initialize orthogonal previously normally convergence previously continue randomly vector differential entropy bss entropy parameterize flexible well desire component see sample per expectation average number via transformation projection differential entropy minimize realization include prior prevent make condition previous uncorrelated source drop term entropy much uncorrelated bss use algorithm show differential enforce bss replace work mix replace entropy partial dependence thus bss flexible partial order dependence source base bss create use mixture mix mix mixed picture source true source inspection http www project synthetic source use source fraction choose sum mean source eigen sample orthogonal matrix matrix construct sample blind mixing mix pm package odd coefficient compare run value non gaussian normality pass empirical cdf normally normal show run sample variance illustrate base bss real berkeley first standardized mix draw distribution noise image orthogonality odd orthogonality without orthogonality figure pdf htbp figure use middle row orthogonality constraint projection show estimate source projection htbp estimate use algorithm middle orthogonality constraint impose orthogonality constraint projection result show fail use upon converge fig detail default setting successful see visually fig intensity exactly well source compare orthogonality model extract source monotonic note distribution gaussian approximation adequate htbp pdf evolution log increase em estimate bottom project density jointly describe show algorithm distributional part distributional distributional root cubic objective instead property check equation next consider model bss bss sources bss difficult one et develop bss problem latent imply factorial latent derive em mix factorial provide exact bss exact intractable propose variational bss posterior bss couple assume bss model ml bss reduce orthogonal differential entropy approximation entropy et modeling factorial suffer work al flexible source al differential entropy distributional observe feasibility source arbitrarily densitie ml factorial source retain computational tractable source approximation variational beyond scope follow application result source simply bss generalize source bss alternative worth bss fig pursuit mix latent source mix transformation source uncorrelate become pp general contrast pp bss sources bss think preprocesse orthogonality solve maximize objective concave maximization problem maximization global method compare run mixture average significantly well would like mention true fact biased experiment illustrate true illustrative publicly visual inspection relatively compare orthogonal density multimodal fig model capture orthogonality perform orthogonal independence correlation dependence limitation bss slow present minute estimation versus ghz intel processor allow encourage explicitly become however dependence bss approximation average become square bss et follow application density illustrative furthermore non zero simply orthogonality requirement assumption unlikely bss optimization vector future acknowledgement acknowledge n medical db lb thm section convert gaussians project vector observe estimate blind bss differential minimize minimize entropy project space simultaneously projection flexible fitting near conventional bss feasibility mix distributional parameter maximization algorithm directly generate unknown projection project gaussian distributional particular derive em similar conventional entropy blind bss analysis ica mixing source signal mixture corrupt latent ica free bss bss reader bss entropy contrast density maximum importantly assume denote al expression flexible bss adequate true near al bss density model factorial advance density addition bss follow however algorithm become computationally moreover al develop latent apply source bss question show apply problem allow scalar denote font bold letter vector upper e p zero bold th symbol denote I multivariate denote ba natural density capital necessary indicate vector expect need estimating model projection need suppose vector form think project scalar scalar realization also matrix column unknown satisfie assume mixture gaussian q q equation project gaussian give project projection project model contrast done prevent onto simplify p write gaussian handle expectation allow along prevent onto gamma similarly vector improper prior choice posterior denominator density solve problem subsection algorithm maximize deal logarithm rather substitute get q density get introduce write write monotonically iteration converge local stage step outline alternate fashion ordinary optimize convex post detect error use subsection discuss detail lagrange tucker w q impose constraint noting give ignore optimization constraint always inactive upon get simplification satisfy condition lagrange multipli lagrange multipli derivative optimality upon simplification write summarize matrix diagonal distributional assumption require dependent maximally source bss difficult couple equation q would require tractable compute maximally correlation loose difficulty bss computation bss trivial fundamental follow detailed bss leibler divergence concept ng variance co invariance singular kl algebraic co eq length theoretic vector p kl simplify mutual linear satisfy q equation see use element second dependence see optimization minimize component write source variance variance constraint encourage nd order term minimize uncorrelated encourage source weight serve balance term note nd drop equivalently maximization minimize mutual satisfy maximize uncorrelated vector minimization bss estimate mixing try answer question approach uncorrelated et al problem second order allow seminal bss problem exact gaussians since flexible sufficient able complicated give component flexible factorial density al integral analytically also gaussian factorial derive obtain et note source describe component step intractable overcome et approximation em summary bss exact intractable
situation team overcome opponent pass team try move penalty area confirm conclusion relational domain whole multi system activity focus team base behaviour agent system team use symbolic sequence relational multi variate environment action characteristic team discover relation raw transform set symbolic b forward forward b forward right b behind c b front forward forward b forward front forward forward right behind university department science di electrical college uk ac research institute research multi agent whole team member systematic verify effective member team observation analyse recognize team action focus challenge team allow symbolic translate multi variate environment sequential team sequence express first logic atom relational meaningful pattern relational team reasoning field intelligence robot environment act task carry challenge always occur design environment create decrease task less realistic multiple capability perform goal possess detection internal planning execution recover engine must income world since outcome environment detect evolve plan either correct team execute task involve difficulty task decision limited robot result opponent agree decision complete robot robot achieve goal address collaborative behaviour environment aim effective team compare recognize behaviour direct team deal agent activity agent team precisely four robot base retrieval base message internal retrieve adaptation situation model action description perform retrieval action individual act allow challenge rich robot pass robot domain require action team implicit mechanism avoid ball towards direction force region etc result always try move alone towards try avoid turn pass avoid ball individually pass chance address represent perform relational enable human understand observed principle could qualitative team concept recognize proposal extract multi file relational describe discover relation interesting relational offer advantage frequent action team team distinguish pass adapt solve snapshot robot point robot robot internal belief correspond game contain retrieve action formally tuple relative ball reference robot position ball position ball scope within retrieve easily relate robot perform retrieval solution one retrieve evaluate adapt case feature index former robot move appropriate position latter goal modify idea modify adapt description indicate function similarity compute harmonic similarity compute cost modify adapt distance position adapt position obtaining adapt global execute manually create file system three real essential minimize speed index list store thus index similarity cost potential select case adapt robot retrieval order player maximized multi interact environment among exchange internal copy base responsible retrieval process distance robot high correspond begin part adapt reach execution rest next execute execution rest receive retrieve arrive rest stop relational mining extract frequent pattern define team language knowledge comprehensive introduction logic programming inductive logic represent atom consist empty symbol symbol argument atom formula symbol apply I symbol symbol apply x ml implication I I l clause term say contain argument ia ba bc indicate substitution subsequence least element subsequence distinguish atom dimensional atom characterize involve represent denote mining inductive programming discover relational mining take belong pattern evaluate support wise lattice order lattice generate lattice evaluate candidate discard frequency frequent pattern actually dataset base start add atom try potential discard store one user refined pattern pattern discard operator background clause contain description relational file able describe team frame successful overcome opponent player represent shape body orientation team compose team robot represent stream consecutive player stream recognize team basic use activity attempt team goal gain ball gain gain player move avoid opponent move move ball toward goal opponent process occur trial place ball recognize contribute describe team necessary take account perform agent environment qualitative world allow recognize event persistence take indicate subsequently opponent compatible previous therefore occur another opponent try ball try event event hold full move ball go recognize event contribute ball pass depend try case robot consider situation robot opponent take system consider event world event event system current represent ball describe coordinate identification relation specific viewpoint perform viewpoint robot opponent front back precise reflect generality abstract sequence symbolic abstraction raw relation opponent box use relation horizontal vertical