root node variable member root refined amount drop stop split node across occur nh tree nh tuning procedure apart choice insensitive choose numerical necessary predictive improve constrain node minimal fraction
frequent statistic lift relative frequency association mining small frequent individual column randomization margin name randomization randomization additionally maintain row margin mining method list frequent randomized expect deviation difference dataset frequent sufficient depict test ht randomization plotted control dotted line satisfied depict find level randomization reason rule swap randomization less randomization frequent subgraph frequent transaction graph frequent pattern subgraph graph use mining readily laboratory transaction statistic subgraph large subgraph interesting select switch edge swap create
cv covariate pair final pl fold give misclassification average hold take hour average smc issue pl pl suit design al probably alm gp instead sequential design context sequential optimize black essence distribution ei obtain integrate maximize ei iterative global optimize bayesian inference account order surface particle posterior ei use student predictive equation let applicable opposite embed heuristic perform random candidate ei direct optimizer heuristic augment include predictive map smc pl ps I initialize
decode code graph product distribution fact variant algorithm long try complexity may grow grow exponentially however prior jeffreys informative provide sense noiseless simulation indicate algorithms noise snr sum compressive set interest nonetheless employ naturally pdf constitute message set node product scale pdf check pdfs normalization note preserve gaussian laplacian straightforward implementation family random symmetric tail successfully process order completely choose jeffreys prior improper improper
like eq complete latent variable density parametrize traditionally denote penalize precisely penalize likelihood like eq notation I order practical optimize manner gauss fact separability subproblem subproblem alternate option helpful keep step level recently kullback successively reasonably optimize
superiority elastic remark group variable lar use permutation house choose consist observation without variable consider variable section well irrelevant concern business build notice predictor present remark length whereas predictor elastic net version illustrate validity adopt sparse predictor case early stop neighbor construction comment definition remark et paris de predictor introduce build exploit previously novel construct multivariate emphasis model even number
sampler mutation avoid computation normalize constant auxiliary smc concept recover set allow threshold merge read journal
start seminal network social researcher network simulate network especially network economic trading network resource mobile public package lot well contain domain domain protein network direct communication internet stanford network package six study describe reasonable include network classic derive survey publish relation dataset spend join analysis root existence member member accept main event take stay member difference year leave ask term relation influence span stay well social response survey subtle lead order past decade researcher truth email widely energy company company united cast management top subsequently account investigation publish online cognitive learn project correct email message contain user datum mail top contain give mail threshold message respectively research activity document classification visualization paper working corpus become molecular biology protein way instance protein physical recent year amount resource collect experimental protein bind effort infer protein role song interaction protein target protein localize interaction iii pca interaction iv collection protein estimate produce part ht method large include identification bind national longitudinal health health united middle focused patterns date survey year wave survey occur country complete background school life activity health status complete school service student home topic select two health home survey student construction valuable resource wave collection occur wave home cover school collect wave I construct amongst student two span oppose previously core core wave iii topic include disease original wave home wave iii participant among wave iii wave wave locate comprehensive survey aspect health measurement access public child complete height period derive appear heart tie member series statistical analysis claim examine world depict network individual explore via longitudinal publish similar focus dynamic behavior time come plausible alternative methodology social answer question network subject width body mass color inside status color tie relationship denote appear processing system conference volume
jj transformation absolute mm decompose integral q parameterize find w b pt b pt q justification j b b random distribute w eq namely j w variable therefore np n acknowledgement grateful comment suggestion lead wish support science et universit bivariate margin decompose copula function finite
player issue home overall home use true value fair particularly set actual know ahead report prediction player rmse n predictive interval width interval full outline couple simpler examine full substantially truly last home year home ie point comparison solely rmse full rather sophisticated external ht ccc interval indicator na na transition consider extended transition parameter markov motivation allow transition display past however suggest prediction somewhat even player long transition extreme force sample
lb pp lb pp quantitative r b pp resp obtain notation apart uniform bound neighbourhood norm topology first claim definition last statement follow immediately cover theory take projection truncation analogously average grid filter analogous explicit speed assumption hold term wavelet mu b random variable assume satisfy infinitely smoothing depend projection satisfy compactly wavelet analysis smoothness construct wavelet periodic period jx jx constant function successive wavelet periodic word sequence one convention range function constitute orthonormal function correspond remain index exact ordering choose characterize namely basis space especially relevant b function understand follow simplicity basic embed embed stand duality respect duality operator nf
conservative remarkable behavior four replication adaptive seem account decrease effective way investigate optimize influence introduce adaptive world study validation adaptive lasso tuning parameter attention future value without proper cross validation yield nice incorrect biased favor strength regularization estimate analytic formula thus make consume unnecessary emphasize depend false theoretic reconstruction gene comparative explore research method potentially use causal presence longitudinal al identify variable investigate partial correlation shift lasso regression promise alternative finding simulation investigate sparse discovery density decrease topology pls high mse real opinion pls suit pls
via however practice detail record generality center commonly pc loading worth pc uncorrelated pc see sequentially minimal possible pc different pc pc direction loading practice typically pc great dimensionality success nice pca drawback pc combination loading often especially indeed variable physical meaning biology gene pc would compose small original loading develop find pc sparse loading nice sparse active research topic decade approach ad hoc post pc pc simple thresholding pc small sparse achieve loading much possible algorithm call find loading explain loading pca penalty pc program relaxation pca recently induce formulate sparse penalty maximization propose stationary method pc aforementioned property pc lose pc pc quite explain attempt optimistic overlap among loading pc method lack orthogonality formulation account nice total pc loading vector connection lagrangian class nonsmooth constrain problem suit method method
perturb dynamic affine introduce perturb like field correspond nash equilibria game equilibria stationary unstable include nash equilibria definition strategy profile theorem evolutionary game completeness go follow corollary clearly equilibrium vanish right stable indeed continuity strictly say neighborhood great fast fast corner nan hence strategie dynamic corner generally nash strategy perform unstable field leave perform never nash unstable interior patch imply stationarity nash pay purely dealing avoid nash randomize already guarantee unstable leave almost surely purely unstable like
peak local image source large wavelet base technique detect ray provide wavelet tool tailor wavelet determine interest bayesian algorithm often gaussian separate source propose way slow typically make priori computationally peak detect error rate deal error focus detect aggregate rigorous controlling error rate source demonstrate ray powerful detect ray source give good error source new multi design enhance integrate procedure maintain evident use competitive detect ray description application resolution datum overview direction study ray powerful ray third extend observe without ray million galaxy matter circle black space answer question several ray infer galaxy universe x ray provide evidence dark matter interact visible matter account universe deep
exist seem poorly nonetheless world often day pc context visual detection discriminant aim maximize skew boost scheme algorithm classifier fig ability research publication dr c laboratory bag act e mail com zhang laboratory south mail zhang com video surveillance face detection effort spend boost used training detector discriminant conceptual simplicity efficiency well term train detection cascade weight property boost separability domain demonstrate outperform significant argue adaboost approach achieve classification detection object adaboost analysis classifier face numerous video surveillance base retrieval detector challenge visual pose illumination condition furthermore typical background pattern
os er probability control uniform edge see clique control os enyi edge obvious section decomposable edge decomposable develop four graph markov structure span use fit model third skeleton way pairwise fourth decomposable marginal copula graph necessary standard marginal independent prior vertex plane walk algorithm employ metropolis approach random proposal sample burn period high display compute six figure compute appear iteration actual vary ht construct decomposable marginal procedure gaussian graphical encode zero wishart mass denote form graph approximate via recommend identity precision independence figure employ hybrid move walk section replace vertex independent draw sample summarize c distribution entirely bivariate marginal edge appear dependency familiar joint complete continuous strictly q illustration q z early example graphical structure associate graphical alpha maximal recover show proposal
dictionary orthogonal constraint update j solve linear present addition fuse problem encounter loss sense equivalent instance using proven experimentally adapt denoise carry sparse code computing solving since quadratic surrogate cost situation I structured group simultaneous code name sparsity group suppose wants obtain active way impose consider norm jointly decomposition define use active group n optimization formulation relatively compute matrix keep variant evaluate matlab interface available project natural genomic efficiency factorization principal component randomly patch database compose varied testing increase represent various typical image color patch norm use regularization experiment normalization factor datum experiment matlab interface lead implementation experiment single core ghz signal try power variable show good low empirically tune since performance intend regular order stochastic gradient share set meaningful algorithm alternative would batch
learnable restrict algorithm noise well beyond require recently show problem learnable reduce establish central moreover related decode code bad learn serious barrier represent simply generator size produce exactly label construction hide markov construction
make ml smoothed accurate ml close move approximate quite guess tree hill like thank third fourth research grant dms program r gm p removed sample burn give hill initial tree summarize column percentage hill tree geometric hill hill drop distance distance map tree nj remove sample burn two summarize percentage hill hill hill avg ml nj ht pt true pt nj neighbor package
process divide source eeg inverse ii example find inversion physical source de mix transformation formulate briefly possibilitie principal component ica prominent unfortunately even eeg successfully sophisticated way meaningful decomposition interesting ica assume trace argue instantaneous exist ica propose mix coefficient model temporal combination instantaneous ica furthermore approach integrate connectivity brain source lasso avoid overfitte interpretable brain remark sensor source connect analysis ica follow connected relation exist detail finally implement bfgs numerically realistic eeg plausibility source model future research direction context
large desire size segregation association alternative present carlo recommend segregation side right sided deviation level fact binomial hold instead recommend asymptotic size leave segregation association middle pattern base circle solid line horizontal line nominal association iid generate replicate observe even segregation considerable separation kernel segregation moderate value observe segregation association segregation association empirical critical empirical ht monte segregation depict replicate right nan solid segregation dash observe even mild moderate value suggest critical power alternative depict density replicate leave replicate solid association alternative segregation carlo power value replicate binomial plot power estimate recommend around segregation segregation middle plot plot triangle alternative three power test replicate base approximation poor empirical association power association circle base plot line table ht power empirical stand point relative segregation size relative
et universit du france testing diffusion coefficient always von kolmogorov chi special alternative discuss goodness fit asymptotically free g goodness test dynamical I initial concern trend basic consideration always simple eq correspond trend coefficient
weight incoming subtree elimination illustrate formula execute summing multiply edge preserve value determinant base straight direct hence ij r correction graph element construction take power define weight weight modify weight b walk respect follow rate gauss h clear capture walk depth computation include walk model hold error conclude construction preserving equivalent
modification situation alphabet order overlap distribution adjacent ii update hold time report open fast lead implement suggest david introduce computing field step sort say calculate convention overall minimization measure observe fix respectively slight abuse notation use calculation format
formula model frequently classical monte bernoulli know ever success show posterior mass r denote bernoulli trial never explore fs base really need form development ir z furthermore I km r toy analyze probability eigen r r r j write order row transition strictly positive ergodic satisfie solution follow eigen finally g fs quite poorly numerical datum exactly chain slowly due move high point follow conditional leave chance back stay translate get bad increase size maintain dominant size size get fs bernoulli
follow sigma sigma gb ram update b tail adjust tail lower b true frobenius norm false true year successfully apply biology rapid pc grow main development algorithm reduction comparison positive false negative grow learn distance
actual build email month dynamic latent trajectory company role first vertex interact role role actor likely role another clique role appear information mean clique yet discover red within many people green clique role reflect management role receive report role among behave possibly dominate role clique connection position source find role stand receiver publicly analyst lead massive internal systematic temporal vector actor people increase weight visualize membership vector entity track mixed evolve role base provide explore behind actor simplex vertex represent cross role active role trajectory mixed membership gene estimate various life cycle underlie determine stochastic point invariant evolve experimentally topology technology evolve reverse microarray point across analyze gene know mixed role gene cycle gene development many gene exhibit transition underlie accommodate new stage role interact role reveal visualization compatibility examine whether
proof b immediate substitute consider model define follow substitute design assumption adaptive satisfie formula assumption essentially sublinear fix norm suppose question adaptive lasso consistency assumption random integer set admissible realization whose row q suppose set define probability estimator section regression denoting imply ij last symmetry corresponding regression vanish theoretical sparsity linear regression satisfy equivalently element
analysis extend moreover side pair mab adversary constrain amount naturally contextual arrival correspond expect arm time metric provide change interestingly across good knowledge mab rare combine payoff bound aggregate temporal notable case per behavior random provable guarantee mab round arm round precisely context subset arm pair feasible distance contextual bandit contextual exploit absence essentially reduce carry mab regret play round interest typically current arm motion boundary treat speed brownian analysis contextual obtain superior provide nearly maintain take context develop provable contextual cover adversarial call subroutine flexible depend payoff plug bandit algorithm mab regret bandit overall payoff payoff introduce explore specifically contextual adaptive analysis clean contextual bound result alternative maintain context lead present adversarial payoff priori priori ability adapt expected payoff present adversarial treat discussion bandit beyond scope bandit background bandit arm bandit similarity another payoff payoff allow away arm assumption distinction stochastic adversarial payoff payoff whereas similarity payoff notable adversarial payoff lipschitz expect payoff realize payoff trivial distinction payoff whereas one payoff
attempt produce meaningful prior opinion device kullback divergence see thing always solution require think reference upon could back information confusion informative mid nature prior often serve informative often probabilistic easily variety topological optimality another much eq regular inferential jeffreys synthesis synthesis hypothesis test rather give proper probability space advance jeffreys
infect latent specify negative follow individual distribute mutually function period may mutually independent place result infection member times homogeneous process individual belong individual accord contact rate finally poisson initially population individual simply ignore epidemic individual community else infect individual epidemic typically behaviour mean quickly else negligible behaviour epidemic quickly fraction infect key mixing describe later unchanged become note various instance fix increase distinct value assume key give contact outside tend branch individual infection non infection branching constitute parameter epidemic branching reflect well process describe threshold
et unified calculus density rule translation conditional quantum mechanic retain conventional theory difficult framework framework bayes use framework idea quantum mechanic probabilistic assign latent latent optimization energy cost suffer practice deterministic em vb sa overcome issue local optima vb learning
would curse extend index result assumption hold parametric assumption list distribution positive derivative lipschitz order component satisfying lipschitz nh nh e derivative function away ensure bound necessary asymptotic analyzing estimator slow deal estimator motivate another variability construct limit behavior estimator asymptotic efficiency linear consistent obtain numerous example exist show pursuit elliptical ng sir latter include correction variance estimation save hold li variance regard consistency need far equal therefore sir r ex j et correspond infinitely inverse jacobian theorem small v notation negative generalize inverse r asymptotically efficient et addition iterate estimate
belong estimate importance represent known representation sparse nonzero index assume orthonormal series achieve cf method cover general orthonormal necessarily small achieve adaptively would construct among general notion reference mind suitably choose suggest also define introduce widely statistical refer regression orthogonal equivalent soft type paper include regression prediction use particular refer vast literature component example literature mixture develop aspect component mixture consistent specialized suggest rely well various mixture complexity
column avg average use minimum sf sf table pair difference statistically bold sf accurate datum come bayesian test detail datum aim c statistic pair bayesian highlight bold pair sign well tie datum positively bias table pair estimate significant medium converge properly biased estimate assess dataset v employ log delay fitness compare square observational criterion k log lead accurate fitness instead fitness log consume try evolve allow poor evolve test inferior evolutionary approach perhaps optical suggest plausible distant regard match low art optical monitoring acquire measurement sf sf
split determine cutoff include feature whereas naive fdr cutoff predictor yield massive program inclusion rate datum set approach gene prediction correlation cat score interestingly differentially gene gene buffer become cutoff predictor pt gene brain datum set thus summary difference fail feature contrast express correlation ignore yield failed feature yield predictor clearly outperform fdr happen predictor sufficiently call pt hc hc hc hc
shorthand particular context divergence edge match write vertex note combination kullback divergence usual kl appendix minimum weight pair class edge graph necessarily union deviation obtain consider number graph notion namely set involve maximal vertex matching specify edge belong cover iii vertex yield possibility claim consequently union deviation bind condition theorem condition assume decoder ml decoder since assume knowledge would ratio maximize begin describe set x exponential st graph minimum conservative failure tie therefore apply prove suffice upper lemma begin deviation bind concern x u lemma lemma complete bind
state also infinite finite range estimate agreement previously obtain equilibrium dynamic exponent fact specific insufficient behaviour long interaction still topic field physics state towards regime pose additional simple lr understanding attract great attention decade g std formulate dynamic predict existence exponent initial parameter subsequently validate obtain variety study generalize
rule p hybrid ridge thresholding obtain penalty design group design pursuit either implement range threshold assume normalize interval extensive performance iteration involve matrix multiplication tradeoff omit iteration allowed reach start theory try start nice need achieve gain find simply start good choice empirically find close building parsimonious course initialization e penalty differentiable warm start poor optima warm start finally necessarily
hence preserve eqn order preserve eqn order preserve concavity q eqn strong strong definition induction assume imply hold preserve strong eq q strongly sublinear preserve conditionally compact cone hypothesis theorem precisely assertion hold contrary word sequence probability eq thus since every imply contradict eqn existence equilibrium distributional equivalence forward uniquely induced eqn must hold extend condition equilibrium f eqn proof step invoke weak convergence would I lemma define denote weak convergence distribution eqn eqn complete ii result eqn establish invariant locally separable metric define attractive proof attractive follow
surprising value lie loose interestingly learn two study might static give good choice system reach minimum summarize conclusion set interactive system solely static c ccc static sake conclusion thing notice choose lot far automatically remove pairwise concentrate sure isolate pairwise step dynamic static interesting slightly surprising error actual collect table dynamically well term instantaneous term cumulative c dynamic er er b er value get picture weighting figure compare interactive decide advance user study select advanced
author thank discussion comment theorem adaptive covariance adaptive covariance sample history plus identity induce theoretically convenient practically good easy choose eigenvalue study eigenvalue tend stay practically restrictive super decay regular contour markov attract increase last original advance review propose date seminal symmetric walk distribution definite accept let ks metropolis prove fx compactly support support super decay regular
attribute ambiguity instead application closure extensive therefore categorical attribute concept relation concept order form lattice relation web
pick obtain subgraph converge towards contrary towards still soon one imagine way relate idea fractional fractional comparable assume deal cast within study fractional cover deriving amount hand obviously use cover argument directly benefit instance cover fractional hand fractional allow iid pac bayes summarize elegant deal dependency occur probably step follow line iid stand mh sample nh z markov inequality least suffice jensen h j h z q z n j lemma z j q lemma instance easily generalization problem learning process iid setting scale mm z circle inner pt z control control z control cm control z
sequential introduce behavior change artificial pls model output residual find covariance input extract latent factor sequentially matrix subtract current extract latent recent pls variant usually differ computation focus pls iteratively hide rewrite covariance stream weight find solve equivalent solve normalize large alternatively load regression extraction first factor subsequent must subtract current give towards pls iterative extract limited case perform main limit pls algorithm remove propose avoid require
population computationally smc backward number cost vast computational spend simulate update population bit order correctly backward square dynamical minimize solution straightforwardly time abc ode closely evaluate likelihood normally supplementary material video video particle distribution intermediate c approximate package statistical environment simulate material graphic computation inference david h centre bioinformatics division molecular college uk institute mathematical college uk department health college uk college department college uk ac bayesian apply bayesian method base sequential monte smc smc perform abc credible interval infer abc range description dynamical engineering delay differential equation vast system particularly biological reliable structure underlie moreover datum incomplete surface quantitative
characterization positive algorithm message pass interested map specify potential compatibility potential px nx j x j x marginal mean I ij else exact variance estimate correct variance condition guarantee generalize dominant current
always f split case assume fused break apart want one indeed show equation know know come go either source capacity flow therefore flow kl equality propose solve want fusion finish ef group valid end trivially set hold fusion converge merged conclude possible infinite cycle split converge
agreement dot propose learn refer widely image promise matrix randomly orthogonal column decomposition svd dictionary set corrupt additive burn estimate matrix iteration performance express reconstruction root actual noise estimate vector zero non zero monte trial depict bayesian analysis k reconstruction combination mix provide source source svd implicitly fix match pursuit omp unsupervise framework propose complete fraction pixel depict decompose patch reconstruct
acceptance however trial suggest intuition seem incorporate proposal acceptance proposal certainly leaf correspondence accept move instead proposal parsimonious variable flexibility capture input label suggest reasonable mh gp region pz x rx distribution set eq slight distribution generalize definition role distribution formula r sample mh begin parameter acceptance z rx rx rx region employ scheme whereby keep propose mutually exclusive block iterated conditioning rejected process yet iteration trade small acceptance poor propose individually change region sensible block leaf likely class predictor small softmax generative label recall zero class record predict label round monte take vote upon summarize posterior proportion class obtain full accounting fully fashion real categorical context
request agent execute task add queue execute queue delay euclidean unit interact request request send indicate agent take step main service average request receive task request network request
multiplication require computation svd outer truncate svd upon multiplication although multiplication also cost multiplication cost meaningful add multiplication dominant calculate truncate svd calculate svd cost rank cost truncate large package request singular value fall current number
solution way instance fidelity consequently might limit deal construction estimator extent goal interesting bias spirit interestingly framework would overview probably point explore many research information code sparse compressive sense latter try admit parsimonious programming author oracle inequality describe view regard estimator application sake make fact briefly always concern problem concern estimator quadratic risk presence state shall remainder namely however identifiability inverse shall idea help overcome many engineering application risk reconstruction scan brain patient matrix image device image vector reconstruct diagnosis interest illustrative problem recognition library interest curve etc stand depend application eigen eigen shape wavelet stand combination vector predictive good prefer subsection risk say organized term paper help penalty practical devote numerical application extension paper typically collection estimator parameterize select
fold display influence exhibit come tendency select influence causal dependency ccccc true glasso short outperform significantly regression noiseless yet advantage influence roc curve roc curve dense interestingly model cccc white glasso
maximum margin hyperplane rely core adapt streaming pass train pass stream first storage make current ball require approximation adapt margin classifier stream suitably center core input result verify center old radius center result ib close construct compute bind bad upper bind adapt update select point belong
common study locate keep exclude participant total snps meet criterion minor allele compute allele retrieve project difference fig explore sample misclassifie create independently pair bernoulli allele control allele allele frequency explore create begin draw simulate draw simulate construct percent snps choose identical individual fraction increment identity individual eqs step create sample individual simulate percent increment generate create sample snps sample summarize introduction sect sect snps allele additionally use sect analytical follow sect realistic computation various distribution figure yield nominal yield vast majority test neither misclassifie member group rejection also threshold sensitivity sect finding sect characteristic choice control minor allele
noise dependence consistent empirical goodness long memory mention discrete nan series et martingale use domain approximate article distributional heavily rely conditionally martingale decade error al model crucial specify misspecification misspecification diagnostic model unknown li li li bp type test assume knowledge diagnostic methodology justify long article unobserved error notation vector p tc denote denote follow distribution nan discuss observable noise proof assumption certainly need causal measurable function contraction wu wu wu nu exponentially bilinear wu min wu besides model difference martingale satisfy bilinear q iid wu z wu nonlinear move average p differentiable number long technical chen decide retain hold pointed moment parameter assumption unable relax view asymptotic statistic lag
shall see complicated theorem posterior convenient popular kl laplace sequel closely relate laplace result effect law mn enjoy dual dual strong shrinkage result regularization look effect regularizer mn equivalent prediction examine integration show laplace exhibit shrinkage laplace k integration k z eqs k eq plot reveal red curve shrinkage toward obtain solve optimum mean point since gaussian mn sparsity kkt imply uniquely shrinkage sparse comparison posterior estimate versus laplace prior shrinkage shape smoothly mn relate mn go norm mean laplace nearly mn explicit compare mn region regularizer bb turn point kl smoothly intuitive explanation kl figure move close norm discriminant function smooth mn offer close lagrangian estimate easily laplace corollary difficult optimize laplace distribution layer exponential admit hyper p straightforwardly follow eq jensen upper kl necessarily kl alternate algorithm detail iterative optimizing outline detailed tb input respect get ia analogous dual
solution suppose terminate satisfy accumulation accumulation observation bound replace x k contradiction conclusion discussion solution problem equivalently converge however slowly large lipschitz frobenius respectively thus converge slowly directly aforementione next equivalently adaptive method terminate go establish solve solution equivalently first clearly terminate total terminate clearly method
player difference great value table preference saddle opponent white h black difference learn game decide close highly tendency white attack normally black opt white opposite pair factor h white side preference central preference side preference indicate preference keep safe white feature iv difference indicate preference absence subsection indicate preference absence c difference preferred move average sufficient capture may high apart balance fix value value normally nevertheless feature
take account tune essentially number regret accumulate define tune fine tune cover specifically complete mab metric induction hold prove general ball support ball constant function ball randomly add subtracting assume ball chance ever ball statistically fraction ordinary ball fix cover ball infinite ball mutually disjoint construct ball pick sequence interval follow support sign define payoff note instance subset uniformly consider value function element eliminate filter whenever play probability jj sampling tn pick strategy bound condition eq imply bound equal select strategy round demonstrate account index contain expect bound probability finitely r jt k tractable full lipschitz expert notion describe space notion relevant notion restrict obtain regret analysis cover minimal cover diameter sufficient cover essential expert metric broad trivial trivial root node child infinitely deep common uniform branching cover finite diameter wasserstein infimum wasserstein distance define probability widely context retrieval log cover diameter whose wasserstein
etc framework take factor traffic stationary vary poisson model forecasting define definite would transformation www traffic effectiveness www traffic time vary role value arithmetic calculation combination solve calculation www tool example www traffic validate discussion discrete paper depend hereafter discrete www request focus
f minimizer move one stop edge occur state terminate minimizer smoothing proof contain start start tell penalty order place constraint whole region preserve close close active happen step change enough subsection discuss change possible happen merge region splitting associate also active algorithm appendix contain complete stop split place
multiclass regret may label balanced leaf reduction binary probability label hard solve operate loss case concern question multiclass binary class basic leaf predict noise root preference utilize multiclass multiclass bound binary bottom single elimination player elimination control somewhat surprisingly single player player play twice simultaneous elimination follow carefully elimination tree time multiclass label construction robust lie set previous work fail adversary number budget error always comparison repeat game analysis error player severe bias elimination elimination player player win play
poisson quantitative assessment slow decrease mention monotonicity comparison decrease blue black derive provide
form random square gaussian mean eigen put concentration
p ps relation application al notational stage going write another evidence account convergence estimator gaussian express nh analyse yield I tn last hand side term inequality ergodic imply relation relation obtain q convergence obtain study order analyse three
show prove chain spread exponentially empirical seem retrieve chain keep mind numerical apply estimator size c c c c c consistent regard secondly size c c constant order
update get formula separately n due factorize perform document separately derivative cm label difficult compute second order taylor expansion py py tr var derivative evaluate cm taylor expansion choose introduce effect shift nice normal extend derivation variational normal see step log partition function involve reasonable max margin however problem use beneficial high order factorize optimization svm lagrangian cm last fully factorize perform document sub plug get dy shift mean nice shrinkage derivation variational problem program exist solve estimate step see second learn lda py py point estimate seek high cm level z approximate factorize define follow dy develop co follow normalization step get update optimize fully factorize separately cm derivative second var py py tr var var derivative evaluate derivative get
irreducible split construction borel depend residual eq unless symbol block simulate differently actual avoid denote practice indicator eq initial part trajectory scheme absolutely borel quantity assume sum estimator fix define basic trajectory excess square error compute q q asymptotically assumption necessary fact prove appear excess k identity claim time classical theory symbol start invoke elegant conclusion quantity
observable characterize system rely tool reveal computation important notion random denote base bit express intuitively variable assume content notion distance log px qx p difference bit code kl expect length code wrong code relate px summarize maximum remain observe deterministic uncertainty provide quantity call little algebra interest randomize protocol tuple mutual
variate vector degree location product hence prove q coincide conclusion define far author numerical classification either real dataset approach implement avoid optima organized subsection case medical area carry see aim unit sample variable obtain accord lambda use variance within scatter index misclassification percentage classify wrong density aim gb g figure summarize cccc lead separate classified model contrary bad misclassification rate
respect subtract may generality bregman similarly subsequently scalar variant nesterov smooth pt pt go regard nesterov smooth dual variant nesterov typical sequence theorem variant nesterov variant smooth exceed state satisfy find differentiable strongly modulus u function nesterov solving address describe context prox subproblem nesterov algorithm multiply result u p norm nesterov saddle done indeed form f frobenius modulus proposition prox nesterov differentiable modulus easy nesterov dual exceed apply dual iteration exceed use relation conclusion
everything estimate large size case small intercept e intercept case term remove covariate difference treatment level method usual analysis adjust homogeneity adjust fit ignore level adjust influence overall parallel test mean value covariate extend specific simulation indicate similar sensitive difference hypothesis test treatment line range covariate adjust give symmetric distribution give situation heterogeneous different covariate range residual treatment covariate similar covariate line covariate adjust extremely whereas nominal level covariate residual covariate get close difference treatment exercise ignore grouping overlap covariate advantage spurious conclusion
lipschitz supremum constant constant condition satisfy analogous assumption ensure satisfied establish connect tail utilize hold sure divide fitting regression type misspecification preserve specifically interested regression summarize sure screen marginally consistency jointly marginally answer reveal easily orthogonality condition orthogonality essentially linear sure important hold basis independence screen hand partial orthogonality marginally select version signal strong follow b bound constant poisson light theorem effort assumption implication many final model covariate jointly normally distribute simplify jointly monotonic addition covariance part proposition monotonic regard direct positive positive word suffice establish sis property
grow network sparse assumption lasso present study parameter appendix denote simplify diagonal value outside pa bn remain effect receive parent restrictive neighborhood neighboring indicate parent node node propose violate show weak refer initial assumption neighborhood well require variable selection consistency relaxation lasso special section adaptive precision well random additional mild variable constant pa control error pa estimate argument present minor modification replace conditional separation next property lasso without consistency sparse cholesky completeness simplify remark element adjacency ps estimate replace pa bn
mle case ii eq variable great location value shape ii censor censor type I great
mention principle minimize likelihood likelihood play role decision consequently always sampling encode coherent method give hypothesis estimate principle coherence constrain two inferential value coherence coverage tend order simultaneous frequentist controlling report normalize uniform measure frequentist situation reasonably confidence measure degeneracy availability achieve either transform confidence approximate remark concern rely solely inference making enjoy utility hypothesis whether reduce measure argue minimize expected respect within outside dominate x xx dp rule convex automatic reduce also recommendation family entropy kullback binomial gray indicate confidence notable irrelevant situation half correct confidence also level record agent payoff difference situation involve hypothesis agent x odd act accept odd would benefit gain unless equal accept function minimization degeneracy confidence
random otherwise go go return parameter desire b easily amount
convex word find extension even every crucially half applicable algorithm problem ball make clearly convexity try hope efficient rotation algorithm naturally set rotation orthogonal insight gain solve optimization begin explore believe appendix derive result begin show satisfie induction update imply update convexity eq put bound obtain plugging qp assume also otherwise solve nonempty
mention moment process sequentially traditionally principle prior bayes alone simultaneous actually question solve solve solution rely realistic method need assumption recover deviation concern posterior conduct wish something piece know relationship since concern product attention must focus maximize prior call explicit important prior traditionally think prior information represent already piece express posterior posterior fact
indeed widely study zhang ml demonstrate well near performance range mention noise improvement numerical multidimensional trace field refer reader say measurement stein simple squared multidimensional appear wiener denote signal ratio whole apply replace standardize interesting whether similar near efficiency result still elaborate estimate base hybrid zhang likelihood easily solver
distribution unknown hoeffding universal universal leibler hoeffding k divergence result test alternate lie test certain alternate shown exponential hoeffding test size involve alphabet advantage due statistic nan test kullback evolve wish say characterize observation favor marginal probability relative entropy leibler observation universal testing information available regard propose testing q alternate suppose alternate lie parametric specific paper hypothesis interpret similar refer divergence terminology within framework view hypothesis identical consequence also establish advantage finite size hypothesis show divergence alphabet
within analytic set thus detail k cauchy contour dominate r fy I hoeffde proposition regularize ex chi department model bound
separated phase regime map ml agree become influence performance characteristic transition avoid machine translation one sufficiently motivated map exponential number whenever ml might implication instance application translation top solution accord heuristic need careful might nearly solution choose directly notion intuitively define ability accurately fact al weak entry emission characterization equal space sequence quantity entropy x intensity
term latent root condition differential root hermitian hermitian verify polynomial invariance put analytic play j k ij I matrix assertion real high let eigenvalue
regularity fulfil dot derivation substitute rewrite formula mean derivation w shift parameter converge limit negligible suppose note mle situation kullback regularity f alternative suppose interior
key cm cm cm cm plain work extend various bootstrapping residual give consistent apply regression norm hierarchical norm generalize locally locally w w multiple moreover extension connection resample boost use estimate interest finally application resample signal processing remain explore pursuit pursuit greedy selection distribute moment derive another lipschitz lipschitz lipschitz constant note improvement improvement selecting give among use leave like appendix design function real subgaussian subgaussian exists normally hoeffding bounded variable amount subgaussian also quadratic form independent subgaussian eq positive r lemma sign j j jj directional derivative along sign far j optimality relate appendix continuity short calculation invertible set result p
strong summary assumption unbounde around fall practice disadvantage live outline tend vanish refer abc glm glm summarize glm give value retain aid abc distribution tolerance glm estimate truncate formula look two review computational genetic tend correlated function reduce analysis mahalanobis identical advantage reduce summary retain sophisticated statistic pls fix simulate summary
episode neuron influence neuron influence episode however neuron delay motivate characterize give brief explanation communication thin neuron electrical impulse across side spike randomness furthermore refer delay ten propagation potential spike spike frequent episode neuron occurrence episode event represent episode event episode spike serial episode inter event constraint fire neuron delay neuron call fire sequence detect sequence spike general delay neuron specify inter assume delay delay frequent episode discovery generalize serial inter specify user specifie
sequence main ratio namely side respect probability something logarithm good whose standard interval grow carefully half logarithm simplify trial test measure obtain previous figure vertical logarithmic red would conclusion unknown compare composite hypothesis nan uniform uniform context martingale turn law iterate almost upper confirm intuition side long enough increasingly walk universe depict value de none reasonably typical setup evidence hypothesis point seem likelihood avoid observation likelihood ratio ratio affect plan plan probability coin stop identical odd evidence decide calibrate wrong acknowledge model wrong may provide evidence anomalous phenomenon continue probability want hypothesis special g make odd turn observe
implication towards space monte paper marginal quantity approach seem naturally provide close denominator estimate output et
leave observation shrinkage site change drastically scenario drastically plot even though size keep level heterogeneity overall pooling tend comparison consider pool toward common posterior variance toward mean decrease statistically bayesian become likely score estimate pool zero adjustment posterior deviation go frame improve ensemble conventional bayes provide precise barrier sure fitting model way software hill fit package package treatment site treatment effect hill fit available well appendix hill site site site illustrate arise comparison national education statistic national assessment mathematic make mathematics fdr many comparison statistically theory average score display well concern multiple fdr appropriate ahead hypothesis difference reason motivation state regard model average state test score year type error rate concern although exactly nan weak classical
control observe around straight conclusion approximation rare moderate tail straight tail tail straight p table increase approximation size assess kolmogorov von function von cm empirical total size million computationally intensive especially replace approximation simulate sample compare performance size table replace permutation slow deviation distance result table converge approximate component contribute increase approximate decrease distance stay appear one preferred procedure comparable reasonably hundred conclusion similar table parameter approximation table well size around preferred counting empirical check around von distance indicate predefine moderate moderate know significance ideally true
differentiable point interior domain augment lagrangian hinge discuss apply hinge slack variable equation rewrite non q exchange explicitly maximize follow similar algorithm analogous broad function auxiliary loss regression define write example h h formulation optimize objective add term eq solve exchange remove eq nb similar thus use minimization lagrangian prox criterion newton computational norm c necessary active kernel c reduce considerably fortunately th files matlab improvement overall parallelization motivate sparse formulation mkl mkl elastic choice previous formulation furthermore efficient elastic mkl consider generalized block formulation nonnegative function example norm mkl give elastic
shrinkage extend consider expect sparse transform make cluster wavelet interaction eq cluster cluster density discuss past help specification lattice coefficient decide two adjacent make total nine coefficient illustrate time frequency examine periodic boundary black frequency counting unless row immediate variation
formally maker determine action vector opponent decision maker variable state illustrate full observe outcome propose state sum tracking instead compete good switch restriction loss task infinitely interval natural connection describe handle propose set task linearly consecutive task similarity action consecutive game far away interpret internal coherence maker assume set logic maximal action task order action higher favorable maker maker high rank need receive within simultaneous action must order maker maker stick action shift budget associate cost
natural measure true environment term theoretic kl divergence environment extend output extension special calculus exposition discrete interaction construct call string length denote string agent formalize
rip stability hold spike separate iid toeplitz know ensure toeplitz sense rip toeplitz construction simulate depict figure reconstruction spike well filter process fix fix minimization show toeplitz bernoulli preferable toeplitz govern equation closely interested note bernoulli experiment ar fix nonzero component ar process sense matrix bernoulli sense toeplitz outperform toeplitz article corollary thm thm definition electrical computer edu realize drive impulse operator compress reconstruct suitable random projection turns indeed reconstruct lp reliably blind de convolution filter naturally
behavior highlight factor aid learner improve learner payoff mistake information publicly learn acknowledgement ef ik grant grant fa ef part nsf proposition non section systems agent act effectively find game converge efficiently nash equilibria anonymous game good dynamic two convergence beneficial many provide agent distribute unable behavior user preference agent strategy however large system agent choose fast user period thousand million convergence sublinear agent provide polynomial exponential number mistake agent payoff agent effect message delay agent learn need find characteristic make involve agent seem hard interaction advantage typically depend agent less useful
convergence walk mcmc describe section property include membership diagnostic markov consider begin last compute z move metric identically converge number fall interval monitoring sequence node stay long enough reason intuitively speak diagnostic chain empirical sequence convergence bit explanation propose view friend message friend page basic uniquely change bit name core network facebook always accept side thus undirected network user join one allow twice network email domain join uci edu email many privacy setting restrict information reveal possibility four refer result setting facebook set friend web page user network membership collect friend additional profile potentially profile display friend collect profile information available method enable statistically elaborate topological measure purely view decide collect interested node rnn extend collect whole population k randomly select l rw total valid k neighbor avg degree median degree rw rw uniform uniform life explore systematic mechanism automate mining per ip facebook traffic call restrictive force modern site enable asynchronous loading web require sophisticated satisfy overview limit memory ram disk collection parallelism machine run share server control number connection track already iii maintain queue account manually
q construct rotation rescaling transform rotation fourth rotation rescale matrix long neighborhood unique order velocity transform scalar imagine quantity system prove substitute transform tensor kk kk kl satisfy solution permutation reasoning equal element permutation permutation necessary scalar separability involve measurement describe methodology multidimensional separability imply transformation velocity correlation product correlation independent source number vanish velocity correlation source
necessity choice multiply concentration concentration choice proposition generality distance dependent marginally sequential invariance dependent crp distance identically marginally sequential decay marginally sequential decay another invariance table absence presence customer customer way table customer customer customer customer customer way link occur first customer customer link occur table marginal consider may satisfied root possibility describe describe result one customer distribution marginally quickly corollary decay distance crp marginally distance sequential necessity probability induce alone marginally enhance concentration accordingly crp customer note customer assignment prior generative z generally sequential follow special window sequential case operator identity traditional np posterior gamma distribute exactly gibbs entail transform variable hyperparameter window rate function section use hyperparameter decay since conditionally case depend function simplify decay normalize continuous might approximately
canonical model poisson indeed canonical link function property contingency margin cell margin cell boltzmann therefore acceptance property hold positive normal write implement proposal independence hasting follow exp rate rate col code figure histogram mix behaviour random density maxima histogram mcmc mean direct metropolis hasting distribution metropolis proposal independence hasting via code sampler evaluate output average lead something range mcmc mcmc valid type remove carlo replication run output hasting biased difficulty converge stationary sense range sampler mean iid walk cauchy prior density check walk metropolis code sd df df df check sigma n sigma rt mcmc valid show acceptable small sense leave window medium scale induce acceptable mean gold use cauchy noise leave right mixture five walk variance file txt sigma pick walk normal center figure book histogram explain nothing top every middle density bottom function condition pair distribution proper proper therefore inner set eq whole follow go infinity include intercept probit version discuss intercept column matrix code I type main lag autocorrelation ci bank intercept book intercept posterior covariate bank magnitude covariate
take parameterized times proportional come draw e index either force precisely pe particular chance must could chance choice agree chance agree subgraph unless degree z z even degree find even degree everywhere edge part value ht
bind analysis upper rate dependence k matrix collect vector supremum proceed supremum relevant packing generalize low concentration step show fixing pack disjoint radius pack agree set cardinality consist claim step convenient every element k k ts satisfy van process n prove claim next combine control correlate intercept covariate b nj p kk covariate contain intercept design lemma allow uncorrelated design lee thorough read paper detailed comment would thank anonymous would thank participant quantile regression conference university mit american american business school conference college theorem conjecture comment etc j bt version acknowledge national science foundation regression high model regressor size regressor grow slowly quantile regression qr post penalize qr qr apply ordinary qr condition qr uniformly compact index drive penalty post qr near uniformly qr rate close condition model select select uniformly quantile response capture regressor exhibit outlier excellent property asymptotic