ball player horizontal horizontal describe environment pass robot predicate trial ball opponent go pass field stop action environment direction intercept opponent opponent relation action playing point behaviour team work action team behaviour extract set able entire team allow strategy global well take thus try ball behaviour team team sequence extract team version simulator create team platform competition additional feature implement message simulator simplify noiseless ball position action perform degree randomness end ideal trajectory robot try simulate failure situation real ball start gradually decrease code automatically exploit default default home go beyond otherwise remain home region ball maintain view consist vs team account two configuration define configuration enter home overlap assign region player robot region player situation avoid define occur come position scenario ball middle field side figure remain home ball middle front decision make execute critical avoid ball set qualitatively situation respect perform since action near goal attack perform trial trial advance approach analyse file game implement able identify level construct sequence team record simulation atomic trial file configuration sequence ability reach goal reach reason end trial configuration trial trial sequence dimension one recognize predicate dimension relate analyse recognize understand team preliminary relate degree behaviour relational abstraction action recognize scenario recognize action indicate possible team table amount behaviour team high team play collaborative strategy try reach move adapt sequence action describe behaviour robot ball try area front opponent act pass recognize would opponent adopt behaviour succeed action recognize cc cc scenario scenario intercept cc cc cc scenario scenario
firstly integral via part secondly b unity ts ts ts k ts ts ts ts ts ts ts ts treat ts ts h k ts ts ts ts signal spread suffice highlight benefit walk reject repetition compare transformation yield version preferable transform statistic indicate intensive accurate weighted df chart version transform df control chart acknowledgement multivariate read minus corollary keyword robustness analyze difference stationary issue analysis often unit nan stationarity favor significant establish define existence equilibrium relationship favor stationarity equilibrium statistically justify test test residual monitoring procedure monitor horizon detect stationarity soon stationary method analyze detect trend usually stationarity see al b limit walk substantially detect change nan alternative apply control stop series horizon convention value design average value serious present characteristic focus article version strongly innovation limit white ensure stationarity modify df time cover local unity alternative df monitoring nuisance firstly nuisance secondly appropriate transformation long dependent avoid least propose root test type accuracy monitor procedure detail motivate carefully series type relate distribution hypothesis unity asymptotic drive brownian motion appear nonparametric assumption allow motivated approach series suppose ar error characteristic root polynomial root imply error calculation ar unit root motivate time sequel univariate satisfying concern impose assumption series functional central brownian wiener strong time coefficient equip satisfie regard nonparametric ensure converge orient definition stock scale notation uncorrelated process exist negative random put common assume common standard normal al unique addition decay coefficient control criterion detection estimator suppose uncorrelated rao standard brownian motion base nan unit quantile want signal root dependent alternative specify check deviation unit df use sophisticated weight current define detection df play smooth distant dominate kernel ensure decrease weight nuisance regularity scale procedure instance otherwise look ensure number get parameter appear limit choose relative therefore restrictive alternative stationary signal start monitor reasonable approach type ensure converge nuisance past datum west specify estimate west q lag truncation west study chart series point calculate estimate less transformation back weakly limit consequently denote ensure rule next asymptotically valid ar unit root statistic regression df df numerator rule know nuisance alternatively limit eq provide functional central theorem define walk nan define property hypothesis central provide weight df series asymptotic representation stochastic ts ts ts ts ts k ts ts ts dr bound variation cf hence eventually equivalent integral integration obviously ts tr ts b k ts ts tr k ts ts function ts functional weakly motion brevity integration ts ts k h ts tr tr ts dr u r dr uniform position numerator ts ts theorem entail dr follow detection require limit result proof brevity omit detail tt limit lag chart estimate e equivalence ts c ts continuous write pointwise nz ts immediately definition schwarz e pa mix decomposition ts tr r r secondly consequently triangle inequality ts ts r yield ts ts ts note c provide transform satisfie transform chart particularly asymptotic yield ts ts ts ts ts ts ks dr dr theorems weight process associate walk k depend parameter note ts ts ts ts ts ts ts ts ts ts ts q fact ts large stochastic yield stochastic
already appear physics aim go group symmetry regard viewpoint hope application let briefly mention predict object assume joint provide previously observe input random training construct unseen object binary pattern input usually describe call handle categorical encode signal etc simply system either quantum provide state respective learn could scenario quantum channel input need classify input alternatively binary coupling want quantum quantum measurement unknown learn training estimate apply intermediate error measurement excess converge size state estimation plug constant state perform minimax capture behaviour key theoretical tool quantum extension roughly say collective quantum system approximate gaussian classical derive strategy benchmark harmonic collective measurement one moreover collective optimal measurement showing optimally simultaneously close template framework template discrimination special size quantum special bayesian difference measurement special symmetry mix quantum propose series paper regard paper scope investigation give classical discusse emphasis problem problem variable subset first second stage present ask guess unseen depend overall measure distribution equal give know bayes classifier choose probable respect bayes fit naturally give optimal known ratio choose high likelihood verify density measure return unknown learning primarily find classifier parameter subset certain excess call q alternatively bayes classifier hand one replace equal excess fact away situation show behaviour bayes area curve area green deduce determine around call roughly speak satisfy parametric learning theory slow depend one principle learn solve procedure function plug classifier aim close paper front quantum well method phenomenon stem optimal measurement quantum counterpart quantum describe probability state denote counterpart state description think measure classical prior classifier measurement x copy together permutation copy operation unitary binary element random copy quantum state whose guess misclassification nothing optimally prior sign tr quantum arbitrary risk q vanish analogous set besides obvious coin similar rate discrete continuous explanation label outcome way conditional box additional appear distribution label resemble datum choose give formulation unknown problem belong available restrict sub define refined risk multiply expect non trivial limit risk well difficulty different capture behavior whole think training consist estimator local minimax sequence call local model le gaussian extend slowly describe version normality simple spin dimensional state unknown methodology concentrate structure convergence model devise initial high label ball local write u essentially unitary intuitively big sphere picture spin coherent spin state collective spin central represent uncertainty collective see law x suggest canonical quantum harmonic equilibrium basis rescale independent limit classical identify von perturb pick first remain unchanged precisely equilibrium bit quantum shift classical local shift define convergence natural quantum trace however quantum channel simply diagonal quantum definition one represent case sequence local spin state restriction parameter channel would comment intuitively provide rather weak rather operational meaning channel secondly devise strategy state asymptotically involve quantum serve transfer protocol local set quadratic form simple prior discriminate label training outcome projective randomness unknown mean optimal case measurement guess state measuring check whether excess eq satisfy state separately basis matrix average construct estimator concentration inequality exponentially plug zero projection projective aim plane expand expression back account rate excess coming term drop uniquely q associated risk need employ normality let local neighbourhood pt pt plane eq shift eq equilibrium state technical lemma normality argument since consider loss local risk aim optimally directly strategy optimally measurement come formulate asymptotically measurement excess previous shift like contain express frame plane angle z l contribution come write optimal jointly measure quantum component separately optimal risk combine additional square add contribution risk depend quantum minimax optimal construct rough system transfer optimal coherent combine classical ball red plane plug mixed set measurement step rough separate observe local classify state taylor contribution quadratic distribution write q component add canonical three contribution consider asymptotic maximum risk plug compare plug r point prior maximum opposite understand require measurement anti directly deal know statistic may measurement bring additional excess add local prior orthogonal training cast additional mean additional factor classify two statistical asymptotically reduce optimally quantum two state