closure variety need consider segment join construct get effect definition kind first note effect invariant follow list binomial term case theoretically markov basis show markov basis effect cccc cccc move cccc move ccccc analog save complicated fact partial result define
possess oracle practice likelihood proper choice selection estimation wang select penalize scad scad able consistently set lin tune consistency investigate scad select yield identical tend cross validation demonstrate include greatly comparable computationally intensive validation organize penalize likelihood section tune parameter bic graphical selection scad adaptive tuning bic cross penalize depend maximum xx I center
x pe see use segment edge vx xx fall boundary region mass arbitrarily edge opposite line vx vx xt rx orientation proportional rx z rx rx ir pe rx pe xx pe rx r rx j dimensional implie fall boundary mass note convex ht xx xx arc pe rx pe pe rx pe pe pe rx j pe rx r nr pe pe pe simplify central thus order paragraph nr pe nr pe vx pe pe pe px pe therefore lebesgue pe pe px pe px pe r arc h hx ix ix hx pe pe rx ii h explicit form mean hypothesis form spatial thus size random geometry invariance result
also q see design sequence proximity initial tolerance augment lagrangian eqs stop tt final newton al prox correspond zero equal general expression note eq proportional element thus let disjoint proximity regularizer obtain moreover analogous soft operation prox operation compute hessian regularization convex norm support group generalize define indicator projection onto lemma finally compute envelope let prox sup prox sup prox regularizer section group regularizer component set thresholding operation regularizer soft identity support envelope prox regularize word update write minimizer show proximal minimization impractical minimization tolerance present rate improve inspire partly however objective check exactly early minimize nx j line arithmetic eqs minimization generate eqs let objective proximity parameter increase converge denote solution c substituting sum side proximal minimization claim convert result norm residual generalize minimizer us denote objective minimum distance minimizer w inner minimization solved exactly follow
framework value despite risk assess framework linear yield expansion decrease variance imply asymptotically asymptotic closed variance quantify explicitly simple population shift loo large make computation empirically minimal loo stability show simulation distribution estimator test bias histogram assess cv procedure unbiased risk conclude cv procedure bias asymptotically good instance model estimator loo cv typically grow belong second explain risk beneficial third small probability near specific cv model efficiency framework directly cv make good cv cv depend asymptotic cv aic multiply loo loo loo bootstrap density show multiply risk framework cv generalize selection estimation model rigorously every empirically regression relate candidate loo framework cv confirm naturally select among despite constant penalization coincide section loo suggest
interpretation consideration much select propose perform principal covariate mixture model top number approach include difficulty argue cluster adapt originally jump discriminate model dirichlet dp global attributes simultaneous mean shift improve significant signal noise lead reduced dirichlet dp mean gamma usage shrinkage inefficient utilize embed step
exploit show favorable poorly interest natural inverse problem eeg localization deconvolution identify much valuable explain predict neural machine consider arise sparse matrix element pursuit denoise brain imaging community efficiently
vertex intuition mention graph approach empirical likelihood control fluctuation cause limited term add low r extension estimate select first show indeed sufficiently exist constant exist exponentially success converse reasonable old graph compose
queue reflect pair line sample pair solve remove pair maximum queue item pair pair queue stop increase search least lift fully implementation detail architecture focus table operation hash table space distinct pair similarly base count input since occur transaction good update require hundred unless cache fraction spend well update approach lie hash use hash table essentially speedup applicable filter reduce artificial mining repository three internet avg pos actor connect uci click stream pos contain see market store traffic dataset generate generator actor set rate
weak prior unconditional rp band star common star star likely highly likewise star latter prior much well estimate want ap detail give star around ap estimate measure triplet nm independently additional finally survey incorporate ad hoc system address goal survey desirable maximize scientific replace less bp rp ultimately forward adapt ap weak h simultaneously practical assume low fit forward section bottom h considerably correspond tm star expect snr model extra variance three extra noise ap bring performance determine two worth point degeneracy mapping degeneracy g distinction approach forward modelling weak smooth unlikely adequate fit lot structure dimensional e newton intend overcome non direct finite issue datum much automatically division independently smooth spline forward naturally actually goodness model low resolution rp summarize limit well entirely accuracy model h star estimate course study average star systematic correlation strong wide range accuracy estimate remarkably variance also affect statistical show intrinsic rp addition report
new variable dependent random prove measure exist apply consist analogous ball support well standard example exist ms cover set obtain gaussian need reasonable bound hold v equality ex h lemma bound kk large understand use k ks u ks prove gaussian h optimal selector satisfy cone hold selector show
c xshift yshift right right c e background width r center width cm text come style circle font plus xshift c xshift c center width cm text text center candidate font inner size e leave b scale leave b c c e xshift yshift leave b leave width center width text come false g rl rl rl e k interestingly number episode enough check node episode sub episode serial episode episode general episode example denote node obtain restriction node maximal obtain obtain dropping node maximal node frequent frequent describe frequent level ensure order set type denote episode frequent episode frequent episode share episode drop episode episode serial episode episode candidate combine candidate note share dropping event three drop event drop candidate episode block respect event block episode combine store episode episode place share event candidate episode episode give rise block episode common episode drop naturally construction episode block sort array type array type episode event type block episode appear since belong may appear later candidate list episode episode organize episode appear array store episode episode episode line episode start try use procedure identical return function candidate equation three possibility closure validity separately possibility save explain decide generate
case difference point particular value hmms easily check parameterization sx recursion recursively coincide eqs practically equivalent indicate limit behavior recursion filter use step potentially limit may sufficient consistency option use work batch mention avoid hmms interval average see regard complexity sx bring operation x j interestingly observation iteration comparison directly meaningful several respect constitute model say gradient equivalent main burden come necessity via recursion state value transition low dimensional cost
case reduce configuration locate ask expect entropy inequality phase r object want find sequentially store see far every oracle close current minimum minimum ask learn know object center htbp object retrieve neighbor times mr ask oracle close hence distance away vice versa need constant object w u r reduce reduce number level time expect chernoff take union hash object analogous locality sensitive scheme neighbor evaluation hash value near search address nearest neighbor near object contrast access object live oracle give object attempt human user capable statement similarity assigning object hope sort object call object difficulty oracle depend inequality rank speak define inequality violate build
evaluate different available notion formalism repeat high observation testing order restrict margin broad margin derive learning theory bind learning algorithm definition algorithmic contribution novel finally mainly output space pair measure set must prediction measure outline step predict set paper confident
score consistent score proportional degree freedom forecast whose compute numerically degree forecast carlo continue multiply acknowledgement author huber discussion reference financial von national science dms university tucker survey express author mm spatial quantile statistic letter coherent risk finance taylor measures international journal forecasting evaluate forecasting principle forecast ed error measure generalize forecast international journal finance wang conditional expectation predictor predictive market reality check forecast accurate finance journal loss classification paper evaluation survey journal scoring pp geometric quantile american journal institute mathematics coherent dispersion business forecast management international overview journal compare prediction probability business economic united york r forecasting international journal management forecast operational journal operational journal business economic journal statistical optimal forecast report quantile international journal forecast proper scoring journal american association hold assess forecast application ensemble prediction surface operational w h economic measure forecast forecasting decision interface college pp quantile j wind journal american scoring c functional incomplete huber huber
constrain number formula covariance maximal order occur pair partition frequently within one tree order tree size maximal product tree small covariance thus tree set covariance tree size denote tree ex u u pair way partition variable one corresponding contribution definition ex covariance sum u k product tree hold partition show n
significance threshold incorporate threshold another source bias snps effect rank estimate biased complex correlation due linkage rank effect allow snp correction however limit I provide fall within clear frequentist significance analysis estimation consider note current association prior move part computational incur inference full acknowledgment three constructive suggestion substantially improve dr association study type institute grant engineer genetic ed follow sample significance threshold propose spike prior possibility chance method testing average procedure bayesian yield likelihood outperform four odd ratio genetic snp
base careful typically testing decide whether reconstruction enable adaptation sparsity check stop thus unnecessary beneficial pooling randomize statistical start drawback limitation sequence technology first heart cs problematic application expect certain long beneficial cs preferable frequency take pool reconstruction pool target efficiently genomic efficiency could maintain large coverage increase future sequencing technology pool difficulty relate randomness discard instance sense randomness view enable pooling scheme carefully design drawback mention contain half may problematic might slow minimize pool pool advantageous assign pool pool design approach experimentally purpose adapt simulate experiment beneficial high coverage thus design suitable benefit cs variant amenable detect next generation sequence technology population interest copy important studying serve reconstruct copy level rather another give present evidence read head map genomic tail map two pair end read genomic rare discover extension read introduce procedure
therefore naturally ad hoc turn comparison distinct share express parameterized function statistic arise scale locally proportional shrinkage test compare improved performance covariance different shape mse simulation repeat shrinkage plotted coefficient small marker figure covariance structure ar entry fig estimator fig
setting find nature combinatorial amenable gradient procedure optima measure neighbor zero otherwise know window function positive evaluation normalize sum sum smoothing ridge kernel except hold mention follow result matrix ordinary regression ridge instance integer eq example ia prove alternative classical spectral inverse cut reference estimator review selection usually select choose family goal therefore drive satisfy constant negligible selection unbiased estimation minimize eq cross heuristic penalization consist form estimation heuristic deterministic covariance imply drop note dimensionality classical penalty lead
obviously contribute remain consider item common available accept vs ir b ib mean get brief proof demand order get demand total number input select cause happen decision claim get complete general program decision section tie arbitrarily use assume prove tie argue discussion proof provide distinct distinct lemma exactly integer competitive compare online competitive online program hold linear program apart offline large program rigorous achieve approach reduce randomly competitive problem random arrival objective distribution input dynamic dynamically update threshold price vector interval current online problem research hand tight actual
method time affine add subspace become work clean outlier work affine make mixed dimension algorithm artificial hybrid rate misclassifie eq pt pt c kf pt kf kf mixture kf local spectral curvature voting matlab kf http http www algorithm http www vision db paper default
assume bound prove explicit hence deviation appear summarize deviation control find whenever universal least lemma almost surely satisfied mn fix almost surely population surely stop consistent covariance cross difference solution discrepancy covariance closed knowledge heavy pls early connect context stands study
inferior fine extension pde wavelet correlate hard denote noiseless denote traditional filtering think pde square standard tv image denoise estimate minimization eq function control tradeoff fidelity image somewhat variation adopt study theoretically pde base image belong space existence establish tv discretization implement continuous formulation tv competitive denoise ideal piece point partial first difference inherent tv
denoise ref also evaluate denoise lie wiener snr different display wiener mcmc snr quantitative mmse well wiener variational approach latter hyper constitute bring interested image observation distribute translation invariant transform decomposition resolution fig noisy image radius fig illustrate denoise mmse sample ht snr db propose kind representation variational approach c mmse ht mmse mmse estimator mmse signal signal perturbation introduce signal prior hyper hyper
kullback leibler divergence domain use distance proximal proximal penalize log likelihood follow implementation r r r penalize point let r th rp nonnegative nonsmooth lipschitz bound subdifferential set standard mapping allow mapping use operate bivariate analysis assumption locally subdifferential lipschitz differentiable locally domain iii convergent nonnegative mapping
huber minimize contamination vice versa huber partitioning range yield contaminate test evaluation censor huber occur set disjoint begin unity unity log large test trial identically distribute directional semi supervise fix significance nan side tail side alternate p n appeal investigate efficiency reader summarize model selection well likelihood minimax efficacy testing suitably standardized statistic hypothesis limit size test efficiency relative twice large computation test strictly label give appropriately standardize efficiency square evaluates imply slightly scale asymptotic relation exponent gaussian distribution confirm density would twice large limit demonstrate guarantee amongst optimality long necessarily retain several
rewrite set projection r u subdifferential case equation equation notice open induced set set lebesgue almost negative diagonal property diagonal nothing imply eigenvalue optimum duality negative outside nesterov cholesky factorization sdp allow write transpose cholesky random cut sign coordinate sample good probability make nesterov remarkable fact future research hope nesterov main drawback former presentation uniform motivate eigenvalue column may require optimum reduce unique binary turn unit
positive negative part subsection give derivation formula ls concept remark even though discussion manuscript count report generalization perfect graph solution derive q combine formula consider expand rh matrix expansion ls expansion l step consequently relate determinant l determinant obtain undirected edge matrix bipartite x l ls diag
provide implement test sequential sampling regard distinct move loop write know complete independence contingency move condition part part loop connect table fix sum extension extensively rate aspect article aspect table v model coincide model sufficient case test connect basic move connect model write sufficient independence free move basis connect table square three via ti two include
storage landscape massive amongst become ever new computational become necessary part treat unique pose modern us notation issue illustrative comprise contain temperature particular period day would component dimension amongst intuitively hard imagine say speed decrease level wind small know value mean contain nonnegative eigenvalue eigenvector associate yield variability analysis panel wind reduction refer relationship sample three appropriately semi describe serve laplacian cardinality positive integer form definite entry matrix evolution diffusion
derive give interaction unlikely construct base fashion tractable mechanic anneal fashion gibbs computational sampler optima example change mechanic cluster assignment draw label label complexity sampler summation choice mechanic replace define omit require sa different expensive less label significantly affect good
point wish state evolve observe lie state express compute overlap gaussians determine maximally point point note evolution quality time whole analyze variation density apparent inspection plot combination need visual select gaussian essentially linearly expand display stage different examine colored plot structure belong follow distance exploration involve perform svd coordinate feature within also employ subset effectiveness use filter visual easy expert method select feature conjunction happen apply et cell patient type expert datum divide experiment expression
interest laplacian visualization reveal soft frequentist ss turn offer improvement thresholde gaussian cb std mse truncate laplacian cb median truncate fix univariate wavelet angle thorough comparative evaluation several repeat estimator outline well review technique experimental level invariant decomposition implementation employ retain propose alternative diversity possibility wavelet multiscale comparable prototype correspond corrupted version scale hard soft universal cs denote correct indicate multiscale tn multiscale multiplicative approach ss db std std median max ss median
e j n I htbp cm reader illustrate asymptotic allow interval compare ratio reference weight cc cc cc n package consist six design except want confidence object vector alternative hypothesis side default reference default contribution
q solution unbiased approximation sequence differ involve simulation although superiority embed briefly model generalise diabetes study probit diabetes predictor concentration pressure diabetes assess diabetes e coefficient associate rely probit probit latent
general prevent computing inspire randomize avoid recover notion definition notion though sometimes representation real borel papers notion computable recall less call great e co space topology suppose countable topological every yx xx therefore topological space computable topological count topological space countable enumeration possibly sm sn topological derive often enumeration b sx notice open space set correspond observation topology computable instance section name computable topological enumeration sn sn computable name similar name usage able computable functional enumeration implication computable partial computable computable increase domain eventually every informally use enumeration name enumeration computable situation establish may refer implicitly relate topology computable topology context order computable enumeration topological standard enumeration computable topological enumeration enumeration give
q reduce segment q reduction th calculation slightly simplify q u claim point remain lie right hand u du value solution theorem also distance shortest pair solution svm vector point uniquely every possible previous word subset set artificial problem must optimal hull tell uniqueness part occur also low path let bend straight point relative confirm finding cube cube h dimensional em vertex directly exact arithmetic programming solver dl vary bend
aa sufficiently loss probably aggregate one forecast case outcome aa theoretical aggregate learn prove cumulative expert expert equal ht read prediction g w work description goal description extract procedure match control ca diagnosis intensity peak diagnosis date take date percentage align peak peak purpose
interior domain interior strictly ok twice fu fu fu fu r norm let prove smoothness plugging prove I r k claim school university university body appropriately sophisticated describe construct statistical decide methodology duality conjugate smooth respect dual analyze potential framework method incorporate recent describe systematic method construct strongly function already derive utilize generalization bound online matrix derive property problem help decide adequate framework derive novel learn tackle learn problem efficiently sophisticated form knowledge share feature across task central understand restriction impose nature impose
formalism accurate se actual formal well true false alarm ti x ti experiment agreement actual dimension se numerous property false measure evolve match formalism success predict indeed empirically mp tune reconstruction provide natural tune mp amp choose achieve eqn refer sometimes mp amp short phase transition tune mp se optimally tune mp evolve se formal evolve property show supremum lie mse
sake completeness simplify presentation sensitivity sensitivity section prove certain anti concentration bound invariance anti hypercube prove anti concentration bound distribution anti concentration distribution multilinear polynomial multilinear work hypercube first bound sensitivity anti bound noise sensitivity follow negative bind expression I follow concentration regular interval length let interval sensitivity regular degree f applying exist kp bias towards np sx ns lp give x
mse three posteriori equation hereafter concern specification use datum poisson example largely insensitive knot allow prior location choose range sort value respectively spline basis formulate instability g trade quickly interval computation prior burn follow update effective update increase number iteration much calculation perhaps quickly fit map use burn edu bar size competitive particularly marginally example generally
bootstrap assess significance create drawing independently distribution prior distribution draw distribution probable record experiment increase experiment actual estimation user subsequently user posterior user prior adversary show biased outperform biased model encourage access use organization organization scientific half month include initial datum include three date access discretized hour per hour
wang associate valuable comment tend monotonicity subsequence assume moreover assume perform infinitely continuous letting monotonic inspection unless theorem global maximizer fast new numerical share simplicity dramatically improve speed improve strategy act global effect multiplicative extension maximum keyword hybrid design approximate encode explanatory extension strategy classical
room likelihood information qualitative inferential reliable selection study sound available close many biology abc employ grateful comment member biology group version grant article bm graphic division molecular institute mathematical sciences college uk sciences important hypothesis exist reality require suitable allow g signal particularly biology
observe nan positive negative equally precede technique deviation see deviation symmetric see clear structure inspection interval include sign quite unlikely occur posterior evenly consistent decrease deviation distribution case apparent characterize rapid count typical right see interesting examine joint immediately apparent origin indicate event asymmetric behind outcome half maxima deviation base short event
present generate system interest markov quantum walk consequence string uniquely determined sketch idea string coincide string vector determined string subsequently string coincide due see therein determined word whose uniquely string string necessarily counterpart crucial map q comprise generalize obviously irreducible derive set strong view inspection let case put
code com p extreme share library incorporate code everything quick include split slow split step split easily turn set specify split merge heterogeneous incomplete integrate true incomplete result marginalization derive em monotonically find noisy incorporate prior without analytical accommodate merge deal maxima suffer incorporate nonparametric modeling infinite dirichlet advance model apply science likelihood jensen inequality continuous case integrable normalize observation entropy distribution take q reduce calculate posterior drop play ij
model particle carlo repeat data dynamic tree outperform counterpart dynamic linear nearly offer mean method sd rf search scheme rank location maximize expect gp leaf repeat mcmc hybrid setting true impossible promise local appropriately iterate towards regression flexible attractive tool hybrid usefulness deterministic stochastic due outline tree search space optimum default input expectation statistic optimum optimum improvement minimize hybrid incorporate idea maximize precision favor scope close give tree ct cumulative freedom n ar ar detail generic candidate location draw sample posterior fit tree leave precision quickly wide leaf seem cm iteration tree posterior grey green blue red cm tree cm exponential solution number dimensional initial location routine routine dynamic tree except
infer appear nearly model use red blue dot dash middle bottom segment approximation segment resample storage linear quantify use importance fitting quadratic allow possibility study contain also well procedure uncertainty advantage underlie appendix detail recent time mean gr like helpful liu mathematics
random method connectivity various metric properly hoc nearly impossible interesting bag move bag bag gram interesting gram move tree similar fashion relax relevance attempt ir anonymous comment great benefit partially grant present extraction focus common reinforcement query short datum furthermore understand justify document algorithm function document formulation multiple question limitation access know ir engine retrieve multiple relevant document
rv rv multiply skewness allow comparable system characterization scalable scale rv rv family deviation rv pt invariant scaling system accordingly exponentially introduce skewness symmetric student input transformation invariant unit shift shift scale rv rv deviation unit variance version rv parameter closeness skew parametrization ignore parametrization case vice versa student input degree parametrization distribution rv simplicity definition scale rv transformation example rv refer family rv look interesting transformation essential f analyze minimum equal physics commonly since define outcome domain exist real branch value include functional identity reference skewness axis map asymmetric output axis event event inverse curvature curvature skewness verify work
analysis advantage positive number estimator assume process pt theorem condition b imply try bandwidth choice investigate carefully admissible term enjoy martingale conjecture case limit carlo improve impose weak moment almost convergence desirable setting typically assumption impose either restrictive growth translate impose condition type surely remove
inherently difficult largely improvement parallelization study statistic solve mcmc course analyse advantage outline brief hope reader discuss challenge sect assess density obtain sect actual consist type ia conclude sect inference expression regard combine information incorporate experiment consist absence information restriction entirely probability normalization constant normalize constant available analytical approximation relate denote function convergent integral mean grow covariance domain indicator otherwise option necessary complex chain production markov chain target many markovian metropolis algorithm give current call denote acceptance q arbitrary applie proposal properly tune proposal small take step support extreme fail converge hand scale exhibit fail sufficiently require multiple number aspect perspective mcmc recommendation schedule proposal propose carlo version importance population sequential manner construct improve estimation fundamental hold include
r sr segregation severe choose large reduce ever complete efficacy segregation variance plot graph segregation agreement order r increase efficacy give pick recall degenerate degenerate get close get ar local supremum l ar ar ar suggest efficacy r segregation plot association small get agreement small decrease increase asymptotic normality segregation large reject percentile specific segregation alternative z plot attain supremum attain power segregation function observe segregation alternative leave plot observe segregation function association percentile e sample level r plot power association
real always use large help estimate formula margin derive simplify conservative nb large cast large margin margin unit uniform mathematical fact natural side estimate distinct sample replacement n count ask give replacement draw vote count chance chance replacement draw drawing drawing count vote chance select count replacement estimate select c begin formula distinct sample set replace q provide detecting size total count count slightly exact calculate replacement employ vote count method appendix post see vote incorrect drawing without vote formula would fix appendix weight work new approach post improvement recommend incorrect confidence enough occur report vote candidate decrease win vote win low contain vote win let high receive vote candidate et continue change error incorrect vote sentence precede paragraph incorrect vote vote number incorrect vote vote win produce report produce sensible recommend ok often vote cast weight desirable weighting within vote quantity vote rather cast put focus contain vote incorrect outcome show margin maximum margin increase quantity size impossible vote contribute margin total
dotted selection divide risk htbp j represent divided argument technical suppose wavelet limit cardinality f k k assumption hold j fu dy periodic dy argument bind p z j hence z jk w n jk z proposition jk proof jk k k j f bernstein e obtain one check jk follow let set integer j proposition jk j jk inspire see jk obtain inequality obtain follow lemma cn pt j
snr transmission direct transmission allocate across capture performance network diversity definition diversity diversity single definition receive time analyze scheme observe tend interference bs density evident interference define translate interference performance system plan transmission bss interference may cause transmission direct connection success probability definition follow x x limit observe definition
work double error peak theoretical precision bit double bit precision matter variation circumstance perfectly acceptable norm gpu useful realize somewhat matter different choice possible along matter web possibility get close unity arithmetic symmetry take inverse double large symmetric sensible seem go case unit zero view quantile precision small precision double precision diverse view line entirely appropriate monte argue simulation independent distribute work elementary eq small invert quantile approximately eq carlo simulation chance produce might influence expectation one imagine dominate copula lot correlation put quantile argue end truncation point high accurately sample special require ignore high mind algorithms region double seem machine already small super carlo go alone small double monte unlikely make poor characteristic error
problem alphabet process distribution completely wish answer question letter distortion code code give per letter scheme code construct reconstruct process fashion address I continuous alphabet source regularity code identification whose gap theoretical shannon distortion fidelity tend code operate code code match source precede rate rate block compression identification vc limitation restrictive g autoregressive source parameter sense diameter priori relax study identification satisfy belong parametric class open subset code infinite assess composite lagrangian regularity distortion code identification length lagrangian achievable code block fidelity converge dimension certain class region dimensional result compression identification capture rich hard affect compression performance decide
verify chen convenient ess pearson statistic formula bivariate spherical significant paper david rule interval knowledge sign monotonicity pattern see application state theorems independent bivariate note correlation q commonly moment rx I moment rank surely pair two sided alternative express odd strictly plot well monotonicity immediate hypothesis
finally learn maximize criterion approximate know conversely focus estimation capacity retrieve class well cluster interest outperform generate indeed class contain retrieve distinguish community appear community explain community frequentist sbm handle topology might structure observe lead estimate class thus class network percentage h c display structure presence central complex
linearly polynomially statement datum relate search often depend dictionary fact take cause scale exponential building train binary classifier enable handle training production generalization execution employ accomplish quickly since boost find gain varied global amenable heuristic show map negative cost improvement increase advantage conventional greedy hardware quantum establish depend propose attractive employ quadratic map naturally
since contain irrelevant estimate parameter transition entire relatively markov transition estimate cross blue line line time change note capture sharp instance somewhat seem sharp might assumption hand quite room improvement design model evolve eps eps eps rise one observe markov traditionally single light degeneracy find distribution investigate modeling variable seem difficult study stationary knowledge directly specify move hope beyond incorporate evolve phenomena existence form merge split flexibility temporal
addition depend far close empirical phase remark different far exploration range independently represent parameter precise therefore phase nan exploration regret exploration phase algorithm smaller far follow exploitation phase full allocation channel bandwidth stochastically usage channel channel reward ki channel result positively detail explain important positively case function positively optimal observe equal ki
sequence pixel imaging l cs nonzero exact cs error exact implementation conclusion sequence fig measurement entire partial cs use set use exp opt give fig exp opt sequence cs rmse stable large enough cs frame keep slowly rmse ls plot l error l number much l inaccurate notice fig significantly outperform cs outperform always fig fig poorly error large cs bad though slowly study reconstruct partly contain know cs relaxation outside constraint sufficient exact reconstruction modify size known extension
result metropolis time adaptive algorithm rr rr rr ac min median max median ir ir median ir ir median ir rwm daily area figure poisson peak term school break occur include harmonic effect peak explanatory variable start term beta parameter explanatory preliminary include eight correspond daily intervention take present monte replication different seed trend explanatory variable adaptive random walk least twice adaptive hasting rr rr rr c median max median ir ir
perform tm limit length time pick therefore set fig besides tm behave trend limit error function tm tm tm versus incorporation tm mc finally distribution tm tm tm tm tm indeed keep tm tm computer fig show contour line decode incorporation detail case smaller slight two tm confirm f tm average accord versus difference huge subsection never compare cutoff enable one relatively number little difference subsection tm quite tm small though tm tm tm though slight tm tm much
consequence spirit coherence coherence similar cumulative local coherence condition recall large eigenvalue weak isometry coherence theorem observe schwarz inequality factor furthermore find important tucker kkt condition context subdifferential calculus kkt q span anti lemma kkt must leave multiply equality q define hold condition moreover prove ns n kkt kkt min ss
learn multivariate throughput technology expression genetic genome provide complex disease diabetes know phenotype aim identify marker nucleotide snps rise clinical phenotype disease across individual deep disease directly express pathway use analysis examine many often co co incorporate association gene exist gene analysis obviously exploit source genetic expression gene jointly trait individual gene module set snp module process single incorporate gene snps search leverage module discover cluster module gene gene snps module detailed module coefficient white gray correlation response highly cluster tree likely influence group tree lasso combine across related leveraging cluster gene gene
beneficial achieve long design fair sharing cognitive repeat monitoring repeatedly begin spectrum access bid history regard secondary activity limited game high bid irrespective entry incur regard allocation make monitoring incur propose outcome transaction term scheme outperform multi comment feasibility exploitation different interest architecture cognitive currently pilot gain communication resource management
recognize surface hand top visible space map remark question direct boundary harmonic regularity question harmonic regularity suppose imply proposition conjecture convention question space g www support high universit university city usa city china setting manifold space image mathematical pattern recognition framework develop theory constitute towards geometry vision appendix metric homology theory geometry extend theory laplacian domain space manifold deep vision lead development develop mathematics vision e space occur manifold paper example recently decade laplacian classical fan extension separable endowed measure tuple define boundary operator geometric special homology interpret finite sample equation finite kernel gaussian reproduce substantial second adjacency threshold picture regularity regularity form solve harmonic progress coincide classical answer theory case riemannian manifold riemannian manifold upper de general topology image lee investigate real basis find homology class evidence homology surface attempt could
exact minimization express involve iteration q iteration become eq produces choose exact result desire iteration concern assume handle restrict ii possibility handle limit note monotonicity play role continuous positive tend increase maxima monotonically proof subtle point
entry rank removal rank stable rank matrix removal generality column write row observe contradict argument one consequence coherence interest arise converse hold row define one entry symmetry stable hand proposition sensitive attribute coherence stability combinatorial notion stability regard generalization construction high stable nice construction quite full clear attain turn typical integer nn consider rely well find suppose equivalently determinant zero square polynomial lebesgue whose column proposition theorem stable consider construct via orthogonal haar orthogonal arbitrary orthogonal haar
within alone induce age acquisition flat age acquisition outside inside light learn early concrete shift within concrete learn whereas acquisition age finding exclude eliminate correlation subset differ substantially dictionary work early dictionary age refine small early rest concrete outer layer acquisition outer remain concrete outer abstract relate outer
execution show speed execution among online method estimation result case make variational consequently spectral search whereas connectivity low connectivity five rand index c thus display blockmodel online variational model fairly bias spectral accurate well adapt particularly partition set densely template realistic consist political day snapshot extract analyze web page represent different identical orient realistic group six community network centre moderate party economic package present organization connectivity correspond dot generate realistic network simulate network accord sequentially minute enhance recent
order kernel polynomial bandwidth random field p bias mean become convenient c c p k expect nuisance parameter decompose note risk come mild get dense bound particular use small asymptotic stay domain improvement bias threshold use bandwidth selector weight generate kernel heat value column mat ern bayes leave plot hard ern exposition class kernel define course estimate result thresholde weight w b tt
leave apart apply algorithm iterate p three map correct one example real dataset efficacy life issue appropriately mark year period core module illustrative simplification investigate trivial stage stage table htb student situation location student student appear detail stage place student situation et situation stage comment membership identify situation discover student situation getting get probability achieve low situation qualitatively student four situation low storing event natural expert
learning perform rather standard hmms present framework entirely device remove traditional human allow well establish supervise automatically acknowledgment visit yahoo partially nsf dms dynamic lead capable otherwise long structure motion high alternative hmms kalman unobserve transition event maintain distribution state condition range forecast finance control video speech detail dynamic application capture small predict
neighbor test simulate variate autoregressive ii compare performance distribution variate simulation either missing give miss random test normally dimension random square poisson chi distribution poisson three set simulation imputation table penalty near imputation variate c c pt c c c pt parameter fold cross validation c imputation perform lc c chi poisson competitive imputation method svd near imputation penalty fact globally procedure reach permit flexibility see cross type prefer marginal multivariate penalty covariance allow observed variate svd imputation full equal base imputation normality outlier imputation compare usually gene array variate normal indeed microarray span marginal array microarray truly accommodate array appropriately correlation last nature information microarray cancer tumor datum miss
often abuse main object error independent equal I ii value asymptotic expect excess say achievable prove type setting idea provide estimate classification function assume throughout sign associate norm tuple dirac mass concentrate I every
person fair win produce winner explain evidence read propose fair winner evidence equally plausible something conclusion sure thing implicitly state get would sure involve contain sure thing hypothesis
cauchy start hx replication set exponential optimal gain importance weight sampling estimator fourth final toy example geometric target one random compute q gain rao probit diabetes year old test health organization criteria collect national institute diabetes disease available
ir rwm dimensional identity adaptive rwm check five sampler compare sampler mean time expect rwm ir rwm ir rwm plain rwm conclusion replication iteration rwm ratio pt ir ratios ir mse ratios mse ratio kp l drift actually depend imply vx vx lx vx lx lx vx r lx lx lx vx vx lx vx line deduce distribution imply exist
light regret product imply I transition product product arbitrary appeal make I low regret obtain cost curvature loss curvature lead example slow rate convex minimizer expect unique regret curvature play role speak exp ensure grow fast g linear grow achieve curvature divergence suggest determine regret shall curvature curvature curvature directly rate first let rooted mirror care notion introduce chapter infinite dimensional space analysis compact present exposition geometric interpretation proof long vector basic analysis take hull end
equivalent strong check since large reader check include equivalent conclude minimum typical follow q follow immediately union row x start number estimate splitting separately q large denote need begin sphere r I case derive sphere distribute mp mp ks ks let sketch inequality special version hoeffding independent surely mean valid coefficient zero indicator zero eq use converse chi distribution moment especially recurrence eq suffice eq part I ij ij identically drop subscript directly little trick always moment formula recurrence formula gamma treat even
versus horizontal computed database band indicate set true evident gamma find different relate anonymous gene profile set gene gamma utilize grouping label information various develop construction ranking hypothesis microarray technology continuous gene essentially count copy gamma ranking extend readily gene tag th microarray replicate library state library say adopt notation library put order latent expectation latent conjugate gamma conditioning gamma shape analogously conditionally poisson response gamma distribute normalize q notice refer sequence arise whereas inverse computation key thing monotone transformation gamma multinomial utility investigation within reason rank section model calculation pattern biological gene expression part round structure comprehensive beyond towards control list
generate triplet briefly train near find triplet target except dataset accord table code code conclude use dataset mahalanobis classification outperform dataset statistically
average cardinality pca well grow extract cardinality b dc really affect cardinality magnitude demonstrate greedy computational pc fix average instance induce figure dc pca scale discard deal dc experiment examine fix ratio parameter solution algorithm cardinality display table algorithm see dc pca ccc dna gene hundred thousand experiment enhance interpretation extract involve usually gene expression say compute expression cccc dataset sample gene microarray first principal consist training sample set sample use widely feature normalize value extract microarray gene first principal total high candidate investigate interpretable scalable study dc cardinality sparse pc algorithm time range result handle dc scalability explain vs mention sparse cm fix cccc cancer dc cca cca dc matrices cca relate task retrieval deal retrieval sparse relaxation cca cca scale document collection document string retrieve semantic language translate space
since fitness ga state also population mutation population actually markov imply mean adequate course genetic inter limit observation chain state observation player make estimation limit impossible limiting without limit equilibrium symmetric call state one union entire although expect still definition average population denote nash quantity hamming bit average hamming nash eq player state nash maximum maximum hamming maximum value bit nash population consist correspond nash
estimator take along monte width unbounded precisely derive burn ensure drift analyse estimator multiple short run allow bound require practically bound many computation integral know normalize feasible approach ensure approximately introduce et seminal underlie discard call burn validity law number
partial quantile polynomially existence efficient partial relation author broad generalized process one incorporation raise issue interested index quantile process quantile quantile build instead probability conditioning automatically condition strictly regard mapping counterpart however interest dependent fix able weak seem even process non quantile property set indice potentially interesting apply generalize nest unlike unknown priori thereby characteristic provide intuition variety interaction key role characterize quantile maximize figure partial quantile b shape partial surface infer band band interval quantile generalize ordering partial example population lead convex collection quantile square partial see representation cc potential partial arise order align quantile example point quantile population well partial align distribution partial px x partial quantile x impact quantile alignment quantile member diagonal draw lead quantile consistent draw quantile seem square alignment lack try separate extreme order illustrate simplex case might suggest deal univariate nonetheless try compare pareto point dominate allow efficient describe order fail nonetheless order binary relation represent value behave therefore consider encounter example consequence multidimensional might payoff emphasize standard univariate
stop epidemic describe infect recall infect consist infect infect approximate distribution period reason first intensity epidemic markovian intensity epidemic period non random version end mathematical contact imply individual explain stochastic sir epidemic section devote large matter possible derive exact simple close expression dynamic epidemic possible recursive final epidemic explain infected infection way e g outline detail formula idea identity final total infection pressure express getting initially get start write number infect exclude possible infect label infection pressure infection pressure sometimes also cost epidemic exactly infected dependence initially infect initially product infect subset infect total start identity transform period step identity period mutually period contact dividing result condition q simplify return recursive formula use sequentially unstable large informative deriving early rely reason early unlikely happen individual conversely happen distinct individual
ignore case gain information seem order stable process main advantage allow instead step attractive purely dependence situation develop multiscale interaction method simulate process focus two extend multiscale might amenable perfect method fitting multiscale circumstance future author would thank helpful discussion prove generalise
risk q inspection right monotone yield minimax wide range space say neighbourhood neighbourhood description back riemannian map nice background aim ng without r g x g unit sphere tuple orthogonal orthonormal frame manifold naturally subset e ex kk ex v tx facts suffice map bundle projection neighbourhood positive definite g yield map bundle bundle np yield invertible see closely manifold characterize admit basis natural manifold normal eq recall g diag onto ny g construction may n n h transpose real hermitian structure
thresholded affinity able train classifier directly minimize rand result partitioning learning image serious ever train performance detecting belong boundary boundary detector affinity performance supervise image field label similar class segmentation require distinguish object probabilistic field parse reason tu module boost level module joint labeling never minimize rand weighted affinity near neighbor pixel group segmentation connect thresholded affinity affinity
state continuous examine expand utility perform htb v good leaf policy mdp mdp leaf node definition du trivial blind policy low analogue blind fix policy trivially greedy mean sampling express form integral approximate leaf expand perform iteration mdp mdp belief form easy mean transition state
first appropriately often dependence achieve nontrivial remark result single ess fundamental quantitative version therefore ess boolean give characteristic theory know intersection uniform approximately solution class elaborate main new structure integer topic sensitivity introduce seminal notion analysis boolean speak boolean randomly choose sign independently bound boolean hardness circuit complexity complexity theory sensitivity boolean yield algorithm agnostic invariance sensitivity intersection noise sensitivity intersection intersection boolean noise sensitivity noise current good sensitivity start noise important towards intersection necessarily believe intersection intersection even intersection remain open restriction see et imply intersection class intersection learnable learn adversarial precise definition intersection regular fall class easy pac time intersection preserve regular accomplish union result towards improve
server might database server easily number start action attack visit attack front server database server allocation edge decision resource might web application kind continuous denote budget difficult attack budget raise must pay attack server run attack surface server must pay edge surface wider large attack surface build height mapping ever edge abstraction justify rate piece software detail amount budget using believe capital decide result hold quantify simplicity purely rational attack maximize knowledge attack maximize evaluate management provide system fix main result term include minimize
fractional control multipli contribution fractional change r gauss markov multiplier suppose want emphasis term choice q denote emphasis eq interestingly choose e emphasis necessarily sensible pick one wants thus satisfy reasonable objective wants put time emphasis sensible eq side discard root result behavior equal emphasis speak want reasonable would make linear emphasis bias term dm contrast variance variance gaus illustration deal design fmri illustrate capture shape response fmri weighting variance choose matrix initialize draw imply equal dm imply weighting htbp dm simulate dm snr glm plot show generate bar unit quantify variance automatically initialize column result initial dm curve gauss imply likelihood observe dm bias htbp cc glm analysis set use dm figure simulate dm snr glm fit dm show dm bar simulation investigate effect bias curve emphasis reduce
take side equivalent switching role j side let hand bernstein inequality bind right jx j bernstein union probability j nd tb j ta c since b note conclude nj n j nj te j n j n n j variable rapid advance technology throughput frequently example genetic functional frequency grow specific np make possible contribute role statistical selection parametric concave selector elastic net mcp simultaneous challenge computational statistical algorithmic method meet
say convergent power norm radius q infimum contain expansion main conclusion f note reasonably mild combine comment estimator precision need analytic eq easy compact connect compact compact every unbounded closed main follow close conclusion c compact unlike regression small precision acceptable eq comment proposition see op
linearization depend although manifold code lie coordinate imply learn manifold nonlinearity coordinate code advantage problem unlabele datum significantly data prevent label coordinate merely significantly simplify consequently label g within interested coding consider linear function approximate estimate ridge lipschitz include svm generalization code expect optimize become optimize immediately consistency arbitrary manifold note convergence intrinsic manifold lie pick local coordinate coordinate code
heuristic prevent optimal performance set considerable progress particular pac free reinforcement remain whether approach history make optimistic important policy computer policy thus online conduct predict learn replace figure show learn work current first promising thorough investigation policy adversarial adapt would search bootstrapping look modify ucb technique computer go class convex combination principled expand power agent generalised define help predicate boolean node false associate leaf node represents reach predicate generalise context analogous retain provide powerful way extend notion example suitable possibility logical tree several extended depth without space overhead symbol case may significantly derivation investigate agent allow property parallel complete core confident improvement allow agent planning ability main single binary attempt attempt example together binary predictor apply technique bit predictor integer character convenient environment model compose possibility construct sophisticated mixture feasible directly approximate ideal well theoretically previously unclear theory agent answer environment monte carlo reinforcement powerful highlight interesting direction performance agent particular heuristic policy intelligence community reinforcement agent thank anonymous helpful feedback communication information technology centre program mm mm pt proposition conjecture height mm monte pt ex new south ng ng com national expect act history horizon act policy ax ax ta achievable reward agent history
inner since throughout take rhs nh n martingale array eq central theorem imply ft show ax x q argument almost term ax decomposition assumption study projection converge apply conclude usual define proposition rest almost surely zero finite compact na n h l since follow rhs define apply get c give q last term rhs
r v shall conjunction thus use allow count tail expression order need replace respectively modification satisfied accord expression since refer add assumption replace small obtain self sum chen turn precise paper allow order moment hand kind especially consider closeness closeness normal reader refer due imply independent analogous value exist shall equality whenever generalize banach use present every banach smooth next borel condition coincide enough statistic one corollarie magnitude numerical rather moderate satisfying term way chebyshev type even additional obtain hold q essence measure magnitude indeed banach satisfied say take enough explicit seem uniform whole traditional significant play exercise e space type type introduce remain mean correct magnitude depend space constant also usually closeness distribution obtain closeness motivation work deep closeness chen improve closeness banach g several infinite existence smoothness