match upper dimensional bound base bind exceed bind finally need depend subset arbitrary metric e distance measure assume dx ball radius dy give empty clustering set hierarchical collection clustering clustering diameter c large radius large radius finally radius discrete radius minimal radius finite find cluster minimal clustering give agglomerative repeatedly volume state distance lie finitely set dy define contradiction u p follow note ball deduce dp volume contradict agglomerative center introduction agglomerative algorithm start contain successively remain cluster minimize agglomerative radius diameter result agglomerative linkage clustering minimize analysis agglomerative center simple adapt center diameter case technique dependency range additive case center doubly cluster problem introduction cost solution three analyze agglomerative agglomerative algorithm minimize I compute clustering denote theorem partition cluster solution ball greedy frequently clustering radius always upper next proposition linear lemma center cluster radius cluster two cluster merge argument center simple analogously merge cluster satisfie divide step reduce increase phase additive term cluster fix center merge computation exist cluster discrete bound phase proof proposition algorithm get apply use lead two expansion q merge introduce term divide merge phase half merge volume lemma distance discrete optimal increase eq cluster exist loop merge use ball radius merging would radius prove consecutive phase ik k agglomerative problem rl minimize difference radius cost refer cost center analogous furthermore cluster always upper bound partition constant break analysis tie clustering compute uniquely dependency satisfie denote divide step phase phase one fourth discrete center intermediate need cover induce apply bind overlap ball radius twice ball contain existence pair bound optimal fix ii dc c radius ball radius one ball ball let merge algorithm merge ball cluster show consecutive let analogously I lemma deduce repeatedly get take exponent geometric convex function line put analysis merge center two dependency show sufficient analyze satisfy certain connectivity property able combinatorial rely merge define connectivity property relate set next input compute clustering input subset cluster another cluster compute input cluster algorithm know let union establishes follow hierarchical union cluster merge computation merge cluster inside absence cluster inside algorithm compute cluster e compute minimize observation k thus cluster ever cluster whose contradict e subset property unique intersect cluster cluster proposition use merge phase refer hold union observation compute clustering cluster lemma iteration loop pair set cluster assume connect use radius two connect bound either candidate radius connect remain either come n complete get nothing show hence cluster still prove deduce obtain repeatedly apply sum cost get u follow merge furthermore z every know cluster another intersect agglomerative I minimization analysis compute uniquely diameter case set cost equal diameter create diameter cost computed theorem partition q hide doubly proof theorem intermediate depend doubly satisfy denote solution two lie ball long diameter able additional proposition divide merge stage merge cluster first stage cluster call translate cluster near merge diameter essentially length translation first guarantee sufficiently large intermediate upper stage reduce stage long sufficiently merge second stage weak argument center leave sufficiently many intersect cluster plus diameter find cost main stage phase bind q fix get ball ball dy I dy ic configuration subset ball dy dy intersect configuration configuration gray gray gray gray gray gray cluster merge cluster exist lemma exist diameter u p mc cluster tc parameter side proof able upper increase number point input compute furthermore imply q expansion e geometric lead remain merge compute phase lemma let cluster exist merge point two k gray gray gray gray gray gray gray deduce apply inequality k u result immediately combine merge analogously problem section discuss cost merging cluster cf diameter cluster get replacement replace cluster upper case slightly lemma proposition become analogously section compute factor make omit present construction whenever choose merge step minimum show e factor least state upper metric base note exceed upper case moreover approximation already show set section approximation low approximation euclidean metric problem cost construction extend furthermore I gap cluster opt n bad possible distance assume figure compute follow lx vx vx merge diameter remain diameter gap choice merge merge cluster length gap create diameter figure go infinity approximation factor converge eight section always impact metric eight point maximum diameter however start merge merge fourth cluster diameter euclidean section low one suggest dimension easy euclidean compute cost construct eight x g diameter merging add one add result diameter fourth cluster remain four maximize cost optimal consider compute solution metric norm input compute problem optimal simplicity sequel unit large diameter diameter hamming start thereby next keep merge end cluster cluster first next diameter keep way round k compare solution diameter behavior power immediately corollary optimal
front allow simplify precision constitute base minimization causal controller convergence bayesian true analyze asymptotic controller consideration operation mode mode finite extra impose stability contain operation ergodicity hypothesis another hypothesis subset hypothesis share formalize proof develop appeal arise context controller approach infinite operation partitioning space operation essentially different operation mode mathematical control solid formulation control theorem corollary university cb pz approach control solution output boundedness ergodicity sure thing intervention calculus bayesian control kullback divergence control fully produce weather solve map appropriate measurement design robot exploration turn counterpart controller simple toy effort center tractable approximation new adaptive control relative attract problem solve statement reformulate dynamic control system causal controller solution rule particularly infer implicitly uncertainty dynamic exploration exploitation constitute promise adaptive guarantee policy develop limited operation illustrate exposition signal set action respectively shorthand like string controller proceed cycle controller stream fully represent action collect history characterized suitable controller stream maximize say many situation probability case prefer policy unknown face control randomly available mp tailor would minimize controller respect controller average incorrect deviation effect give cause minimize observed action give controller minimize q constitute causal calculus worth result define treat hypothesis context htbp diagram useful illustrate symbol highlight collect diagram dynamic state choose transition thereby choose subset direct illustrate diagram especially dynamic endow operation see give sake simplify interpretation existence policy happen stay determined stochastically variable realization draw pm pm pm deal introduce unique realization decompose sub form average divergence play realization inequality clearly divergence process interest go well increase complexity e specifically growth realization mode qualitatively realization hence impose stability requirement ergodicity possible tractable light insight say htbp boundedness key construct point realization divergence inequality previous bound well inequality rearrange identify side yield inequality eq pm want vanish characterize true hypothesis prediction region differ accumulate eventually enter htbp long enough mean execute risk right policy motivate operation contain sub strategy sub divergence grow demand execution guarantee controller operation vanish operation controller divergence apply stochastically include gm result final mode core indistinguishable
relate elementary way function list function l utilize theorem recall property therein thus index cdf define equal statement cdf reflect cdf weight indeed establish section therein decrease non function weight cdf consequently show submodular non note assume decomposition utilize fix existence decomposition bound h demonstrate equivalence decrease decomposition hold pair aforementione trivial otherwise define imply decrease depend position show submodular theorem next illustrative know mathematical convex shape usefulness example offer mathematical four function complex supplement mm literature reference therein known expectation weight w easy x h mathematical graph specifically xx xx lemma role establish strict monotonicity function section helpful determine certainly seven log prove equation x follow whenever follow eq right formulate prove establishe result x establish hand side complete easily check fact non follow follow p prove function increase monotonic somewhat surprising increase depending impose strict monotonicity require speak visible loss variable surely pair concern look seven imply satisfied thus make roughly speak closure line leave continuous even look somewhat nevertheless seven every continuous everywhere increase contradiction rewrite hand cdf measure right continuous almost right equation expectation coincide say almost contradict thus function increase side equation zero positive numerator leave equal contradict assumption conclude motivate end importance continuity naturally look seven continuous every w seven every connect function care random seven satisfy moment proof assumption satisfy x seven continuous every assumption fail singleton elementary omit satisfied continuous example crucial cdf order collect section shall omit elementary positive hold upon converge positivity verify check imply log contain interesting decrease namely hand increase check equation show show equation right equation see lemma conclude decrease calculation rewrite imply take denominator positive side conclude proof acknowledgment partially support science research reference risk apply york additive measure ia w transform random economic finance journal management submodular north weight calculation mathematic economic weight pricing functional application north american risk estimation mathematics economic u concept electrical north c f partial submodular lattice mm mm mm statement section definition loading wang motivated concern monotonicity loading loading appear literature increase interest weight loading answering establish loading consequently herein methodology loading monotonic c make conditional tail wang chebyshev uncertainty mm author mail triplet literature random know net negative note otherwise class net need accept risk remain order meet various financial large define consideration uv expectation well mathematical chebyshev integral reference utilize economic finance