see side proof approximated state property context procedure run subset contribution degree loop alternative fact normalization choose leave side prove give formula let show
suppose previous systematically throughout expectation main goal characterize test power exist locally candidate calculated pose employ throughout paper vector respect use calculate basic integrable hold nothing particular distribute everywhere everywhere condition power treat non various assumption property follow side
actually forest continuous object edge number statistic bic edge search decomposable measure decomposable determine add add preserve one stop component start forest isolate isolate change join true bic bic edge step example object number default model edge start empty decomposable figure forest tree graph object first consider isolate join evaluate cpu ghz gb ram running bit
probit error compute easily denote complementary function rbf derivative jj kx variance site show task point illustrate b together explanation vector corner triangle interact triangle edge dimension negative corner discussion issue often machine rbf tend thus derivative point axis htbp consequently define sophisticated try try explanation value thresholded resemble bayes logistic regression sigmoid vector density index k approximate want explain assign omit use ensure orientation assign summary practically classifier high care available may describe length certain
maximal multiply correct sharp approximately spaced factor polynomial gaussians equal next rapidly shortest close error ignore
harmonic counterpart probit sampling sampler recommendation approximation reference acknowledgement mod de universit paris centre de en paris survey cover importance sampling reversible jump survey fundamental dimensional model method bayes avoid importance approach survey single carlo assess recall probit
write n vector contain deviation detail omit notation finally lt lt lt pi nh finding bias error difficulty quadratic functional observation instead variable
repeat relation illustrate multiple point query share set score rewrite logarithm dropping constant detail dataset sample model assume link q pair decrease order integral present carlo compute compare query similarity sort due variational logistic impose initial approximated reasoning interaction function general decomposable present indicator give entry call decade structural mean node similar connectivity seminal introduce strategy role indicator block ij integrate computational exploratory expect relatively short alternative bayesian formulation merge binary relationship create pair marginal treat merge really relation fail dependency conditioning marginally nevertheless computationally evaluate section provide existence link useful generate automatically automate selection relational combination cell specie web page web relax requirement possibility treat formulation database
view structure scale conclusion ensemble agree ensemble however visible tendency variable largely behave turn indicate structure largely phenomenon size score much indicate structure transition width probit expect fit account substantially asymptotic eventually understand phenomenon success mean polytope projection face course precede rigorous receive pair site hand belong lose event sparse face ensemble quite small ensemble nan reader discuss nan denote ensemble find find sparse nan provide heuristic apply algorithm random sparsity record column sparsity level record sparse indicate common draw matrix contain zero case dramatically remarkable may explain bad intersect support become consequence increasingly common instance lp moderately yet asymptotic value although effect compare c ensemble indicate close constant tend zero rapidly value ensemble unable interpretation initially interpretation driving effect behind ensemble position chance rapidly report place happen two ensemble rademacher complete long analysis basic principle science validation building construction provide rademacher excellent reason point exponent study rigorous exponent lack fit argue lack mean long assume believe occurrence argue however instance validation actually validation hadamard special happen analysis
bias naive bootstrapping tend advanced accuracy multiscale bootstrapping implement follow section multiscale propose outputs multiscale bootstrap multiscale test conclusion gaussian call dag
difference ks former chi snr confirm finally aggregate ks single operating aggregate ks subsection apply performance context linear discriminant covariance class separate hyperplane density ica asymptotic bandwidth get nonparametric regression estimate european organization aim provide weather forecast channel km center degree every affected cloud processing thank band problem cloud infer presence image discriminant mask produce endow mod aim cloud mask pixel mod threshold mostly deal spectral band generation investigation statistical water perform mod water pixel pixel mod truth evaluate water divide randomly use estimation capability split pixel
location tb estimator gmm panel function estimator cluster calculate bias gmm clearly bottom panel removal bias measurement panel galaxy cluster degree camera enable acquisition observation calibration contain measure object galaxy magnitude star survey comprise galaxy dr description location selection publish elsewhere galaxy rich galaxy galaxy galaxy locate galaxy center completeness dispersion cluster spectrum count galaxy cluster varied therefore varied critical radius mass cluster weak scaling range next cluster color correspond measurement galaxy result red represent galaxy color component likelihood get new original continue analysis subsample component measurement
step expansion start contain large possible introduce exploit bind perform hilbert add kernel proceed device expense property scalar delay initialization beyond adversary act fashion case adversary arbitrary requirement term realize see copy product occur see past versa formalize martingale technique favor issue expectation key bound correlation case limitation improve assume lead guarantee increasingly via q xt hinge loss detection three loss huber except rescale value depend impose smoothness simple want show lipschitz piecewise occur delay
within performance shift oracle scenario quantization multiplicative ill condition standard scenario normalization case condition incremental rank context recommender base rating rating test mean absolute mae mae ij predict rating item rating incremental present four column rank use miss entry predict neighbor yield incremental c incremental last movie rating last incremental look user point approximately although make difficulty deal decomposition manifold low reveal enough give good local high
iv expand eigenvector output implement implement dimension expand j return signal jt j jt algorithm contain zero eigenvalue become long sec signal become wrong reformulate sort eigenvalue svd covariance find svd eq solve j line singular zero precisely row exclude keep zero exclude svd lead expand central proposition svd
vote situation batch vote ip vote eight case batch rr ip c b upper bound batch eight kind batch chance require count count different control risk incorrect count batch draw chance draw let batch stop
inequality remain triangle inequality nonempty string term xy xy reach respective element obvious reach drop moving argument observe reach additive asymmetric like exclude degenerate measure want important density distance computable broad total asymmetric element term admissible admissible list constant additive verify list element define
parameter turn get p p posterior ix jx lf l case v f ix v posterior add see correspond gamma tr ibp gene network account information topology mcmc evolutionary unfortunately converge integrate factor infinite component treat take account hierarchy standard ica propose fix assume mass
point require case free generalize spectral relaxed case consider scalar canonical bernoulli satisfy odd central standardized analytic moment burn begin look like quadratic evident standardized generating great mean multivariate matrix definite seek moment eq moment connection self response draw distribution exponential sufficient special setup family take q see either unit logistic least easy standardize analytic eq norm fisher
uniformity figure illustrate idea f rise uniformity restriction stack partly perfect second note enough formalize henceforth great uniformity variable satisfy condition prove put q integer prove fashion z still accept exceed previously assumption make regularity adjust r iff admit pareto
notice tu estimate constant bound incoherence denote minimum start prove together imply f lower prove us inequality ij ij remark follow n e call six calculation tu ns tu proceed condition contain diagonal reveal f cn tu f analogously combine na constant f leave prove thus fx ij ij bound u f bound velocity yx value respect geodesic parametrize th thesis together imply thesis cauchy schwarz let prove
interpolation many present accuracy expect single rely random variable position pick multiple orthogonal distance estimate deviation process generate trajectory display indicate indistinguishable zero estimate accuracy fair constraint table c component component want reliable filter reconstruct path reconstruction large reconstruction mean step table display x number particle
theorem exercise proposition remark theorem theorem division bank school business university new south south article estimate marginal correctly propose mixture normal copula avoid normal copula flexible copula marginally adapt normal improve normal use empirically copula behave much third dirichlet process normal constraint comparison mixture encounter mixture dimension imply model understand marginal mixture normal mixture normal moderate multivariate mixture propose normal marginal transform marginal normal marginal define copula normal perform copula copula normal
cd rw sample half width interval roughly rw half rw half half width time large rw act rw bad replication rw rw comparable comparable rw surprising similar chain component two gibbs sampler scan metropolis gibbs rate outline geometric enable identically distribute certainly study component enable uniform ergodicity true ergodicity ergodicity setting seem rate think especially study either geometrically ergodic mcmc often geometrically ergodic implementation section superior dimensional every case real literature practitioner acknowledgment health award city york conduct full
set inactive gx sd kk property terminate iterate update terminate optimal final iterate know gx u behave exactly termination statement compare minimization nesterov coordinate sparse invertible entry density prescribe generate finally small discuss smooth case nesterov however performance differ compare presentation code matlab code code accordance algorithm terminate duality gap sm randomly size sample covariance matrix give four objective cpu time give sm nesterov scheme ii namely variant minimization substantially experiment nesterov approximation equivalent
correlate ability hc appropriate accommodate method independence statistically speak nearby extreme perfectly remove standard hc utilize datum design exploit good wide setting context paper correlate brief detection describe strength boundary sparsity strength correlation able precisely construct lower special toeplitz asymptotically hc particularly extensive literature hypothesis dependence include control randomly despite appeal convenience uncorrelated correlate special exact detection complicate tight either subtracting amount difficulty measure etc adjust recent decay rate decay add inference hard specifically model noting see intuitively add hellinger hellinger distance diagonal review concern diagonal message cholesky positive polynomial decay rate writing correlation matrix uniformly matrix turn well sequence rate proof ready correlation boundary fix sequence error test key new higher bind hc really reason summarize structure hc coordinate build correlation new statistic establish connected innovation
make arbitrarily thesis sample tight second constant together desire thesis case z max application interest approach present similar obtain variety circumstance rank matrix entry perturb produce approximately entry reveal reveal position rl ij analogously respectively position convenience recall algorithm analyze normalize decomposition guess minimum standard descent description technical reason cost denote condition may reliably follow say reveal row column represent set
candidate task scale well amount determine prediction observation process scale window introduce redundancy impose rescale element combine scale evidence sum order investigate evidence across sum sum compute
leaf appear associate weight multiple appear every possibility grow tree obtain split pick tree node form weight loss weight quadratic practically adjustment prediction first latter hundred tree final measure calculate function fit tree cross interaction advantage interaction extension cover nh efficient basic nh space minimize empirical typical minimal maximal interaction depth partitioning would weight vector form empty optimization difficult solve correspond feasible partition also computationally latter demand partition sense suppose part exactly helpful observation belong receive empirical indicate whether part exactly demanding understand equality constraint yet still due relax take relax
away inequality define candidate specific generate mining frequent consider overcome problem cause support hypothesis may vary scenario part considered split spatial datum operate present contingency association find adjustment multipli pattern second adjust well resample set case raw pose explain test assess number gaussian value nan measure draw constitute dataset parameter amount covariance semi therefore proper simulate mining hoc third number start generate pattern value test dataset store method depict correspond ht negative method control
sample obtain clutter pattern take pl slow individual marginal mix show square numerical rmse predictive via pl repeat test sd rmse pl pair pl standard sided pl mcmc tune long narrow come expense prior comprise initialization fold gp initialization consequence sample identically neither require fold latent propagate helpful play hide dynamic treatment pl need total probability q second come conditional label quantity student integral monte collection student
modify enforce final input extra way determination large chapter sort relevant coefficient achieve undesirable propose modification compressive employ prior improve approximation motivate algorithms initialize recursively unchanged simulation algorithm cycle length snr pick generate normalize decoder output solve involve matrix distortion snr denote procedure performance decoder define convert db db performance simulation eq indicator true specify stop reach simulation sure algorithm stop pass schedule reweighte present
tucker optimality particular show robust increase understanding strength author discussion since small set maximum incorporation may rigorous space alternate proximal precise alternate proximal relationship discover analogy motivate density define kullback leibler maximize proximal wise implementation proximal rr proximal maximize log continuously
method selection lot ensure right stop pn observe remark pn also figure stop select elastic modification lar one mark green square selection link elastic lar fail true illustrate elastic turn lasso select often produce close one conclude superiority net elastic prediction penalize result provide drive choosing illustrate simulate lar confidence secondary f observation regression noise suppose collect label corresponding
ensure particle uniformly stage mutation degeneracy path introduce extension article impact scientific
domain distribution observe bind protein certain exchangeable control ultimately connectivity enable rooted notion information theory focus per se keep focus modeling establish issue list chapter detail graph model partially scalar difference exchangeable goal space position association community membership exchangeable graph binary string carry semantic meaning induce connectivity principled parametric fashion regard social conceptually separate science direct graph dyadic keep track link neither base rate popularity effect incoming add indicate model probability normalizing purpose one observe mutual enhanced factor problem representation identifiable interest os enyi direct appearance reciprocal focus solely version study likelihood dependent reciprocal edge parameter constant summing correspond subscript minimal set iterative major recognize lack asymptotic fit procedure likelihood example ad hoc deal e recently basis generator conceptually direct form incomplete contingency loop redundant standard show relation rational contingency analyze use analyze relation multiple generalization aggregation identify network bayesian typically refer set quantity partly individual refer variable quantity serve treat popularity fix think popularity effect develop paper reasonably take multivariate fix effect spirit extension involve effect distinction additional level high come approach work monte use version implementation network total dependent share undirected normalize triangle star star star triangle three generalization work three model maximize generalization frank maintain structure count pseudo bad sensitivity number edge change sensitivity variant major double counting overcome nearby estimation model describe associate use combination distinct mode configuration zero topic empirical root method relevance carefully construct package analyze package mcmc estimate current formulation formalism undirecte denote node clique parameter clique potential advantage em feasible expensive strategy monte lot methodology develop undirected primarily learn image os expect degree os enyi formation finite lattice os explore focus utilize physics style model fix sequence sample idea representation th
copula finite mixture margin latter mixture disjoint jeffreys proper jeffreys margin result proper author choose fall approach entire nonparametric copula take space copula bayesian adapt methodology inside mix weight pt lemma iw c w since elementary row get mi mi mi mi jeffreys proper follow partition mi parametrize doubly selection doubly polytope contribution jeffreys difficult
variability position position surprising corner home middle dot posterior prediction home mixture curve figure player full player past examine past player additional prediction home run home age plot middle poor home plot posterior draw gray dot mixture posterior indicate happen consistent player relatively see case substantially year player go non regardless year middle status sophisticated home major age home home information across player primary interest home hold
kn kn gm nu k assumption claim dominate convergence discretization inversion concern use consider inversion discuss projection follow coincide variable expectation extension p dimensional p p endow theorem k u assertion finally convergence addition k measurement satisfy limit hence assertion practical inversion datum provide device implementation monte algorithms let borel pe p give next substitute pe p algebra consider probability weakly theorem measure weakly measure compactly wavelet scale suitable expand variable prior next definition follow deduce ii independence p notation stand observe finally almost surely negative sure expectation hence discretization invariant sequence operator proper converge surely sure convergence weakly assertion embed realization accord quantitative result describe thank difficult together
estimate graph consider mind proportion select five method decrease considerably obviously proportion gene set validation split lasso connectivity graph six connect datum depict violate high lasso lasso figure five six vary also method assess reliability good subsample lead set exclude subsample five successively consider include repeat computationally expensive candidate time edge score degree agreement proportion assignment r agreement measure finally denote average denote agreement expect table absolute compare set display shrinkage probably rely subsample cross split approach less differ dramatically run approach shrinkage far one
usa proposition author discovery zhang principal pca widely dimension pc linear interpret drawback propose achieve standard lose pc orthogonality loading explain attempt optimistic paper uncorrelated pc loading novel augment lagrangian nonsmooth suit regularity subproblem pca several synthetic random respectively demonstrate pc correlation pc orthogonality pt pca lagrangian nonsmooth classification processing reduction numerous science biology face handwritten code gene analysis essence aim find combination orthogonal capturing augment function minimizer parameter detail stationary gradient nonsmooth close convex suitably subproblem lagrangian method random result pc obtain substantially outperform explain pc orthogonality pca formulation lagrangian nonsmooth problem nonsmooth close applicability lagrangian pca concluding remark vector assume symbol resp diagonal sign th entry nonnegative denote far clear semidefinite write whose euclidean associate explicitly state otherwise maximal denote denote real closed set formulation orthogonality loading formulation generality commonly load entry
ad hoc efficient mainly good dynamic expect convergence dynamic player game propose dynamic relate possibly perturb algorithmic evolutionary theory say see visit corner mix interior equilibrium player player correspond pure simplex pure strategy strategy component unity corner strategy specify mixed pure strategy correspond play classical write write denote game payoff random whenever profile know get play pure resource denote algorithmic game resource player singleton subset pure strategy space subset cost player pure load sum resource pure strategy cost player profile pure strategy
source help maintain control source varied mean pp simulate filter high maintain false derivation background approximation satisfy acceptable infeasible commonly enhance assumption pp scope develop simulation produce numerically assume rectangular pixel simulate maxima value pp plug value procedure equation wide pp image source pixel pixel want believe number computation increase build give area formula sized area remove square value account fall difference area get source default worth shape give tail benefit simulate value instead handle wider dotted line information solid dot mild gets small consider get run simplify since background thus computation simulate percentile nan ab maxima big image
adjust stage stage cascade collect positive previous cascade show use curve cascade classifier result seem report early perform want rate false specific cascade huge success time detector lot underlie boost limitation adaboost skewed li detection wu ignore scheme adaboost another train strong base divergence therefore haar publish preliminary face greedy sparse discriminative classifier manner face patch reject quickly reach patch face patch reject cascade detector lead fast cascade context soft cascade develop follow cascade contribute efficacy algorithm cascade adaboost forward choose proper coefficient
global stochastic search intend region still bayesian decomposable monte carlo specify law simulation construct prior think need offer exploration meaningful prior os undirected place prior decomposable specification allow vary feature solely place informative direct graph use agree feature potential markov tractable encode certain jointly copula jointly reduce easy case dependency distinguished dependence novel section configuration euclidean induce sampling modeling early subset benefit approach yield estimate graph undirected parametrization flexible family prior parametrization intersection system specifically approach induced graph graph hypergraph stochastic framework concern form theoretical concept concept distribution set resp say graph undirecte unless complete possible inclusion respectively collection clique two vertex incomplete disjoint nonempty path must nonempty collection subgraph complete
aspect almost surely act suitable warm computing initial iteration lar k warm learn th column update dictionary solve soft thresholding observe column empirically become much slow lead lar homotopy whole description implementation prove experimentally thresholding providing solution robust require stop update warm advantage learn store solve th keep one optimization block concentrated make coordinate correlate concentrate effective algorithm convergence invert impractical present building discuss simple enhance predefine case video stream situation examine simulate I set reference experimentally speed carry xt natural require store past block sized store impractical still exploit remove cycle auxiliary replace carry old information old weight aggregated decay forget date propose line algorithm practice apply iteration condition convergence even show understand become j c version could also speed draw
certainly encode find circuit iff correctly predict output theorem proposition observation conjecture mit mit complexity produce quantum show hidden generator find hard quantum generator quantum quantum repeatedly classical outcome perspective outcome work question learn
hill optimize tree nj improve hill hill particularly encourage core red hill minute depend nj long hill bayes tree close directly maximize contrast observe popular consensus minimize bayes estimator study exploratory hill give path metric hill close true tree nj cases hill initial tree hill encourage future choose type theoretical distance geodesic bayes distance connection hill try general interesting hill bayes
remark autoregressive ar frequently series original procedure past lead error information spurious causality include observable pairwise autoregressive causal coefficient g measure probably argument cause say influence series present alone th lag causality indirect cause eeg interact model eeg represent effect model innovation term accord innovation spatially uncorrelated assume source simplicity number invertible exist sensor pca innovation impulse coefficient thank innovation bss statistic would like preferable apply ica directly domain ica infer brain model mild spatially eeg super activity
triangle combination number stand stand r numerical line refer degenerate line degenerate limit nr replicate right number top mt rt rr monte carlo generate leave right r table example mt rt table case r discuss increase table expect c pe mr st nr stochastically hence desire multiple triangle triangle wish investigate segregation alternative realization identically accord correspond correspond triangle use define triangle sub number edge triangle point plot figure arcs iid construct become asymptotic binomial mr p ht arcs hull point circle leave right gr j mr r equation desire top plot give look since binomial hold increase curve become however order approximate normality triangle base point unit square present histogram triangle normal gets approximate normal top gr gr j carlo replicate curve depict j carlo replicate finite
find power alternative alternative discuss asymptotic trend unique solution generic deterministic present goodness von kolmogorov test limit moreover consistent alternative test classical test identically distribute simple von kolmogorov respectively bridge describe help
z final include subgraph submatrix lead quantity appear tree copy mark ij include marked edge let adjacency covariance node mark similar argument use correspondence totally backtrack expand express backtracking time appear numerator appear denominator totally backtrack see subtree tree power vertex node show embed could give intersect marked edge contribute power lastly power
interpret opposite overlap singular lead convergence quantitative maximizer algorithm eq proposition appendix proof eigenvalue deriving assume final see consequently remark write satisfied value since overlap sense conservative overlap recover theorem optimal depend satisfie minimize compare proof
h yy yy dy xy dy dx show adjoint inner product kx dx dx dy xy dy q conjugate markov reversible operator equivalent rewrite show eigen development course eigen eigen conversely eigen eigen se author play role extension chapter compact main relate da spectrum follow proposition also aside share eigenvalue dominate transition remain reversible draw draw draw finally draw product conditional explain da imply easy indeed start half never negative fortunately note marginal exists actually base therefore develop refinement spectrum chain space compact eq eigenvalue order characterization self e
covariance valid bernoulli probabilistic state configuration undirecte present simultaneously prove b else sim prop c create datum probability sufficiently sample presence composition fundamental level propose inference model random associated arc derivation variability arc bayesian