r convergence tie sake attention investigate possess uniqueness strict convexity e conversely strictly strict uniqueness whenever tend sufficient contrary bound conversely state td ty show interior origin interior hull interior accord separate unit criterion constructive visually interior check solve attain condition accord happen hull existence origin segment attain reference converge unique minimum cover objective algorithmic isolated point program result summarize function minimize cluster constraint regard possesse solution tend dimensional experience al offer evidence algorithm work well high separation quasi newton surprisingly crucial treating programming extend even program inequality special constrain quadratic algorithm slow quasi newton acceleration dramatically acceleration matrix program sufficient convergence mm mm nature absence theoretical descent make technique diagnostic behavior programming fill stroke department box usa derive programming generic geometric inequality hyperplane simple converge boundary one interior occur easily quadratic simple branch optimization geometric programming chemical equilibrium mechanic circuit maximum process finite power may subject geometric type program geometric program computational hard program convex method contrast many recently unconstraine programming imagine geometric connection arithmetic derive new generic device mm possess advantage unify operate reduce minimization problem advantage ease compete geometric generalization introduction rest review mm derive program mm penalty section principle surrogate around iterate constitute second minimize denote minimizer fact f g remarkable stability strictly speak art mm choice fortunately invoke inequality coefficient geometric number mi scope mi handle support hyperplane z multiplication negative coefficient surrogate additive attain minimization surrogate function derivative equal x mi integer rational function solve accomplished root positivity constraint transform iy ix second mi convex possess argument continuously differentiable implicit stationary determine consequently worth mi support apply put back position already detail composition demonstrate iterate appear objective x x x x x intend initial value relative pre f mm mention furthermore constant slow way accelerate publish quasi newton acceleration reduce necessary iterate function acceleration fall convergence criterion condition require worth parallel parallel graphic processing hardware device separate min f p iterate leave mm condition low mm condition extend algorithm programming parameter develop couple x penalty depth relax positivity enforce achieve multiply side example constraint x become method constraint whenever toy choice x course power fractional achieve status minimize usually decrease iteration mm exposition far manner generalize result parameter positive converge summarize procedure traditionally minimize unconstrained unfortunately suffer error instability mm involved iterate property conditioning cause previously mention acceleration largely penaltie
indicator metropolis candidate allow near graphical sense advantage posterior indicator innovation time use ordinary model stage accomplish run processing usual way burn iteration store posterior distribution easily auxiliary k post compute way model posterior iterate value jacobian transformation rao method improvement need chain indicator form transition normalize chain discard burn code reversible relative likelihood assign prior model visit start discard burn estimate posterior bayes close agreement sample chain estimate steady base fitting logistic return length indicator five model il distribution logit standardized element htbp part choose leave specification update store fit particular prior prior coefficient case jacobian model repeat draw fit length discard burn indicator tune visit appear rapid half chain bf prior give lead steady simple jacobian identity matrix factor assign move p cc u appropriate gibb within full conjugacy constant indicator categorical chain rapid chain datum straight forward exact solution appeal bayes equivalently model barrier implementation post offer partial argue elsewhere choice efficient thought simplify must offer construct investigation benefit consider view determine representation efficient posterior direct distribution simplification move move pair involve space induce default construction lead inform weighting monte carlo ordinary implement use mix remove fitting model bayes example reversible equivalently involve model provide reversible jump chain output technique bayes factor generate joint indicator common instead criterion usually outline alternative alternate categorical relate careful recover invertible g k km km jk k accommodate
iteration square iteration row exploit highly lar implementation use qr efficiently row work recall strictly great qr compute dual solution qr factorization idea behind fairly straightforward complicated require least square point start fully regularize path toward moderately one compute path terminate could large leave operation approximation quite accurate time coordinate leave furthermore path lar lar discuss section find problem reason lasso critical piecewise preferable efficiency discrete example another interest simplicity separable therefore coordinate descent various sparse wise coordinate dual connection lar motivation lasso path via find lar strictly full state lar return continuously decrease correlation residual refer lar path lars path path elegant dual dual leaving drop precisely compare path show path familiar show lar diabetes color enter vertical ignore dual leave lar equivalence lar ignore primal lar notice rearrange get column residual correlation residual appropriately absolute current maximal among absolute match realize prove lar set lar recall freedom interest fit difficult degree fit arbitrary corollary freedom section freedom degree freedom interest formula come stein unbiased almost differentiable state easier use stein checking check almost locally affine essentially take divergence treat easier express fit onto polytope projection onto contraction lasso already filter derive transform rank recall coefficient concern second interpretation come possibly outlier reasonable simultaneous perform give x nonzero coordinate fuse fuse fuse fuse group order number outlier vector lasso outlier count appropriate difficult visually may knot important criterion like bic problem employ assess estimate choosing obtain one critical solution residual check use simultaneously norm seem fuse single group freedom end cubic degree cubic degree freedom knot ahead cubic trend knot remarkable property search fuse knot outlier shrink penalty back toward lagrange towards less greedy fitting group speak fact fuse degree right simply think replace give solve combinatorial property entry reduce constrain discussion fit freedom subset expectation problem fuse trend special case develop path instead lagrange solution original continuous piecewise linear respect original sign furthermore dual degree perspective fundamentally tie fused reveal freedom underlie graph corollary present direction website several describe possibility fuse project simple wise may therefore iteration lead connect leave boundary geometry investigate could path lar lar ignore dual leave boundary modify forward stagewise limit follow lagrange attention optimization problem complementary interpretable novel insight acknowledgment suggestion consideration thank lar help make wide choice solve dual lasso greatly lar derive unbiased freedom generalize turn introduction seem mathematic engineering regression predictor omit intercept solve lar solve also early piecewise path tuning case change slope large path efficiency aside lar characterize insight notably lar establish freedom lar exception enforce instead pure coefficient regression depend typically geometric fact choice know lasso entire also prove freedom fit note author propose relate annealing follow begin motivate penalty fit transform regular need section lagrange serve sake clarity build section fuse lasso case add one loop design familiar outline path case path lasso lar lar refer lar probably version lar lar lasso turn easy procedure exactly unbiased utilizes vary interpretable trend section save detail defer document application wide exhaustive illustrative example motivate place split main often signal soft coordinate equally exist gives highly observe signal row geometry solution fit adaptively structural begin address complex fuse lasso position straight adjacent fit setting coordinate genomic number copy order genome copy believe nearby copy copy valuable mean development tumor take natural extension neighboring pixel noisy arrange vertical difference pixel technique actually special carry field science electrical engineering toy color color intermediate datum solution piecewise idea define adjacency edge hence graph fuse fuse adjacent figure application underlie edge graph node include state edge take map reflect low measure log dark reflect high red noisy infection exhibit solve underlie connect state share west mid west south east north east color among region get live north east west south east certainly state region infection reader fuse penalty typically norm penalty identity actually carry yet trend call fused technique give setting unknown recursively define piecewise show cubic equal piecewise cubic respectively choose fit spline knot adaptively polynomial estimation nonzero jump piecewise fuse regression spline spline operate knot substantial literature place implement represent global smoothness local signal trend potential property localization field largely wavelet wavelet smooth popular compression wavelet perhaps formulation smooth orthogonality many desirable generalized long quite answer approach attention traditionally center synthesis suggest outperformed fused penalty outcome image penalty fuse explain contiguous region brain trend filter engine response air ratio engine interaction linear intercept slope subject intercept fit observation lie bin slope cubic course low trend last solve fit engine cubic filter line interval bootstrap unlike previously majority come point might set outlier exactly coordinate convex relaxation transform lagrange let rewrite coefficient matrix outli fit circle read natural generalize transform lasso discuss invertible row lasso lasso except clear onto back generalize lasso lar author establish solution fall case fuse lasso wavelet several satisfy admit fall summarize next dual nice arbitrary penalty essentially problem compose rewrite q constant immediately nice constraint problem original unique constraint duality see write respectively relate last equation tell dual look complicated moreover look restrict attention derive path give relationship focus simply though begin study case use build algorithm general strictly row efficiently path stage u box origin box fuse turn state fused coordinate q proof connection lemma lemma state say coordinate general fuse correspondence therefore lemma intend explain rigorous argument correctness section describe solution path apparent detail path piecewise slope boundary stay eliminate consideration know path move boundary maintain contain currently sign boundary interior pt start direction next complicated define index index use subscript index paragraph precede treat order list consistent unconstraine clearly sign lemma optimization every objective square estimate interior boundary solve call maximum respectively study property rigorous argument consider simple interpret path initially path primal picture path piecewise linear change slope piecewise obvious algorithm continuity also general primal coordinate path dual version primal path become fuse picture suggest primal coordinate coordinate fuse complicate dual prescribe primal prescribed lemma path check fuse dominant unfortunately fuse lasso boundary develop strategy path checking check boundary interior coordinate path boundary define leave path boundary idea call coordinate one leave maintain sign start compute next leave coordinate leave fuse detail technical view tucker optimality kkt subgradient function overview condition satisfy leaving happen interior coordinate continue interior propose boundary give time yield q boundary next leave boundary examine express leave th boundary coordinate first leave next iteration critical leaving happen verify visit kkt derivation leave property terminate return continuity hand subtle continuity appears recover transformation path continuous linear nontrivial nan solution slope help solution boundary coordinate know fit onto space several reason use along geometric argument prove primal slope turn slope achieve primal correspondence value coordinate various interpretation edge assume without otherwise recall set two
represent intensity pixel expensive one optimal case like plane still far perform bound behave metric learn build metric compare shape point set surface utilize key distance use interpret reproduce kernel hilbert rkhs moreover become tool compare point uncertain provide kernel distance similarity spatial uncertainty object claim kp q setting k algorithmic main fast set run take general reduce convenient representation point distance various surface tool map dimensional preserve believe map popular machine learning attention technical relate standard space unit associate curve tangent vector surface form identical measure replace kp kp type kp fp consider reproduce compactly map reproduce kp two kp p product function simplify approximate ignore far apart tail kp q xx kernel surface weight introduce error input result formal kernel curve surface correspond computation I measure imply algorithms approximation normalize additive normalize diameter orient surface point illustrate surface decompose orthogonal axis kp dp kp dp specify without unit present simple technique reduce continuous discrete grid shape point g correctness alternatively random weight weight work technique remainder assume weighted sum put observation yield ab b b pt point size ia ib ib ib weight contribution edge pair lemma pair w I ignore desired describe p shape reduce computation calculation immediately yield algorithm shape euclidean algorithm result allow kernel computation rkhs dimension unfortunately dimensionality approximate directly dimension directly projection shift fast gauss gaussian produce probability diameter ambient essentially implicit work shift fouri harmonic probability g hoeffding last follow union pair q manifold diameter building domain tail net adapt trick replace recall guarantee work operate domain fast improvement gauss brief full taylor sort come sum error degree dimension require map approximate advantage dependence lemma lemma problem apply euclidean combine feature near preprocessing set extract specifically binary denote total family subset p want notion value kp kp k vc notion fundamentally tie range directly combinatorial dimension subset every cardinality show give way relate space random heavily optimize construction complexity tie vc range realize cardinality show range space show construct thorough run exist exist construction algorithm create size recently provide algorithm method central prove algorithm claim approach context ball technique matching packing vc range size attain sample beyond show range align x th sample kernel instance variate kernel skew link variate gaussian super level set sp error approximate approximately time fail property scheme error sort super sort count simplicity assume differently negative value possibly sort sort place begin place since super empty construct partition reverse clean empty consecutive point sort point reverse empty place least third derive super level present negative p kp ks eq gaussian covariance query convenience weight deterministic weighted create unweighted one construct link kp sampling dependence appendix optimize subset algorithmic theorem link n notice link extra work gaussian trivial standard kernel transformation rotation vector r origin preserve translation rotation minimize rotation tr depend thus kp optimal translation adapt rotation parameter translation begin analytic pp kp kp n dp g g p imply guarantee translation lemma must close need kp kp pair know kp q w pair point produce runtime correct q find k describe translation rotation align origin perform rotation origin location choose rotation ignore extra modification ensure give pair q ss r p rotation reader possible construct
search yield constructive pattern number evaluation rand dt dt ds si ds ds ds constructive search iteratively add name respect randomly feasible computed control amount randomness value correspond greedy construction nonzero global optimum asymptotically begin iteration adopt feature selection save prove pattern great confidence level equal system validate particular achieve obtain accuracy conclude real lc system fisher class set label firstly method relational optimal lead adopting use search na I classifier performance real class relational discover label sequence firstly efficacy strongly sequence task solve adopt local embed I world fisher reasoning fundamental intelligence indeed may lot contexts day science sequential application video planning biology modelling speech recognition etc form structure different problem pattern mine firstly capture investigate assign however environment mine extended multi relational mining involve lead exploitation powerful formalism classify adopt adopt language markov hmms simple relational able description contrary paper tackle discriminant learn value form learner obtained adopt mining frequent pattern use boolean strongly represent carry task optimum however search perform obtain solution explore filter select select adopt preprocesse heuristic intrinsic characteristic ignore learner paper name public relational non I subset construct power compute na I hence probabilistic outline discuss present section follow experimentally mining lot mining effort design face restrict pattern involve predicate early domain highlight bioinformatics represent domain effort extend predicate involve category first logic frequent boolean tackle relational probabilistic combine author construction firstly construct adopt language discriminant feature logical combine search algorithm refinement logic framework operator sequence one indicate atom base language correlate tackle account combine approach combine relational description logical relational field design sequence sequence flat alphabet logical develop generative promise logical successively exploit handle flat symbol structured probabilistic finally extension alternative hmms dependency discriminative relational learn relational briefly report relational take account formalism mining implement probabilistic classifier construction capability selection approach logic review symbol symbol symbol atomic symbol say atomic x occur clause constant define substitution applicable obtain multi relational mining report briefly recall order atom separate consider atom sequence express dimensional relation dimension may atom dimension event represent dimension indicate closure step th direct dimensional prove operator appear multi relational clause ground concept iff identity object identity term represent object step system construction obtain mining frequent make lattice order search start level lattice lattice evaluate frequency generation phase monotonicity pattern pattern frequent none approach start step try discard frequent storing refined pattern discard phase basically operator pattern knowledge clause satisfied pattern particular type language predicate formulate bind variable must specify specify relational class whose label use pattern refinement pattern atom pattern add atom pattern dimensional add atom exist start length pattern equal atom refinement step final pattern classify unseen space relational single relational predict build component otherwise discriminant function discriminant involve discrete value conditionally binary define independent yes pattern expect th estimate count relevance determine assume conditional write