summarize initialize outer converge iteration inner loop ib worth role compatibility make compatibility mix evolve describe section need differently first appear equation replace kf initialize converge current estimate solve accord kf procedure smooth follow use logistic normal inference pose constant summarize initialize converge ib smooth notice marginal variational marginal optimum optimum validate algorithm well evaluate early investigate major laplace adequate estimate membership well fit role iii fit network role role role compatibility real life show three row cell estimate project represent truth truth estimation link grey color actor row display compatibility synthetic synthetic actor actor connect mostly actor mix actor simplex corner indicate dominate role actor corner close edge membership actor lie near simplex mixed role actor possess role synthetic membership compatibility diagonal
model situation plain fail restrict eigenvalue linear noisy noise design either throughout I object parameter infer pattern refer understand also follow problem choose convenience penalization attractive method provable large stability necessity lasso rather exclude case exhibit exploration computationally selection relax impose towards goal analyze lasso originally analyze progress high scenario paper lasso lasso initial estimator behavior submatrix whose index thus variable condition necessary
ball rather point lipschitz metric constant full access similarity information practice believe issue application via specific oracle answer query time horizon advance regret phase round typical point point space phase duration duration copy contextual mab payoff problem take adaptively maintain space region contextual complicated algorithm differently notion argument subsequent express contextual notion pack refined version take payoff guarantee follow form unless otherwise pack rt rr small guarantee suggest guarantee term include dependence bind eq dimension extend pack packing subset include optimal payoff pack number type call optimistic arm pair small packing likewise contextual optimistic contextual payoff contextual guarantee dimension theorem enjoy trade drastically dimension contrary prior essentially take account defer horizon round ball similarity add active call ball active activate initially active ball radius similarity select active arm accord arm activate forward fix number ball arm algorithm relevant definition defer subsection ready break tie arbitrarily activation rule rule select relevant let ball radius
numerical enough opinion indeed illustration probit demonstrate precision approximation discussion accuracy overcome difficulty bayes present single advance far impossible computation introduce model reasonable use simulation tool principle produce attain replace relaxed calibrate easily simulate method calibrate well choice graphical model illustration popularity opposite diversity promise researcher advance want try choice tool modify choice
posterior distribution detail contrast temporal epidemic inference final complete temporal impractical analytically likelihood overcome augmentation simple behave independently literature level discuss practice complete inference latent impossible period final invariant choice e ball greatly realistic choice affect infection see numerical sequel latent simply fix infection parameter latent derivation quantity start fairly allow explore interest mention threshold infect individual infect infected individual individual transmission assume occur contain individual summarize j infect adopt imputation describe approach nevertheless enable address question correspond
two space sa concept statistical mechanic temperature decrease temperature gradually landscape high temperature change see sa global practical temperature physics annealing attract alternative anneal analogous quantum fluctuation help optima low novel obtain generalization vb density motivation interestingly
yield note ne pn complete employ central relevant component useful nonparametric estimation simplify proof main proof lemma together lemma rate sequence r cn nx nc x put ff x hand condition hoeffding hoeffding together lemma assumption lemma c proof proof write e gx nx x use lemma gx eq side x side lemma note eq similarly prove assumption noting remain need cn verify calculate hence lemma obtain enough lemma obtain prove complete condition separate note pn proof suitable number obtain tr r sr lemma reference model analysis new york linear
mathematical sciences complex university university e paris paris france support nsf dms sparse penalization high recover density yield inequality offer drive construction substantial complement theoretical simulation employ dimensional mixture finding finite function estimate functional infeasible candidate large gap give intermediate identification component recover large since emphasis tuning penalty regression section inequality show adaptively factor class number wavelet together employ introduce computationally construct tune choose candidate validate study inner
plot k plus level large empty bar small plus symbol moreover appear case mean performing pair case pair pair sign comparison second number imagine column sign test rank good mse mse pair less c c fig find datum delay regardless gap row ci measure result follow time data delay bias place optical table fig group statistically see table show result well also fig suggest capable compare artificial generate methodology performance fix sf covariance sf sf sf several delay day unitary increment
sign regularize cat cat comparison square group cat maximum absolute value second constructing eliminate provide class conventional prediction validation primarily offer correction rate fdr emphasize construct control fdr contrast classifier aim identify confidence eliminate fdr see subtle important distinction differentially decide gene similar false confidence nan false feature retain discovery instead include argument fdr pt correlation
connect permutation connect permutation permutation state background variant need vector manner datum say summarize decoder widely statement family kullback kl form stay relatively edge control follow control leibler divergence pair apply follow begin denote denote subgraph edge remove uv follow lemma lower bound let convenient shorthand st begin imply remainder calculation low choosing shown always lie since st completes begin ensemble graphical family st st equal result field non kullback divergence mrfs uv st uv uv x st uv since st u uv v st kullback leibler probability bound order ensemble grouping base
support chen li fisher g mod e w da de g chen chen z li mat k mat mat mat l mat c mod b de dynamic critical behaviour dimensional long law system uncorrelated state infinite dynamic concept system al extend lr decaying ise relaxation preliminary numerical std dynamic
tight reduce iteration algorithm nonconvex attain exact discrete penalty satisfy p unique pp simply ambiguity first suffice satisfy q q detail though iterate characterize g pc expansion j j subsequence challenge model sparsity desired occur real suffer problem beyond meet challenge group rigorous allow group predictor nonconvex include discrete penalty examine estimation joint bioinformatics improvement estimation useful technique everywhere method contrast differentiable characteristic however dimensional component discard
borel banach space sup norm fx bb banach space norm radius center closed element bilinear operator operator markov exist linear markov space subset weak weak convergence metric markov uniquely probability review kalman observation gaussian vector matrix sequence uncorrelated signal kalman covariance pair observation drop observation randomness bernoulli arrival probability arrival protocol know signal kalman give recursive eqn matrix update accord unlike classical element emphasize arrival probability analyze governed eqn fix equip far property complete equivalent canonical eq denote path generate cb banach fix follow appendix dirac concentrated notion boundedness boundedness strong
ise expectation single system close simplify model unary front learn finally perform inspire foreground stroke foreground notion either operate super simply foreground robot image tight black segmentation comparison describe user specify stroke place next interaction history policy position label center second connect absolute current truth circular pixel away motivated user mark precise also take account analogous user learn interact mark stroke pixel low marginal hamming amount hamming pixel hamming expensive act advanced actually centre user
probability hold student cover contour grow tail article lower induce first propose asymptotically like eigenvalue bound verify modify define adapt covariance practice affect scheme particular large small eigenvalue covariance mix though projection avoid behaviour section first improper behave show set uniformly continuous laplace practical preserve belief ergodic result adaptive slightly parameter proposal fix advantage bad proportion take positive value
differently attribute site interested help optimize site example formal concept user economics www site site site correspond external group medium
stationarity dependent fractional number eq exact consequently q expression last appendix define measurable z z p q lead adjust end first fractional cover tool mix derive mix question decompose iid use encounter graphical consideration connection relaxation fractional cover besides remark multiple posterior empirically tight bipartite ranking good iid bayes rather contribution competitive likewise interesting bayes translate question remain kind learn perform obtain generalization connection make order thank plan investigate direction study note u work bound iid bound cover rademacher bayes possibly tail meta except randomization propose fractional bind subset
solve loading apply component variable pls framework calculate bridge pls whereby criterion unit may loading number see concentrate optimization problem fashion penalty component sparse iteratively procedure pls bridge pls proceed lead factor loading pls coefficient describe pls pls procedure u u rs control financial direct induce achieve exhaustive select correct select thresholding apply perform difference zero simple expensive portion binary search method penalization multiplied guess normally achieve less equal large number operation space expensive sort another pls
perturbation one therefore careful irrespective moment experience th toy axis population poor posterior update whole population weight abc perturbation equal whole parameter abc smc work big abc smc correct resample ess sis smc algorithms sample step perturbation require sis develop novel allow dynamical provide model allow describe decade conduct optimization develop extend parameterized bayesian simulate annealing find various system estimation extensively financial biological system maximum test nest investigate way bayes factor deterministic criterion selection couple multinomial logistic regression tool parameter estimation extent selection kind ordinary delay without estimate incomplete reliably paper sequential monte carlo abc
convex move slowly follow initialization diagonal possible non next th error h perform obtain loop perform inner loop single loop passing become necessary good loop efficient loop possibly column shift note pseudo construction drawback explicitly
enhance variable incorporate also inactive show solution piecewise slope change give solution possible therefore necessary keep currently associate coefficient inactive look subgradient general fuse fuse whole substitute precisely set keep condition ff f f lk j ik definition change reflect adjustment mention addition also formal definition
wu active source l avoid source describe denote eq detailed conditionally basis nan space distribution q fisher column iteratively conditionally draw sample von fisher look carefully hyperparameter probability paragraph accord achieve q ig source mix parameter detailed sample mmse estimator empirical average total mcmc chain collection easily hyperparameter ba consider comprehensive distribution probability source observation vector
stationarity exhibit stationarity stationary undesirable much unnecessary obtain advantage trend partition gps region leave interpretable inversion require separately gps already bayesian outline proposal tree carlo leading cart due may utilize class variable softmax two way gps gps region partition propose interest fast monte summarize full modification efficiently latent variable outline shall section begin partition cover prediction focus integrate structure categorical framework discussion categorical clearly interpretation feature section method flexible offer inference discussion nonlinear decade real input formally input gp define mean correlation explanatory define decompose underlie strictly identically predictor call term represent write gp function variable help remain frequent concern instability sampling necessary analysis popular parameter describe decay increase underlie isotropic vector differ result process still structure
still decrease adapt simulator simulator start decrease entropy stable depth understand system general beyond try metric verify stability reliable usefulness methodology performance underlie instability significant drop successfully instability paper
prove stationarity firstly unique accumulation point boundedness subsequence infinitely z contradict subsequence stationary point nuclear norm behave soft norm surrogate perform wide variety situation produce model error situation concave penalize regression popularity analogy problem penalty propose concavity penalty
want estimator framework come valid vector orthogonal replace please simpler positive look basis subspace respective vector basis matrix respective vector complement either orthonormal regular generally parametric regularization power depend smoothing operator nonetheless simplicity generality remark closeness estimator constrain appendix latter number put obtain collection want impose filter operate new operate reconstruct linear pass band band stop etc various application low g filtering processing white improve measured peak interpretation e image collection linear like image indeed image therefore goal low risk assume configuration kk say achieve low predictive insight risk influential explanatory minimize configuration linear remainder address application since efficient tool allow good recover start address issue ideal quadratic
instantaneous ar none possess measure receiver operating allow regime positive measure performance area auc curve auc value causality toolbox particular instead coefficient logarithm residual ar interaction exclude score divide score
mnist vs operate one vector interested know pass achieve accuracy streaming pass note pass kernel mnist vs turn hundred pass report htbp investigate varied stream deviation accuracy order permutation still perform single mnist limitation approach exact albeit thing secondly deviation
false grey line correct width along sample amongst inspection result distribution lie shift due draw false positive sect participant come population observe snp independently population false fig appropriately sensitivity specificity considerably improve know positive lie individual toward threshold analytical sect construct distribution pose substantial difficulty analytically need know width conduct deviation size sect amongst snps cf sect first population population also slight compare column table population correction simulation appropriately population move two unit misclassification nominal sound estimate alone hence upon assumption match sect correction sample size context unknown context size readily snps distribution nan either
look aware context density linear zero lack specification context smooth test optimal reader note fourth wu proposition work statistic white prove variance statistic parameter horizon process martingale additional impose moment white implication nonlinear uncorrelated martingale view lee chen nontrivial martingale discussion n critical speaking procedure asymptotically limit bp valid presence unknown estimate variable parametric model uncorrelated treat section interest cf reference goodness long memory time see autoregressive move backward z pz z ar polynomial respectively interior martingale iid pn tt initial q specify expect residual behave j follow simple omit detail smoothing type error impact limit relative reasonably phenomenon exactly critical critical
network important modeling lead max simultaneously dual structure structure output enable incorporation partially label detail two explore iterative scheme bayes procedure mn demonstrate sparse map real superior promise prediction benefit explore algorithm formulation tight adapt translation association genome trait framework interesting semi supervise hierarchical hierarchical structure plan relax since natural structure principle plan mn desirable perform suggest regularize markov stable desirable acknowledgement university national natural foundation china discussion discussion support nsf fellowship support national science foundation grant foundation project cb cb microsoft fellowship mn lagrange straightforward mn another formulation transformation derive problem dual variable lagrangian hand side zero true infinity I proof theorem corollary constraint another non dual normalization constraint rise lagrangian cm get dual ip mn solve appendix solution conclude ip rich normal mean optimum argument algebra mean optimum proof corollary present derivation divergence posterior divergence corollary solve quadratic
minimal resp real denote partial element integer general covariance first namely adaptive spectral smooth subsection provide framework inverse throughout follow addition however whenever establish problem unique optimal see sup fx x diag q along easily observe fs observe strict conclude present introduce terminology objective solution estimate
similar increase decrease specify w initially set gradient momentum w typically choose rule momentum give decay somewhat td subsection proof concept task learn result evaluation test try discriminate record game methodology discriminate player discuss subsection limitation progress carry feature matlab search however choose use matlab working challenge implement matlab carry windows ghz processor game notation file record module implement variation alpha prune minimax return game testing choose feature
mab bandit exist prove mab fix either tractable every occur bind countable furthermore show infinite metric tractable lipschitz mab exist payoff strategy reveal name strategy payoff could payoff reveal round select receive payoff payoff reveal strategy bandit version obvious specify triple space borel payoff borel induce product payoff lipschitz present metric algorithm measure expert formulate receive infinitely expert tractable even tractable full mab expert tractable tractable opposite ask space become completely characterization pre compact moreover upper even tractable compact consider view interested et dimension expert problem naive therefore natural ask lipschitz prove characterization low novel notion tailor expert space isometry tractable version feedback expert term feedback lipschitz expert feedback fix isometry tractable lipschitz tractable metric tractable feedback tractable min theorem point technique joint two identify topological entail topological topological isolate every call metric notion topology topological good perfect follow lemma tie topological compact countable iii theorem make mab complete space double feedback expert feedback expert lower possibly borel history past
update rule subsection divide theorem describe last mainly discuss already prior gamma observe theorem assume anomaly www traffic context follow cc update simply consider distribution assume prior poisson contribute constant newly remain conjugate discuss power model depending forecasting satisfy one mapping
spike spike leave show parameter satisfy interval run algorithm spike however baseline estimate solid variation penalty leave improve baseline right baseline index value influence new vertex connect idea baseline region baseline everywhere since contain observation baseline spike join region region knowledge spike encourage vertex join choose vertex influence cause baseline demonstrate algorithm suggest neighbourhood exhibit area
multiclass loss binary prediction multiclass scale regret classification composition reduction classification root undesirable regressor create multiclass bound dependence feature state raise winner repeatedly player loss start loss winner winner continue remain construction elimination see elimination good could single elimination imply concept introduce notation divide tree inconsistent motivating sensitive multiclass prove dependence return cost multiclass experimental filter multiclass present family integer control tradeoff minimize small setting multiclass early powerful adversary fail sort ratio large number call large label classifier define extend case multiclass binary
view speak proportional simulate entirely dominating suffer jacobian acknowledge turn thus turned translate compare
new inequality serve form gaussian happen useful controlling
see taylor al scale neighbourhood consistent h need estimator use standard expansion example rank symmetric without impose without point eq pn us rate q equation instance I pn l slowly example remarkable independence
otherwise assumption herein dependency level dependency level n local dependency level observe q sufficiently consequently pl finally let markov contain ts
method movie datum towards max present section several improvement extension horizon due nature optimization advance max margin variational incorporate field tight collapse variational gibbs moreover experimental incorporation expressive integrate utilize model advanced margin would efficient acknowledgement nsf support chinese nsf foundation project cb cb cb basic follow topic proportion word topic assignment n unknown manner take entropy discrimination principle call joint cm get cm evidence define intensive regression cm cm nonnegative multiplier fundamentally passive current tend measure closeness p infeasible compute apply develop alg term posterior solve optimize factorize cm form component lda formulation since thus expectation co variance affect step let vector factorize plug cm framework ki write normalization cholesky nice margin extend laplace derivation method normal prior quadratic programming via cm three see fix topic proportional normalize derivative cm tr
eq normal distribution account arrive conclusion big one task bounding appear geometrically chain unbounded defer next chain direct tight I definition uniform geometric ergodicity markov rest typical situation ergodic property acknowledge time problematic execute al intractable computing set clearly geometric bounding variance well reversible independence metropolis chain uniformly ergodic uniformly candidate translate spectrum contain lead spectral self adjoint see case
completely biased ensure message randomly select plot path two different first vary probability estimate check variation sampling uniform correspond crowd identity reveal identify technique symmetric protocol currently protocol assume source randomization security protocol arise due poor randomization identify provide wrong check formal correctness guarantee rely trace protocol promise technique extensively protocol project
outli outli outli confusion misclassification student small misclassification concern robust classification http www ac uk five measurement express rw mid line maximum body group class direct variable use reference add variate th table well read begin central initial without nan outperform grows slowly reach unchanged error finally remark equivalent suitable concentration last concern beta model compare relationship subset individual moreover schema consider remark bic
least compare use randomly instance variant nesterov efficient key programming smooth interior trace dimension multivariate relationship multiple response common observation response explanatory response explanatory coefficient independently classical square response widely response explanatory factor express column factor dimension decrease correction partial least formulate form linear factor regression proceed equivalently square able accurate estimate may relationship factor choose discrete nature unstable sense change determine loading advantage trace nuclear fan force definite fix idea
covariate adjusted residual proportion difference v n mc mc significant detect I I mc mc proportion interval whether nominal interval proportion either check significantly table conservative nominal see conservative nominal however replicate case level case replicate error symmetric tend normal desire term normally covariate mean term asymmetric tend covariate adjust residual specific mean covariate adjust stable desire nominal test desire error different error covariate extremely observe method w mean considerably similar case agreement estimate different usually case w acceptance rejection region k acceptance rejection
exceed condition x f g hold give utilize rank invariant strict monotonic able screening suppose nk nm size simplicity technical nk n nm sis generalized fan screen reduce size include sis procedure sis sis apply final avoid focus sis difficulty simulation generate standard variable double location proportion standardize correlation matrix variable simulation present normally setting lc median simulation size realization empirical zero covariance decrease behavior sis two sis mle sure screening choose difficulty computation burden addition sis worse complicated procedure like scad sis complicate screen scad scad regularization table logistic regression
choice long penalty biological order temporal causality correct order dimensional estimating entry diagonal also row tucker proof weight estimation iff j ij j th fix value pa pa j pa pa consider show similarly suffice follow show exist eq pa event lemma use pa pa pa ij pa pa ij ij pa pa pa without unique event problem j pa second lemma pa since imply verify suffice jx pa jx pa pa pa pa pa pa pa pa j
fix iteration propose literature point f extreme largely apply engineering environmental see parametric distribution widely approach
p p x correct involve parameter frequentist asymptotic asymptotic generalize quantity frequentist asymptotic prove result function bayesian interpretations pearson framework cdf confidence value thus interpret potential relationship framework aim frequentist report function component pearson alternative bayesian fail guarantee interestingly originally pearson main frequentist interpretation confidence proportional bayesian report interval disagreement imply implication coherence compatibility posterior coherence necessary dropping requirement bayesian enable coherence achieve pure toolbox statistical inference enable nuisance prior incorporate frequentist circumstance measure decision whenever information parameter observe independent confidence exactly function independent incorporate information decision apply combine cdf value bayesian level level function inference demonstrate hypothesis point mean give versus observe numerically probability value
therefore rejection inferior section new rejection evaluate envelope keep
inside ball produce center radius minimum first type guarantee seem scale follow solve obtain unknown center eq suffice multiplicative like scaling scale merely become ratio appears appropriately choose polytope problem translate rotation allow furthermore face intersection hyperplane polytope point want convex polytope polytope help equivalently radius center usually ball express maximization maximization rewrite mn identify set
detailed form lagrange expand rewrite inside eq multipliers substitute solution substitute divide substitute lagrange multiplier substitute integrate q rewrite side term likelihood exponential effort put name piece call give normalization notice bayes canonical section particularly process depend constraint ask constraint whether process simultaneously sequentially collect purpose infer x x process matter
full efficiency proof outline notation everywhere choose classical series assume everywhere without loss observe time well spectral g sum mat ern keep mind coincide constrain spectral simplify us b characteristic define sense square extract plus extract like important article w r w throughout denote integral index resp resp estimating identifiability encourage expect
follow bound loss generality law large boundedness n jx quantity result eq differentiable scalar decay exponentially combine using directional write n boundedness quantity moment apply lemma n know write hence taylor expansion directional also know large continuity together establish orthonormal span unitary proof chi square vector give freedom c pt mm corollary proposition pt pt pt nsf grant recommendation material author necessarily publish decide make impractical involve alphabet also divergence grow control remainder concept include ml formulae section term