property appearance claim surely neither degenerate I q use z condition use nb n gaussians power gm h matrix use function get q assumption I notation ic q cauchy normal conditionally assume moment lemma bx nh albeit note induction w enough inverse eqs provide random var latter small eigenvalue hand induction r vector limit converge apply eigenvalue limit lemma definition tm lemma eq hand prove proof side combination converge hand notation induction hypothesis term h modify hypothesis r hold prove analogously last cr ts q deduce limit lemma r almost enough finally note definition contribution last lemma point since use induction hypothesis surely assumption corollary write distribution surely finite drop pseudo bound degenerate follow hold surely exist constant almost surely double grateful support fellowship nsf present heuristic amp standard message stress proof reader intuitive understand amp connection belief propagation rewrite great reader side message exclude guess na I correction produce vanish z I substitute negligible writing notice type expand term obtain analogously eliminate analogously q equation section theorem taylor next cn e generalize mean lead condition therefore ne ne always use truncation weakly bound since v v ki ki v bl functions l kb recall lipschitz full absolutely lebesgue nx pz compact thesis fix matrix complement formula vanish constant constant lebesgue vanishing law imply exist constant z thesis case constant independent gram schmidt construct orthonormal define schmidt thesis follow plus minus plus plus lemma claim rgb rgb rgb rgb rgb reconstruct incoherent linear numerical dynamic term rigorous foundation indeed hold asymptotically gaussian message class sparse fundamentally different standard short underlie context spin theory compress reconstruction vector small noise reconstruct start convention succeed cf normalized way nf ia finding class formalism evolution claim tradeoff amp se coincide appropriate stress e amp boundary determine polytope reference finding argument paper proving se hold system limit matrix prediction random prominent pass proceed argue approximate pass limit appendix argument amp rule derivation necessary help reader familiar pass develop intuition important analysis evolution sequence sparse graph locally graph algorithm indeed complete bipartite evolution regard sense analogue evolution dense sake graph instance study discuss proof theorem symmetric mention propagation bp gaussian refer generally none exist rigorous bp seem remarkable odd standard evolution work limitation standard literature begin miss show relaxed let everywhere differentiable sequence exist shall constant type explicitly constant properly adjust sense index weakly x nk kt surely finding eqs find finite principle theorem ix z n rely assumption particular hold instance broad heuristic argument clearly insensitive detail entry online supplement suggest state evolution even broad domain open description since involve bring play modification copy correspondingly vector eliminate follow dimension write bt recursion matrix iteration develop intuition central easy entry converge exercise eq entry together evolution draw independently across argument fall dependent dependency negligible system iteration several notably call object great standard optimization application study behavior dramatically system simply hand term lead consequence evolution fact keep evolution rely exploit density decode exact tool threshold construction threshold major locally like special case david study assignment speak increase neighborhood converge bipartite like nevertheless admit rather subgraph bipartite concrete derive case indeed eliminate argument outline argument sophisticated lemma tree impossible graph fact local limit bipartite graph vertex node root simple spin assignment prove context hundred publish replica user detection performance coincide rigorous foundation along concrete principle replica insight specific replica method compress rigorous limited statement replica provide sharp predict multiplicative determination constant practical application group base replica foundation work mind apply predict iterative algorithm generally evolution warm namely evolution scalar estimator lipschitz amp read theorem variance minimize square result recall mmse compute u random motivation mse predict case asymptotically mmse mmse law wishart calculation mmse variance asymptotic evolution result already derive e non vanishing assume converge weakly soft non nevertheless square choice nonlinearity indeed substantial g threshold sparse soft prove state evolution deduce yield sparse amp coincide popular argument optimality evolution way improve class idea choose construct mmse distribution mmse simply phenomenon model code division signature frequently theoretical giving signature I random independent justify signature far naturally square algorithm read state evolution recursion signature dense signature mmse expression method recursion deduce mmse receiver point instance apply conditioning technique develop spin relate idea simple recursion reference conditioning convenient somewhat namely I consider scalar evolution first argument minimum generally problem algebra technique avoid conditional instead ts conditioning event gm h gm equivalent linear play random linear projection formulae p appropriate gm g new I tm refer explicitly control law number array phenomenon vanish large term recursion claim describe amp regard sequence recall lipschitz everywhere differentiable derivative context abuse analogous sequence vector fix obtain f th derivative finite dimension via lipschitz function vector whose empirical bound nx order amp amp define equivalent mathematically recursion track current recursion coordinate similarly therefore I value calculate basic characterize conditional distribution write projection onto column define projection column r coefficient cf go transpose product measurable word trivial algebra surely consideration large represent subscript similarly property variance let define recursion column orthogonal eq lipschitz exist variable
category c ssc trivial effectiveness corrupt world analyze stock greatly company financial discuss lrr interesting albeit stock category stock exchange center world stock divide york stock exchange center category generally stock category basic consider stock category subspace stock stock categorization stock label historical price stock global exchange market divide exchange category stock within top rank one category price york price obtain unfortunately historical price stock interface stock price accumulate stock stock stock span trend price state previously sharp drop stock strategy stock price average stock interval plot normalization adopt pca reduce dimension stock pca pca market htp ccccc markets ssc lrr sc stock market method statistic sparse make perform well market raw uncertainty bottom sub stock category stock mark stock experimental sufficient class categorization impose datum way learn essential corrupted mm algorithm convert prove practical sc model give rule achieve motion promise stock contain many uncertainty phenomenon solve multiple bit consume lrr especially recommend learn rank direction method component principal sum low affinity consensus robust cluster subspace discussion property strictly hold claim infinity accordingly sequence every bound exist convergent subsequence also convergent rely decrease get since h theorem liu member recover play role various processing name sum recovery traditional directly utilize reasonable enhance intrinsic although propose effectively mm function iteratively surrogate fall algorithm successive apply typical robust principal rank lrr principal lrr apply motion segmentation stock rank log well sum heuristic compressive structure network efficient rank e typical approach handle preliminary sense nuclear lead structure community consider learn formulate world corruption recover ill pose address optimize optimization vector little use exact intractable approach make try minimize envelope via convex norm heuristic regard practical powerful learning complicate corrupt dense promise paper approximation heuristic enhance recovery mainly conduct main representation generalize advantage solve surrogate taylor expansion next replace upper paradigm reweighte accordingly possible prove framework finally converge point paradigm adapt application use compare principle pursuit handle many rank tolerance rate propose extend verify removal face video second task power sc goal performance video test besides highlight robustness stock historical record art improvement include three fold sparsity solve reweighte stationary point extend stock demonstrate structure powerful area vision remainder paper review introduce discuss address low subspace segmentation discuss conclude part work recovery lrr reweighte recovery consider low matrix real world device sensor due low topic simulation introduce reweighted improve performance corrupt low representation tool desire general original correlation residual lrr regard exist lrr much robust public inspire lrr recovery paradigm effective solve optimize therefore mm fall field mm tv compressive reweighted lead practical portfolio management processing reweighte prove work reweighte approach work reweighte semi could relative alternate due possible capability method heuristic theoretical prove approach state basic impossible solve usually intractable make alternative one try replace envelope nuclear norm via accordingly relaxation definite programming define summation essence verify point replace norm formulate therefore stand convex constraint envelope concave indicate norm ask whether might log sum signal prominent term indicate close approximation therefore encourage sparsity heuristic formulation obviously differ norm use log instead typical powerful enhance unfortunately cause convexity function concave case non define replace hard one maximization fashion repeat iterative construct mm algebra taylor e ij ij mm convert reweighted call denote update construct surrogate besides summation nuclear solve adapt two part corruption pursuit derivation matrix formulate reweighte weight place nuclear impossible nuclear inspire add equality solve single objective close proximal alternating since alm relax instead lagrange multiplier e accordingly problem alternate direction convergence work minimize minimization update rule minimization unique close eq value immediate solution summation address gradient discussion reweighte mn ij z k kt recover low precede comparison robust simulation low rank independent simplicity rank rate comparison lin inexact solver inexact solver indicate highlight effectiveness record cost second cccc frank times frank e obviously exactly high table provide htp basic low vary variable feasible feasible verification accuracy repeat time median recovery regard much get conclusion make fit boundary apparent could difficult task fails experimentally verify conduct respectively coordinate count report part occur assign converge need resource accelerate criterion apparent four step recovery recovery second phenomenon advantage reweighte powerful conduct real suggest stack face subject conduct extend respectively face accumulate rank htp original face b face faces fig dense face image effectiveness apparent leave dense highlight significant visual face recover face recovery correspond remove use evaluation video list fig htp normalize frame convert benchmark video use theoretical recovery limitation divide recover truth remove corrupt three video besides apparent keep dense sequence illumination make isolated noise part foreground keep many advanced concern without mixture comparison evaluation false foreground correspond machine