impossible latter identical identically unknown know despite possible get tail present necessary contain important bound series analytic secondary research
agree presentation introduce notation strength skip fig via viterbi quantity regime jump remarkably indicate govern jump monotonically decrease decay jump dominate sequence virtue dot proper support prior increase towards maximal jump regime small leading dominate simulation obtain viterbi algorithm calculate directly continue value noise entropy naturally regime entropy usual transition decay point
st mh upper triangular h h h h svd dx I dx nu tv td nu tu td normalize arbitrary phase row let thus iid easy deduce density I
infinity already convergence well statistic put remark double wiener process function er von statistic statistic converge free limit form close condition fulfil tx eq mild regularity put strictly
bootstrappe pair empirically bootstrappe residual n ki iy bootstrappe strictly bootstrappe show behavior notably pair residual currently bootstrappe model replication large need regard various correlation sparse sign analysis carry tend show lasso asymptotically behave support index element optimality regularization path involve c solution condition note essentially hinge make quantity characterize noiseless speed relate q commonly simplify design consistent procedure satisfy weak lasso figure middle condition design observation relevant explore theoretical behavior extend current relevant ensemble weak various rate currently refine relax stability e probability average one local currently derive condition sign loading usually high regularization lasso zero hold asymptotic meaningful design would sign
glm perfect posterior depend smoothing may support fair adjust artificial gap quite distinct posterior frequently away true reason step abc outside support propose prior transformation form low upper prior complex abc introduce glm much less observe rejection abc glm posteriors posterior glm prior range uniform estimate maximal consistent human classify
rank w b p node table pattern find appear frequent neuron chain connection subtract count leave occurrence modify new occurrence middle calculate respective come conclude false apply procedure subtracting count strength great correctly true connection demonstrate high frequent episode spike upon future l technique place neuron connection neuron potentially connection produce spike methodology furthermore connection track circuit cognitive formation usefulness methodology analyze neuron record analyze al two repeat discover track
case martingale decrease admissible martingale analogous discrete greatly et al technique make mathematical finance article idea composite hypothesis possibility measure evidence composite measure nontrivial simultaneously martingale goal simultaneous produce demonstrate martingale simultaneous test simultaneous almost illustrate simple fair suppose random variable horizontal logarithmic power unbounded direction consider non frequentist require interpret probability merely principle section sampling r cox corresponding hypothesis hard sensible widely composite second composite follow hypothesis composite bayesian simple hypothesis example supremum ratio implication inverse supremum likelihood ratio value picture discuss composite composite represent former uniform latter calibration track supremum martingale relate reach high value conclusion sample evidence conventional finally back
smc impact single certainly demand e degeneracy particle contain partly de paris project mc author particle publish journal
hand within simple might fairly instance several phenomenon behavioral problem several might modeling trend outcome suitable outcome return treatment site eight type cognitive test stanford year primary year intelligence child allow site contribution treatment allow outcome index site site parameter contribution effect simply model variance test allow vary last piece plausibility exchangeability also display test score outcome different year age interval display intervention appear site outcome phenomenon outcome figure outcome correction linear example eight estimate towards adjust year type shrinkage cause conservative eight stanford score display hard arise model outcome exchangeable group model procedure comparison different comparison large variance procedure potential tuning multiple comparison false discovery hierarchical truly zero sense prefer issue term type social program equivalently interval wider prefer
project grid independent central chi cumulative dy dy z dx note transformation formula convert integrate nan consistent simplify jj important degree freedom sample plot blue base million row third use value count figure blue solid line leave middle matching square red n match l kolmogorov von nan gene na na na mean na na kolmogorov von nan matching counting count comparison use kolmogorov von hypothesis matching probability tail matching ii l distance chi chi match chi cm k chi counting chi require significance cm department university department university school il
step duality gap unstable drop qp increase proceed get hard fluctuation duality duality monotonically cut behavior cpu rapid decrease kernel huge computation per relation speed kernel spend dependency size repository duality tolerance kernel random report logistic hinge logistic increase median random train fast method however clearly fast roughly scale unstable programming iteration kernel good regularization increase efficiency block result converge small scale number good loss algorithm generates iteratively loop super linearly inner newton block elastic mkl mkl connection norm weight block norm give optimization elastic net numerical number scale
space context necessary dominate location minima rather global maxima location minima lower close dominate reduce dominate process maxima essential example dominate intensity location dominate eq start version accept process carry obvious naturally interest generate step give rule rule give ease notation pd jk
add direct loss equal finally vertex loss tuple action path generate tuple weight equivalent accord exponentially e connect consecutive task discuss exist consecutive short path tuple accord dirac mass proceed vector auxiliary maintain round state round action state show weight draw conditionally generate note equation put mass proportional action hide correspond previous task sm repeat draw space particular draw proportional realization store perform update number pair complexity negligible naive parameter action pair similar argument example transition problem
regime perfectly predict iff go countable going tailor design give interaction interaction realization convenient much kl construct minimize total expect kl divergence kl observation arbitrary divergence
ty justify z ty tx prove try component x te pa I rhs notice pa claim decompose term ty q call submatrix comprise index tight follow proof denote column cauchy distribution subsection iii satisfied least tail probability bound ty ty ii ty last pa x condition long arise system refer wavelet pass reflect coefficient reflect output refer trace record filter include nuclear train communication impulse output measure noise detect approach decomposition scheme
thousand implement minimal agent learning fail solution anonymous approximate equilibria game frequently design theorem system game confident learn become response agent guide beneficial second system something link depend send importantly distribute system agent play somewhat hard game efficiently game behave agent minimal converge algorithm round stage strategy agent choose reward current condition stage dynamic guarantee learner nash poorly number number agent many game converge dynamic game human weak common game interest dominant agent strategy agent result agent converge agent agent speed order magnitude play game
introduce weight impossible many purely procedure scale hierarchical simulated importance resample problem concern weight effective ultimately versus trade initial sampling practical case use trade produce nod facebook contrast bias facebook focus topological aspect one happen topological node facebook differently networks node law regime approximated power cover decade behavior formation degree degree cluster connect connect node degree vs study similar network summarize plot pearson coefficient possible explanation stop boundary high likely friend user friend phenomenon connection near node average facebook report clustering tend node fig privacy privacy bit privacy facebook setting aware level user privacy aware l l ab united co city bc city facebook user user extreme end facebook users middle seem concerned split english speak privacy facebook report finding present privacy facebook fig leave setting modify facebook modification friend facebook user factor friend classify b k fig degree find tend modify setting trend make concerned carefully facebook easy privacy c b average neighbor finally privacy privacy friend clear positive correlation framework recommendation contribution several rw bias unbiased efficient offline truth rejection implement finally facebook unbiased sample facebook characterize structural facebook publicly publication uniform
fig plot subspace bss procedure manifold separable must state invertible figs fig source version generate scatter plot nonlinear transformation nearly effect noise number information analogous multidimensional human speech content bss procedure capable recover speech content recover reformulate velocity space state space conventional attractive reformulate bss mix unique permutation independence velocity physical compose interact deterministic avoid difficulty bss previously bss ii recognition engine recognize state
suggestion marginally family subset entire construct contrast crp throughout relaxed version use notation collection customer number distinct customer iff crp customer among crp identical crp marginally invariant ij set take place customer assign value unique customer iff point partition crp decay note customer occur th distance crp customer divide unnormalized rewrite linkage crp traditional dependent crp decay produce traditional customer marginal crp separate customer fall within traditional crp group whose customer contain customer customer since marginally sufficient dependent distance equal customer may assignment suppose suppose crp collection distance marginally share customer customer customer directly distance crp marginally identical writing leave right possibility distance keep multiply either generality must claim proposition proposition decay proposition dependent crp result function marginally invariant
examine feasibility uniform even simulate simulating fall within accept reject step accept geometric conclude trial draw surface probability round first round accept algorithm normal bounding target exercise involve r n bound pi prop sd sd ratio prop sd mean sd rapidly moderately sampler conditional gibbs slice sampler chapter exercise distribution uniform conditional slice invert take exercise thus produce closed solution reproduce analysis exercise change deal simulation long beta interval necessarily confidence around establish exercise coverage devise rather conditional schwarz partly along time constitute captured observe past location conditioning make unobserve generate would bring choice proposal block thus far extend definition take thus consider mixture value define mixture distribution unless show deduce way trick ball case remove allocation ball represent ball case equivalently extreme speaking partition normal distribution unknown subsample weight therefore prior pair normal allocate via already find chapter exercise distribution exercise influence q lead tt sd step step sd sd em em em sd em sd em increase log along log start eq function give parameterization simulation conditional n I implementation simple mcmc allocation gibbs mu mu
field easy sense easily follow build create add edge choose field create merge merged node fashion seminal approximate instead ise edge indicate encode edge follow take subgraph value sum configuration equivalently normalize
convenience normal consider location q diagonal minimum uniformly coefficient constant hold l chernoff tail eigenvalue design maximum thus concave variable bind location regressor independent assume moreover satisfied since ff hoeffding inequality eigenvalue bound zero nonlinear impact k b f k class van lemma hand consider vc vc intersection basic class vc vc vc form van convenient positive matrix characterize behavior maximal regressor establish hold q converge uniformly suffice collect van n r nr p conclude converge conclude converge rely give asymptotic covering regressor negligible relative sample regressor number significant regressor quantile arise processing become widely investigate penalize acting demonstrate fundamental demonstrate fundamental forecasting thus excellent forecasting even growth total development review exist contribution develop result selection quantile since inconsistent quantile norm qr function near rate quantile cf quantile index cf hence complementary fundamental excess forecasting
either contradiction diagonal belong imply row contradiction column nevertheless open fact table element simplex inclusion proposition vector definition e divide j unit want find scalar study situation recall system prove chapter normalize generic equation quantity zero divide thus
bic selection lin fan wu regard selection tune drive edge edge true search upper search region find wang result model associate bic define suppose know correct two result construct working wu result identify coefficient theorem fan wu tune p ij ic penalize correct probability tend g fit essentially likelihood graphical denote ij theory model
vertex h segregation possibility definition association alternative definition alternative particular segregation triangle use segment segregation correspond away opposite edge argument segregation alternative normality relative mean asymptotic normality obtain degenerate r normality segregation universal normality base statistic segregation alternative value since segregation statistic underlie asymptotic critical sided segregation segregation test test nan might nan less mean alternative pe rx rx pe rx x rx pe rx rx pe rx p pe rx rx pe rx rx pe rx pe rx rx p x association consistency provide investigation asymptotic local segregation alternative graph likewise segregation provide work appendix segregation segregation arc detail figure segregation choose segregation segregation r sr suggest use association r sr imply
tb pg interior point ip iterative limited newton dual augmented discuss text row condition condition well parametrization see text mean extend beyond regularization exploit exploit sparsity intermediate solution efficient show bind subgradient optimization minimization method pg ip exploit efficient directly g stochastic subgradient problem strongly however since base fail problem poorly condition challenge information tackle poorly condition quasi quasi challenge differentiable regularizer deal fact easy many regularizer remain coupling show section consider strategy method manner operator forward need every current size since reduce ordinary gradient consequently loss term differentiable batch setting approach maintain throughout reduction compare interior sense difficulty size problematic poorly condition naive iteration need f minimal approximate curvature efficiency poorly condition denote guarantee per see member qualitatively low term rigorously study converge linearly course convergence cost nevertheless qualitatively mild intermediate effectively minimization optimize plus generalization exactly insight write problem else reliably employ duality stability optimal must trade
theory variance remark universit sciences et de france estimate simplicity exist performance relate emphasis distinguish statement rigorous conclusion accord likelihood least rely available call cv strategy generally selection cv risk error avoid overfitte popularity cv mostly come empirical cv entirely confirm prove fail depend identification question picture precisely aim question cv cv keep goal framework choose cv overview mind classical procedure cv risk cv framework discuss finally focus cv procedure conclude survey common observe throughout except purpose mean quality basically accordance accordance framework fit detailed statistical framework useful quantity relate purpose exhaustive aim excess kullback leibler aim predict predict multivariate sx minimal excess equivalent correspond finite discrete prediction contrast minimizer denote consider instead look classifier basically confident classical convex
update clarity exposition shift shift well brief cluster differ attribute model attribute standard deviation share sample process concentration paper rigorous notation since use least normality impose component discrete measure provide mechanism shrink towards derive specification mixture intend assign mixture follow
case include generalization loss inner david discussion ms support aid rt mathematical mail ac signal number unknown observation primal reweighte differentiable non series exist minimize component initial
choose regular ss without reasonable bind success il wise upper subgradient mean l n q n ss expression eq sample min q min denote expression ss bound line via defer version explicitly root
dominate roughly transaction item distinct transaction lift consider linearity expectation similarity indeed item efficient paper read measure sort carry track transaction belong compute sort list os pair transaction fit pair memory item sort trivial occurrence item os tuple transaction tuple write disk sort tuple transaction use os transaction sort support transaction fit inequality memory page phase os read total si os sort pair pair os os dominate cost previous expensive size imply transaction subset
effect residual h precisely star tm moreover suggest degenerate resolution rp investigate degeneracy estimation try try plus width degeneracy degeneracy amount observe would unable ap apply trivially star thus significantly function e g fails represent uncertainty accurately star panel dot connect show triangle red cross initialize converge run star near tag spectra various plane degeneracy star panel initialization quite offset due three wrong solution initialize poor practice reject turn k bad predict spectra indistinguishable star bottom panel band bp rp dash star convergence ridge solution initial final spectra agree spectrum degeneracy degeneracy reflect identification ap near star suggest degeneracy systematically spectra predict spectrum adopt original grid mahalanobis simplification inter degeneracy star arise nan hypothesis spectra degree freedom observe give less correspond word star degenerate square unnormalize one lost significance dd contour star grid get distance plot degeneracy star grid case word strong degeneracy plane noisy one indistinguishable lie contour
random relevant result map variable crucially exploit bind contribution let integer definition thus subset vector apply powerful context complexity measure extent uniformly isotropic constant independent copy immediately finish twice upper bind bad event constant condition proceed treat deterministic condition describe define formally design satisfy hold proposition constant theorem optimize theorem
lot reasonable serial mode mode general time c minor candidate generation serial episode episode satisfy maximal generate stream episode parallel episode run algorithm episode episode serial episode episode recover serial episode mode six episode embed result large similarly partial order embed episode episode see run threshold almost mining order burden candidate level level bound high font inner sep pt style c parallel episode xshift c cm b right g right font b xshift cm yshift c node node font c xshift iv c yshift cm node node font xshift node vi xshift yshift cm c font c g yshift node node font width cm episode xlabel ylabel frequent episode grid reduce xlabel ylabel episode xy rest run rest c embed fig vi iii mining robust frequency eliminate episode episode constant algorithm embed level generate event embed average number event datum stream signal refer come various episode stream episode verify ns tn l number simulation see say less signal candidate successive candidate level increase run go candidate episode variation observe keep run embed increase increase number paper discover frequent episode partial episode available discover serial episode present algorithm occurrence episode care time candidate interesting order serial episode serial episode episode order episode inherent episode one general discover partial order effectiveness extensive paper consider episode partial episode serial episode algorithm serial episode discovery extend present episode
grow grow likelihood converge rate towards behavior batch display leave parameter associate summarize box compare variability batch iteration range thousand independent em estimation small variance obtain use size online preferable despite slowly decrease batch provide choice guess size scheme poorly figure step pilot improve simple large size thousand figure cause slow reduction bias estimate contrast choice smooth reliable require observation forget l matlab batch forward
know neighbor direct respect construction query object oracle direct close query exclude neighbor learn near near deterministic must ask direct list direct neighbor ask compare every current candidate equally identify denote outcome answer ask probability tell answer branch star neighbor equivalent assume branch locate contain identify ask question direct neighbor object close question exclude direct ask direct know indeed choice inside direct way choose direct direct ask way neighbor consequently bit ask question store approximate phase answer question hash base application retrieve hash distortion underlie collection face angle distance similarity human human proximity respect human object make statement probably lead formally aim efficiently oracle object query ask phase answer
margin illustrate simple six order line value iff incorrectly picture label validation value use equation picture label conclude assign value p fix measure value label confidence q permutation value fraction permutation bound sequence proceed throughout prediction validation calculate
square norm error mahalanobis al rigorous depend restrictive continuously differentiable forecast al observation univariate setting et forecast stochastic definite respective hide apply forecast say resort score q wise derivative multiply one half whose element strictly convex standardized homogeneous extend general bregman representation rise scoring stein notion quantile score particular condition course could general type situation propose fr fr median traditional case ideally forecast quantity event al forecast evaluate name mean predictive consistent functional consistent scoring literature forecasting forecast develop theory notion consistency scoring expectation ratio expectation quantile forecast function mean consistent bregman form attractive homogeneous scoring evaluate volatility forecast recommend superior power west test ability depend direction desirable theoretically quantile forecast assess power appeal interestingly family square score absolute percentage median severe seem consider consequence score fair comparison compete employ score consistent scoring rule hence protocol point forecast predictive forecasting literature forecasting realization volatility notion forecast participant ex score employ target name name
table margin row count infinity bound number infinity expect valid asymptotic entropy variance integer sequence determinant lattice first tr dt matrix te zero multiply notation comment much prove particular third iv fourth require characteristic center cause determinant lattice possible let ik determinant large contrary reciprocal consist integer lattice reciprocal original reciprocal determinant reciprocal lattice
demonstrate via simulation connect robust iv phenotype status quantitative know effect size reasonably precise percentage phenotype explain snp expression snp phenotype unchanged table great estimation great association little I close choose big confirmed effect effect misspecification resample flat unlike mass via spike well balance setting clear meaningful standard base twice small study threshold snp nucleotide report discovery initially use selection threshold phenomenon winner curse winner curse gain major factor failure five three winner curse recent paper dedicate curse argue reliable
appears reduce effect simulation consider pool include approximately minimize individual possibility modify thus pool individual mark small together former preferable qualitative success matrix recover evidence set f preferable beneficial number panel case coverage read efficiency reduce dna pool oppose dna section price read black line incorporate cs without recover minimal prominent still number becomes keep beneficial number many drop hence add start enable number effective although read add high allele treat individual increase add naive black perform present identify allele deal rare display naive rare benefit put identification vast amount cs rapidly believe major advantage tailor may cs fashion use cs solver try keep optimize likely formulation incorporate allow sample example input signal consider post distribute frequency low deviation treat allele independently rare allele also
eq error specifically improve rao whose square iterative approximate iterative closed refer approximate although constant provably even well adaptive mmse fundamental accurately matrix year covariance matrix attract interest include microarray forecasting image brain activation fmri method motivation common unknown coincide maximum minimize mse
generalize heuristic projection propose estimator see reference apply selection drawback maximum criterion ridge support finally aggregate final present follow compute penalty optimal selecting algorithm practice grid embed thank possible case maximal jump entirely jump jump elsewhere state union obtain ridge regression parameterize assumption hold assumption happen true exist theorem assumption actually derive oracle section remark depend almost lead unknown look less possible ol estimator every applying inequality treatment compare generalization problematic great care deterministic oracle refer
probability learn price p sample program primal solution treat union distinct distinct price otherwise exactly price bind price show feasible primal optimal program high probability proof base two observation satisfy dual solution program ib hoeffding bernstein without replacement manner low observing hold solution feasible therefore objective online optimal decision period contribute relate optimal linear program program offline q solutions py dual weak duality p least follow event happen decision take uses fact horizon require claim propose improved price price history learn precise price rule dynamic set dynamic proving competitive intuition behind
purpose kf project onto span kf kf report table initialization despite probably even though low noise outlier superior kf kf kf two online good experimentally order typically exceed storage synthetic clear advantage study component outlier intrinsic much work extend
construction ensure n immediately lemma control error ensure compatibility control n violate borel inequality essence detailed continuity argument due limitation sketch iteration reach almost dependence obviously ensure almost control quantity iterate obviously initialization monitor locally alternate cg approach rewrite iterate cg
superior performance investigation total scad variation pde quite denoise modification success failure largely characteristic image scene color solid object exception tv denoise variation less near significant utilize spatially weight procedure obtain show computation tv piece development explain although priori numerous exactly imply shrink estimation piece wise shrink propose smoothly deviation scad become statistical image task tv
carlo run sampler section ht sampler two perform kind redundant frame conditional orthonormal histogram know reference wavelet pdfs tight frame translation filter norm frame coefficient value hyper suppose mmse estimator base frame report ht c automatically appropriately monitor base scale simulate chain use basis fig stop burn period computational use intel ghz architecture two propose sampler term simulation sampler fast sampler reduce
proof give generalization monotonicity sequence useful straightforward belong successive iterate decrease stop establish statement tend infinity subsection lie closure tucker type establish kullback distance notational alternate kullback lie particular first optimality assume subsequence small belong compact continuity proof make tend deduce claim decompose directional
method semi supervise denote select supervise force reference base automatic recognition accumulate list accordance censor additionally method oracle anti oracle hand oracle small anti oracle distance notion recall case g ax ax ax er experimental fig choose sup hour architecture choose use query indexing institute technology scoring evaluation trade demonstrate operating point quality rt rt rt sup sup supervise semi select table select observe error close change ax instance recognize incorrectly system recognize make consequently level prune word rt data rt phone error word way comparison may supervise candidate recognize contain word comparison standard rate reference necessarily well force alignment instance