line outside normal decide available negligible computational hypothesis large subset accept likely subset visit processing model reasonable amount several integral joint probit probit represent independent iy latent volatility volatility variable thousand period considerable complexity simulate similarly nuisance inference evolutionary mechanism capture capture mark capture informative description chapter open episode associate involve unobserved indicator capture experiment mark subsequent second individual stage use improper prior n q summing create indeed summation dataset example chapter relate european cover year observation france capture completion require near class label large corresponding unobserved usual denote delta motivation full conditional constant closed call summation number respectively generic approach posterior procedure go name bayesian functions formal monte carlo effort posterior effort grow square assessment central stem integral integrating include effort independent importance n return sampling formal leading infinitely way surprisingly choice wide generality include fundamental poor choice lead use stress curse dimensionality contrary numerical carlo always parameter space derive satisfactory importance become difficult get satisfactory function dimension probit formally approach poor equal ml estimate ml estimate approximation specific probit posterior importance former weight much concentrated weight miss constant histogram simulation setup diabetes exclude integration constant compute replace normalize version respectively normalize converge constant normalised weight posterior need weight large close unlikely closeness relate importance give effective q completely importance weight evaluate iid direct comparison sampler probit size primarily resample sir derive distribution use weight importance technique careful break recent literature extension adaptively monte evolve either sequential importance population connection early literature principle criterion simulate behave population give technique dependence remain unbiased iteration recent discussion therein sampling importance resample major past simulate currently iteration decrease effort generate n choice open convenient proposal update abc produce approximation population purpose optimal multiple mixture already test bayes factor monte rapid decade prior nan hypothese proper nuisance carlo elementary approximation consist simulation indeed n consistent define importance support two importance estimate importance significant importance approach n come respectively apply upper lead performance method connection se since use approximation bridge posterior quite since common bridge thus embed bridge density equivalence obtain clearly obviously performance depend pseudo establish asymptotically marginal harmonic estimator generic harmonic markov generation core exploration weight lead representation include hence dimensional gibbs short introduction mcmc simulation monte alternative differ issue output distribute target distribution irrelevant quickly happen fail converge prescribe iteration result biased markov correlate iid metropolis truly offer universal simulate markov explore proposal q internal irreducible respect distribution associate visit whole distribution converge propose sometimes reject relate accept reject generate q simplify else sometimes accept diabetes walk mle iteration mle also chain figure behaviour algorithm describe rhs visible rhs half reject lead acceptance rate walk diabetes leave graph first component chain straightforward hasting recover standard strong limitation attractive algorithm present generally strength sampler break simulate successively dimensional systematic scan effort tt return hierarchical multidimensional note probit base variable sampler aim indeed conditional depend produce produces take estimate walk hasting figure mix superior metropolis hasting gibbs diabetes mix performance local classic hybrid replace non metropolis hasting hybrid solution probit depend proper alternatively simulation metropolis step move validate random find trial dataset contour last satisfactory surface posterior probit mcmc source mean good therefore difficulty hasting include proposal reasonable visit scale stay local several modal modal region jump sample mixture multimodal secondary spurious stem run metropolis prevent markov mode compare choice mode evolution walk hasting chain mixture hard determination curse dimensionality increasingly part intuition modal weak complex impossible proper hard increase unless difficulty tune I adaptive difficult convergence discrepancy chain line empirical acceptance reach like inherently modal lead scale scale produce iteration adaptive mcmc right preserve ergodicity implement g create independence preserve path proper ridge image valid basic ergodicity adaptive monte coupling construction weak large mcmc tune adjust logarithmic result well algorithm impossible density produce situation latent analytic impossible handle additional joint cause mcmc illustration inverse give equation case computation population wide rejection sampling effort I return free eq level p summary tolerance hasting sampler computing effort rejection get generate perform sequential alternative method abc carlo key issue simple subproblem algorithm begin simulate density leave abc mcmc black diabetes illustration probit describe abc density tune bound simulate euclidean predictive mle therefore avoid predictive good fit incomplete biased choice model interest rather design anneal parametric process choice available benefit conversely area method challenge partly la paris mc acknowledgement universit case associate paris email chapter chapter aspect approximate procedure recent carlo approximate considerably choice method algorithm computation choice coherence inferential etc perspective function prior construct usually involve optimisation implicit branch concern early laplace probability development year approximation chain sequential monte extent statistic make nature object involve challenge community opt solution numerical indeed specific bayes simulation essentially computer law large major build entirely upon approximation advance area chapter discuss connection
divide normalize check forecast insensitive testing evaluation figure original aggregated wind clear wind power change low variability spike wind power first short forecast hour ahead model often wind change throughout year day period give fit study cycle play cycle wind due air wind sometimes train help decide exclude cycle aggregated wind series wind nonnegative unconditional aggregated peak normal common transformation wind include root demonstrate wind datum logistic transformation wind approximate individual wind gaussian generate density forecast wind aim rely monte carlo approach iterate easily reason forecast transformation wind figure result approximately transform first difference transform volatility small autocorrelation qr past consider qr select minimizing criterion maximize likelihood ahead forecast obtain step forecast gaussian standard move density aggregate wind jacobian ty z h ahead conditional forecast second wind however ahead forecast handle ahead ahead ty conditional ty innovation dynamic evolve ahead density require stand function give conditional horizon approach stand function variance model additive may violate value final forecast reality ahead approximated function characterize scale parameter simply step forecast depend estimation conditional proxy model gaussian well good density normalize aggregated wind truncate successfully model nonnegative parameterize truncation true simultaneously method exponential successfully forecast volatility forecasting smoothing give framework ad forecasting forecast process simultaneous forecast corresponding process enable ahead forecast iterate exponential smoothing simple wind constant need estimate refer conditional parameter scale correspond smoothed give update accord step ahead iterate highly lag autocorrelation forecast forecast simple smoothing forecast case obviously conditional essentially distribution exponential trend trend stand ahead forecast identify forecast forecast forecast gaussian innovation ahead forecast ahead give estimate variance case explicit maximum nice property maximize ty continuous rank score much large amount slightly forecast minimize forecast efficient ahead density use next parameter scale smooth equip ahead forecast square wind smoothed series update parameter forecast due ahead forecast highly correlate include account forecast unfortunately extra negative approach smoothing negative allow logarithmic small equation forecast equation except take replace insensitive size forecast ahead forecast identify smoothing summarize smooth estimate logarithmic parameter describe smoothing apply write write update equation exponential variance unlike asset expect volatility distribution truncate due truncation distribution obtain transform forecast calculate forecast asymmetric wind always calculate forecast conditionally assume conditional expand taylor performance forecasting approach forecast test minute hour ahead horizon absolute mae rmse forecast mean ahead forecast forecast mae respectively ranking mae mae rmse forecast hour outperform forecast hour interestingly almost identically phenomenon smooth scale parameter logistic wind mae rmse explain investigate forecast capture change volatility forecast discuss aggregated wind forecast aggregated consider simple good power wind sense krige predictor spatial datum process context calculate spatial temporal covariance hour point forecast generate wind aggregate normalize divide aggregated aggregate performance course sophisticated interest one discuss persistence forecast forecast forecast continuous score show strictly score tool forecast forecast analyze forecast affect extreme order resolve calibration forecast density forecast ahead forecast cumulative indicator ahead forecast persistence forecast constant forecast forecast density forecast mean ranking mae rmse forecast forecast summarize contrast investigate conditional variance probability ahead coincide forecasting calculate th quantile deviation show indeed generate forecast calibrate indicate description volatility time note slope figure conservative large spread ahead conservative confident small calculate calibrate calibration reliable description forecast slope opposite valuable evaluate condition period capture propose confident density risk risk capture volatility variance essentially give realize variance large value times diagram show demonstrate step forecast model period two confident spread give moving realize early n right line deviation generate multi forecast wind generation normalize wind fit forecast show generate hour step truncate aggregated wind underlie exponential forecast forecast truncate although approach use truncate alternative forecast several first performance smoothing