relaxation problem propose max also advantage eigenvalue eigenvalue relaxation remarkable writing content follow section rapid lagrangian duality globally binary third section semi recover relaxation inexact deal binary primal solution give norm eigenvalue strong mean term maximal eigenvalue relaxation experiment method chain anneal metropolis gibb present recently relaxation far encourage duality denoise recover special pass second effort sdp inner real e hull closure denote diagonal relaxation quick duality collect role present
doubly entropy become bethe gm bethe free incorporate multiplier enforce look set quadratic observe contain among integer interior rely description set perfect refer bp approximation follow enforce belief enforce mf energy one example mf entropy explanation mf fact edge perfect properly solution bp homogeneous b show homogeneous blue green
table support methodology fit zero arise many practical study one incidence table viewpoint interest markov chain minimal may connect table cell contain square move connect table entry connect one find connect connectivity satisfied model markov restriction connect table organization rest preliminary fact theoretical table minimal connectivity one table contingency version discuss square conclude paper remark preliminary basis
recover original unfolding although symmetric eigenvector definite pd analysis necessarily case diffusion map neighbourhood vertex among point versa kernel map sum subject diagonal consider laplacian semi definite calculation constrain reformulate whose consist exclude diffusion map proportional laplacian comprise important point either map laplacian pca modern discriminant kernel multidimensional scaling extension embedding cloud quickly induce computational thereby motivate introduction increase use across field exploit redundancy often relative summarize subset actual turn accomplish via nystr om om upon
primal institute university cluster anneal schedule find assignment sa furthermore easy sa implement popular typically optimization relaxation np simulated sa promise able optimum slow schedule temperature schedule cluster sa find reasonably good solution schedule mechanic anneal novel add sa annealing see
property subset question association geometry analog heat operator evolution start hamiltonian method summarize quantum formalism vary detail evolution wave behavior quantum clustering begin state hilbert view datum move original distance change many center instant cluster association reason visually explore approach begin know window euclidean gaussians scale view observe maxima determine cluster maxima prominent location maxima clustering take hamiltonian potential function inversion harmonic center original expect minima maxima minima frequently turn minimum display maxima estimator associate potential simplify identify effectiveness two wave wave
combine proposition case recover moment normal likelihood tail implicit shrinkage illustration shrinkage show shrinkage zero asymptotic turn rule fit soft thresholding piecewise match odd symmetry functional x eq slope numerical indexed yield figure laplacian mmse compute numerically soft piecewise approximation compare numerically idea straightforwardly may unimodal mean shrinkage haar coefficient maximum identity multiple may detail equally set aggregation notation location haar result shrinkage operator furthermore exponential yield unbiased thresholding iy I result case white variance
form clearly quantity measure easy follow chi fisher formalism possibly equivalent eq r us concentrate equivalent lead give g eq name center chi center reduce distribution quantile build course would classical fisher formalism comment robustness express g kind
numerator posterior density alternative posterior define joint guarantee representation bayes marginal posterior impose uniquely impose bayes everywhere hold apply formal bridge conclusion whose uniquely define version therein satisfy differ density implication variable
topology uniformly ij consequence computable computable de computable vice joint generalize topology unit bound immediate integration continuous function borel relative access recover uniquely moment show mixed evy inversion characteristic function xt finite dimensional inversion direct obviously instead computable mixed moment mixed building polynomial pointwise polynomial approximation rational polynomial computable show uniformly x n f computable effective pour coefficient polynomial computable integration computable moment versa computable moment uniformly computable consider coordinate computable uniformly computable establish eq lemma computable converge pointwise indicator dominate convergence sup p linear moment supremum exchangeable sequence de distribution uniquely show distribution relative claim computable x let joint computable order furthermore preserve obviously lemma argument succeed result mix c mix uniformly co
theorem distinct contradict outlined problem express way either solving secondly appearing call polytope convert polytope convex polytope show dual cube distance point construct general particular exactly eq unique hull write combination definition cube hence convex actually system uniquely linearly know zero apply contradiction assumption construct maximum summarize short cube ray optimality suitable vary value cube dl point first section define p prove construct occur path correspond exponentially support
strict final categorical aa useful well combination follow aim investigate whether disease follow month stage triplet patient triplet interval length expert aggregate decrease accordance law peak sense peak sort peak frequent combination combination etc empirically coefficient aa error strict show aggregate good include combination
fan von strongly simplicity x square dual definition xu fx fu f xu xu u x fu f fu equation imply point modern impose restriction much understand nature implicit knowledge impose norm tailor margin suit sparsity deal systematic work regularization norm methodology property notion ability range batch impose complexity online central behavior lie tangent strictly characterize norm rather convexity understand duality strong smoothness approximate linearization function expectation sufficiently smoothness enable focus characterize derive rely previously specify systematically decide base underlie matrix much recent usefulness
evolution equivalently evolve partition se complement se recursion evolve sparsity guarantee mse subscript evolution depend essential simplification prescribe sparsity moment threshold proof along derivation explicit expression precision numerical evaluation canonical problem formal mse region optimization formal truly thresholding would imply allow solve hope last five possibility replicate behavior iterative property phase extensive iterative algorithm nonlinearity phase small lp base
necessary present structural boolean noise boolean bound sensitivity polynomial define regular oppose polynomial ia ii multi say multilinear work multilinear notation unless state degree polynomial p multilinear polynomial either px ss clarity exact extend determine play role intuitively regular variable high coordinate polynomial multilinear regular qx regular loss invariance anti concentration degree multilinear follow anti concave anti polynomial qx generalize
sampling mle considerably slow mse importance approximately find c map l equivalence spline one interpretation coefficient example model extreme modelling datum pair distance measure point formulate coefficient interpretation detection number location perspective carry birth proposal equivalently express framework retain interpretation high note current datum truncate poisson truncate change inspection sensible interval occurrence point mcmc iteration burn
rule estimate adversary observation explain discuss practical neither precisely obtain amount prior distribution possible adapt world employ point process decide discriminate employ rule compose shall omit discussion user discuss adversary model decision adversary shall rule perfect information adversary know priori adversary generate oracle adversary user concern adversary uncertainty
standard idea near may conceptual optimality closely problem censor wang yu support special california author like li david van introduce grateful problem vertex simple monotonically never algorithm slow strategy b propose vertex exchange effective consider work name combination include multiplicative ingredient contribute effectiveness neighbor strategy intend
able explanatory delay differential dynamical application availability simulation routine application system thousand parameter development believe exploit make improvement come adjustment iii develop kernel exploit inherent substantial molecular species uncertainty employ sequential range describe pathway simulation ability employ exact mathematical analyze complex describe model allow compare ultimately rely
starting give discussion initial event manually quite insensitive initialization yield similar keep mind discuss outline mt cumulative distribution cdf interpolation interpolation calculate quantile value analysis serve zero avoid issue vertical interpolation axis cdf polynomial interpolation calculate cdf would however spline inverse simply linear calculation interpolation quantile adapt presence potential curve beyond hasting would proposal maximize
show unique due string statement finite uniquely value term language introduce unconstrained translate contain yield yield ideal generator follow polynomial map constrain model string result attempt novel notice explicitly state adapt language function process view module language area amount column string function unconstraine hide
demonstrate detailed velocity interesting evolution disk deviation close gaussian velocity expect simple model disk dynamical bar shape region galaxy part disk rise velocity b similarly steady effect coherent infer signature directly observable need direction branch along line motion rise apparent background distance apparent shift star traditionally pc intrinsic star systematic background source angular motion proper velocity velocity star line around uncertainty star purely velocity measure line star distance reliably star diameter
regardless scale aside leaf would split prefer option posterior surface thing approach involve grow move proposal whereas sequential change swap present cost desirable global filter observation dynamic leaf combine l pass split leaf predictive covariate somewhat subtle point would predictive efficient impractical seem distinction dramatically concentrate option leaf reduce functional likelihood leaf regression ease evaluate prediction free fortunately subsection regression move example process gp leaf impose provide propose herein posterior inference lead reversible particle sequential thus dynamic tree nature expensive lead preferred choice application modeling leaf node mean eq leaf force leaf set framework scale leaf likelihood allocate leaf student leaf plus shift partition essential probability marginal marginal
produce look extent occur jump hard infer occur position number b plot black line green blue dash analyse log probe surface number change interest detect furthermore segment model thus idea change position remove analyse analysis possibility underlie allow model whenever plot variety way
relevance especially relevance feedback kl yet consistently one explanation separate word similar ad hoc expansion advantage work integrate coherent manner instance document explicitly redundancy could account preference similar walk advantage query underlie retrieval noisy leverage know know query successful ad hoc retrieval view perspective ir work interpret query expansion ir terminology importantly expansion hoc statistically language describe
pt central moment rv since central moment th central moment eq accurately skewness define dividing moment respect give rest expand expression skewness odd contrary taylor around indistinguishable first approximation skewness coefficient unbounded denominator numerator skewness dominate rational tend whereas one numerator determine behavior use skew skewness restrict sample commonly likelihood mle present input output function branch allow scale x w cdf pdf rewrite transform depends necessarily coordinate approximate mle know back compute uncertainty likelihood z w think transform dependent support location depend crucial asymptotic mle depend confirm gaussian behaved show unbiased however rarely kind estimating build output aspect skewness estimator see detailed discussion back iterate
kernel illustrate covariate transpose rwm posterior compare plain rwm rwm example easy b hold ny hold apply measurable example covariate presence absence disease variable chain discard burn contain section generic actual appearance toeplitz use follow martingale pt notice lemma imply similarly g ns
importance closely match estimate use importance equally suffer select accordance fitting carlo possible solution aim produce regular associate normalise sample approximate update term divergence kullback divergence relative select large density parameter specify usually latter preferred tail tail vast density mixture considerable flexibility wide posterior follow population monte carlo sample produce self normalise output improve increase density normalise importance induce extra variation residual systematic sampling generic gaussian arbitrarily parameter result proceed recursively iteration normalise counterpart function derivation expression simulation sect appendix kullback divergence time although target density improvement shannon normalise kullback good agreement sample non importance high weight ill fitting also estimate normalise tf fx importance formula derive normalise adaptation chain simulation consideration autocorrelation chain approach fig great depth next use student
increase density solid segregation dash empirical critical critical empirical segregation estimate nan solid monte investigation segregation alternative replicate mc separation nan density carlo replicate notice function skew segregation monte carlo segregation skew almost skewed segregation alternative empirical empirical empirical segregation carlo experiment six estimate nan case segregation symmetric occur skewed segregation dash level critical significance furthermore estimate increase decrease segregation alternative leave function power test standardize experiment value empirical power nr carlo investigation j r appropriate yield desire segregation carlo critical segregation function circle represent
limit roughly unit potential somewhat level manually unit uniform size number number need incorrect outcome voting report count store design make delay public release candidate mail publicly consume way mail batch sized unit article risk limit post insufficient initial inefficient insufficient likely detect well unit incorrect inefficient manually another constant pair insufficient win calculating produce insufficient proportion vote total upper sometimes actual insufficient poor insufficient sample state vote occur cause incorrect unless expand initial understand logical implication level paper say case manual calculate sample analyze win candidate even mix calculation precise methodology save computation resource calculation sum within margin win candidate separately win candidate pair separate multiple initially discrepancy conclusion less unnecessary failure manually count stack sample post confirm instead zero confirm win vote limit post risk wrong winner incorrectly desire summarize post detect incorrect outcome win article calculation size unit single winner treat candidate equally margin error ensure allow win candidate win candidate weight conservative method generalize sized improve material development count voting would vote specification provide count
combine argument suggest take quality typical example figure possibility choose already show take uniform goal choice compare hyperparameter usefulness take depend wavelet average repetition display wavelet deviation thresholding estimator keep wavelet behavior minimum laplace minimum small indicate take estimation level threshold perform generally display small oracle introduce resolution large confirm reasonable satisfactory j test nj htbp nj deconvolution performance wavelet deconvolution penalize collection contain integrable band adaptive band limit localize perform select make realization display
bs locate square specify figure well match perfectly diversity otherwise gain select channel selection l z gain equal throughput transmission throughput tend intuitive throughput increase use analyze consideration location two diversity alternatively noise interference limit otherwise system select exponentially conclude gain select good overhead compare help
capability build tail data gpu ran kernel formulae pass result preserve none formulae deal effectively value significantly generation extreme argument move compare quantile code quantile remain cpu improve scheme gpu gpu dramatically analyse rational gpu initially follow computation way gpu benefit improvement old form sde evaluate payoff make relative substantial increase gpu computation lack use provide understanding hardware etc optimization also would course base different distribution pre exponential normal evaluation benefit time work precision modern gpu need somewhat us satisfactory monte almost ok sure one repeat work generate concern associate unity issue concentration thus double unlikely candidate method feed high assess series formula suitable http en probit series code cross precision agree
objective encoder decoder controller reliable decoder encode redundancy decay scheme universal aim immediately ensure compression compression organize basic code list regularity result three suitable regularity direction contain source alphabet separable borel index marginal whenever carry subscript e process absolutely avoid clutter omit whenever measurable supremum partition gray density respectively dominate pf partition measure ball marginal coefficient observe split equivalently memory alphabet assume space letter distortion x countable binary string equivalently bit encoding block encoder precede composite denote compactly source process letter reproduce expect rate string distortion convenient distortion q lagrange multipli control distortion trade geometrically intercept line passing carry subscript
increase decrease end rt defer r nan hypothese corollary conjecture pairwise make piecewise illustrate approximation near various far tight course interval state another corollary similarity il corollary solution whose approximate dash immediately imply bound expression variance fact theorem aid pattern shall satisfy may open closure three transformation
hold model fix vanish old old hold maximized give increase sbm assume factorize node variational em variational look conversely variational sbm intractable full modelling decompose eq problem variable look tractable q factorize q n approximation equation bind initialize vertex euclidean distance assignment parameter variational cycle value small
tx sd think round weak produce validation possibly much empty solver draw dictionary round always find global equal objective low classifier algorithm rather try decrease may construct add classifier outer initialize current opt outer tx w hx tx outer outer outer classifier qp per adaptive weak yield accuracy apply
likelihood make prediction correct additionally interesting likelihood compare predictor predict split provide reasonable learn room specification graph sequence run graph type benefit evolution r transition dynamic outli initial behavior explain usual choose exclude fit type statistic transition magnitude validation start include plot magnitude play play involve two expand include explain section person support support person person person look link time effectiveness weight possibly co support could proposal happen
accord hence observe receive accumulate average long reward notation highlight average depend addition exist default correspond policy assume step make determine phase confidence fully turn sufficient algorithm suitable let instant terminate exploration phase policy three four policy follow finitely default select remain phase include otherwise confidence policy say empty exploration basically policy policy hence purpose stop even neighbor challenging course consideration suggest
c study robustness modify cs measurement case correct relax cs gaussian simulation change measurement varied normalize rmse expect cs cs regularize modify useful recursive reconstruction sequence clearly signal support contain application cs reconstruction estimate compute initialize I modify support wavelet index measurement e need measurement slightly ensure reduce increase increase enough alternatively support keep add element condition number l estimate l typically l h compute either empty compute cs tt knowledge reconstruction reconstruction something generate reduce regularize
iterate current density processor kp kk block processor carry draw chain draw marginal likelihood often marginal importance metropolis hasting scheme discuss use auxiliary particle comparison several datum illustrate flexibility applicability particle filtering acceptance compare define acceptance metropolis proposal sampling divide scheme generate autocorrelation lag lag acceptance factor measure effectiveness interpret sampler attain draw depend whether equation transition autoregressive outlier distribution truncate shape particle apply yahoo finance datum
directly connect detailed usage publication initial contain reaction light cycle nucleotide dna sequences complementary basis one side reaction produce repetition encounter sequence dna subsequent available reaction cycle test positive chemical reaction fraction copy call average incorporation mean copy next test response incorporation incorporation measure negative account usually description consideration differential incorporation effective description consider slight adaptation base sequence base explain usually cutoff keep definition rate incomplete plot direct look test fraction sequence test count instance position value direct path start
involve discuss subsection approximation condition conclude kkt condition exploit way albeit adjust restrictive relate illustrated figure condition compatibility look substantial room eigenvalue compatibility selection lasso condition zero perhaps strong compatibility verify compatibility approximate analogue realistic cover situation nonparametric slow contrast linear regime compatibility population regime general formulation compatibility condition well need
previously conventional association tree lasso mostly recover report indirect evidence grain module perturb article call jointly leverage response analysis snps activity co express smoothing originally induce lasso view reduce false part gm nsf research problem quantitative trait mapping discover genetic gene level investigate capture hierarchical response node response internal leverage recover relevant structure systematic coefficient manner membership employ induce penalty simulate show pattern run agglomerative expression scope compare hierarchical gene correlate leaf internal root internal represent subtree level common type effect strong correlate gene small height tree among correlated cluster tree great height grouping available
bernoulli generality secondary common power secondary assume bandwidth channel bandwidth exposition htb cost account spectrum monitoring sense control channel past access model access user always pay monitoring regardless decision begin whether stay spectrum option bid bid vector component q indicator function maximum bid spectrum incur accordingly spectrum access stay game transmission occur
significantly homology equivalence dimensional homology large close homology bind characteristic tie heavily homology throughout give promising area summarize point apply concrete particular al homology class systematic quite seem naive attempt argue rich object beyond real spirit different either via question decomposition arise failure extensively g property algebraic discuss generality complete calculation laplacian rarely possible rather homotopy invariance write essentially section theorem must point assertion moreover dimensionality theorem prove helpful measure result homology subsequent take homology complex homology endow product keep mind domain euclidean space lebesgue borel assume certain additional simplicity impose later section concentrate rich notion locality instance define complex map space construct confusion operator co formula sequel recover operator define set gradient linear integrate right give complete map side first term co define l inequality inequality side finish proposition theorem variable put
heuristic explanation observe value gain design adjust place design increase weight gain increase convenient small remarkable place literature algorithm attract attention appear case probabilistic inequality know assume see broad aim condition ensure certain
reconstruction proceed definition probability number bound complete finally rf threshold operation rf hold stable bind algorithm notion addition exist reconstruct specifically reconstruct know great setting interesting tradeoff direction sample corollary proposition conjecture axiom grant project motivate compress sense reconstruct attract theoretic variant main reconstruct number impossible reconstruct matrix number entry specifically stable matrix satisfy incoherence propose pursuit reconstruct factor away information limit compare sdp perhaps importantly sdp keyword randomize pursuit randomize subject arise frequently
particular belong next section hierarchy hierarchy order alone understand internal word small dictionary function ht good bad dark natural multiple variation word loop empty vertex subgraph strongly define represent c vertex derive natural model graph criterion world turn degree degree seem respectively connect remarkable source correspond strongly heart cyclic analyze dictionary relate age word word
connection like cluster assign green seem prefer assign cluster approach membership integration possibility relevance publish elsewhere provide cluster basis development online connectivity green red green gray w gray red green gray red gray belong family prove claim stand link connection aim estimate incremental recursively update current notation ij addition use addition new q n z update relate em purpose maximize difficulty factorize base strategy replace computing efficient alternative base monte approximate tractable node extent n
parameter mat ern present closed attain degree old smoothness behave ern prove covariance appear derive spatially vary second kind spatially vary mat parameter determine covariance problematic feature difficult separate scale local therefore desirable interpretation suppose geometric positively identity parameterization simplification smooth recognize mat parameterization page smoothness amount simplify prominent observation simulation automatically motivate theory equation local w therefore minimize exclude fix ignore form one
prior give stage stage dirichlet contain situation stage regular stage attribute unchanged stage dirichlet stage prior dirichlet prior child child dirichlet consider node child terminal child situation lastly child order stage root node leaf clear probability justification stage markov event analogous modularity property colour margin definition prior situation partition theorem trivial prior u v n leaf place terminal stage
nine place person action hence represent value test length normalize state nonlinear consistently autoregressive particularly long range dataset nonlinear model range contain history dependency incorporate k trick filter map reasonable classical recently time parameter estimate prediction system dynamic incorporate nonlinearity dynamic nonlinear sensible together observation purpose maintain formula maintain exponentially difficulty
difference leave incorporate svd basis incorporate right singular singular length svd covariance whereas row share differ imputation formulate computational limit variate application imputation maximize exploit maximize one coordinate considerable mathematically behind briefly discuss computational strategy consideration computationally suggest foundation calculate procedure example bayesian step address imputation method covariance multivariate notice penalty inverse covariance kronecker leading reason method repeatedly see imputation propose strategy present review part miss develop imputation mathematically begin seek indeed likelihood em e variate trace matrix let value step form pp variate structure imputation correction jj jj jj jj ii analogously form maximize along solver multivariate gradient penalty since differ term take eigenvalue decomposition eigenvalue lasso apply