length set estimation model set take datum remain perform approach model forecast computationally forecast density transform forecasting problem choose forecast testing forecast summary general non particular forecast aggregate wind wind challenge reliable wind generation individual wind interesting nonlinear jump may long low wind individual power mass forecast wind power forecast portfolio wind location development include process truncate modify version site would importance system wind forecast aggregated power explore neighbor wind acknowledgment wind generation author taylor little associate suggestion quantitative finance engineering fellowship project ct multi forecast intensive wind intensive avoid transformation normalize approximately describe datum forecast iterate approach forecast govern forecast two underlie forecast truncate generate forecast power ahead generate forecast slightly approach computationally length attractive approach normalize wind forecast wind power wind wind store risk period wind wind wind drop power uncertainty wind addition wind accurate uncertainty wind generation penalty maximize wind speed forecasting increase research wind speed wind power forecast literature forecast year emphasis place quantify uncertainty forecast forecast performance forecast recursive quadrature significant nonparametric forecasts series monte carlo simulation approach intensive forecasting model wind wind power wind easily major power vary also difficult quantify power curve forecast wind power forecast ensemble forecast generate weather forecast day ahead approach wind forecasting focus direct modeling power difficulty lie wind
virtue ergodic think select ergodic process stationary ergodic weak ergodicity allow n n ergodic independently stationary ergodic put either tend respect sample length formulation close assume different gaussian hide markov one distribution fix model likelihood cluster clearly formulation problem homogeneity whether different cluster problem classification three three parametric I easy indeed ergodic sense stationary work asymptotically upper mixing cluster rather empirical distributional distributional real sigma probability space alphabet tuple could omit difference tuple triple etc formal definition useful tool various although summation minimal cluster assign cluster thus cluster point cluster unknown simply put together certain calculate base rate incorrect cluster quadratic general distributional replace distance theoretical preserve theoretical problem lead practical compression way combine consistency promise direction alphabet work consider extension multidimensional borel sigma volume b b kb kb kx nb tb ergodic occurrence distributional word volume count weight fine metric q three infinite distribution stationary ergodic idea analogously grow converge cumulative frequency converge lb jj lx l vx un prove statement symbol different unknown stationary ergodic partitioning partitioning target set number know asymptotically require sample ergodic perhaps assumption finitely cluster select statement loop calculate iteration need pairwise calculation computational apart computation precise algorithm grow infinity replace partition estimate sequence consistent integer go observe follow asymptotically consistent statement sequence number infinity computational l hand proposition burden precision cluster term however control far know advance cluster assumption stationary unknown number impossible two distribution impossible decide weak consistency come unknown make expectation rather model assumption unknown extension multidimensional range mix coefficient formulation informally stationary way assume stay non call many process irreducible markov finite coefficient reference therein input cluster far measure calculate pseudo obvious level select joint one parameter moreover bind provide consistent fix finite also joint partition output way finally mix coefficient process sum correct answer bind asymptotic term practically parameter multiplicative factor individual expectation take would frequencie expectation assumption namely follow speed decrease strong without define come advantage beyond stationarity ergodicity basic clustering initialization easy main establish mention suggest compression spirit direction concern optimal course rate length grow partially education research region national project theorem
result jt neuron represent change provide summarize specify relation encoding neural activity activity support extract activity bayesian fire profile evidence excellent neural calculate remain next focus highlight utilize pass restrict introduce assume linearly relate one another appropriate update minimize entropy respect weight implement specify without work capture evidence consistently pool support u national science ci e university conduct eqs fig fig neural network derive represent network dynamic require determine calculation source deal consistently inconsistent evidence neural distribute network neural code artificial ability quite numerous neural probabilistic either train interpretation explicitly strategy constructive explore representation neural encoding challenge alternate suitably broad end encode probabilistic allow calculation accurately usual activation formulation explore analysis original implement analog quantity investigate population extension develop interact neural population additionally stochastic boltzmann belief network machine neuron manner enable give connection neural architecture attempt encode readily approach match instead activation pass nonlinear depend encode impose neural network analog normally result value network usual form neural neural input transform activation summary procedure generate network procedure acyclic model arcs strength influence probability additionally take causality arc pointing cause causal belief arise cause effect opposite arc first upon graph separate node node comprise parent direct parent random decomposable denote decomposition neural network match desire organize bayesian facilitate probabilistic utilize naive wherein assume place network accurately encode rather take firing rate mean describe neuron random activation work temporal encode neural state determine interpret transfer function rule describe activation time fire recover relation decode minimize difference density
truly permutation partition observation handle limited conjugate distribution case normal pick mean scale possibly obviously open discussion without consequence prior fast acknowledgement chapter book follow take mathematical trust statistical support participant contribution technology exact analysis parametric mixture conjugate relevance limitation complete impossible conduct datum complex conjugate use keyword prior family poisson mixture reader want beginning mostly understand automatically datum structure crucial simulation advantage also former available relevant reference due build upon thus denote scalar product vector purpose chapter exponential multinomial representation natural function literature variable observation one end family extend prior prior conjugate complete amount conjugate prior different dirichlet exponential since fix simplex conjugate prior q family case poisson mixture conjugate gamma posterior gamma observation posterior normal gamma indeed allocate component square difference convention derivation correspond complete development obtain posterior consider expand complete configuration sum except process multinomial truly observe upon missing conjugate component compute availability estimate sense switch posterior since available obviously discriminative effort summation partition allow form distribution posterior equal hyperparameter spike identifiability proceed fairly purpose illustration sensible correspond important feature term act like rather nu ji jj nu ji ji update ii update correspond column add last plot shape seq le type seq type considerable posterior miss much much memory size pressure also observation show first table poisson make assess sample zero explain value pair dataset poisson miss storage comment distribution happen occurrence feature ccc sufficient posterior pair poisson term requirement turn dataset observation large since posterior marginal bottom marginal bottom remain compatible range completely component figure marginal remain nonetheless symmetric mixture prior dirichlet use note indeed consider also available constant allocated allocate sufficient poisson namely count particular book apply follow I
proportional conceptually relaxation far focus elegant devoted collect specify dynamic covariance consecutive ask condition support time recover make precise letting indeed keep indeed prove carefully control dependent time evolve discrete subsampling system model coincide uniformity reconstruction interpretation roughly regularize closely relate follow develop produce use corrupt successive configuration elementary sufficient significant gaussian reconstruct binary ref regularize logistic structural focus naive present guarantee discrete fix produce accurate scaling whereby interested slow degree freedom fast relevance address fix number dimension relate graphical paper mainly difficulty analogous high concrete check specific class vertex laplacian symmetric outside entry diagonal denote well negative semidefinite eigenvalue fit network connect strength result simple q least recover achieve take trajectory length polynomial logarithmic system determine section numerical synthetic confirm consider set draw independently random square observation row average leave depict success curve vs sample time trajectory equation success probability converge evolve portion row apart additive coincide indeed precise see model distribution determine lyapunov context refer notation shorthand discrete sign large logarithmic network particularly derive limit confirm intuition produce large fine limited correspond configuration namely finally equivalently x omit write hessian outline main dynamic state concentration lemma prove outline mention proof compact condition recover supplementary condition hold square sign continuous provide likelihood check configuration hold checking condition regard expectation respect stationary non proposition proposition supplementary material relate ij dd finally corollary follow subset proposition imply second guarantee obtaining impose proposition recall distinguish adopt couple supplementary large claim prof leave task continuous time couple slight abuse check initial continuous discrete define respective couple drive let follow standard motion fellowship nsf award nsf grant dms addition q relation take support kkt condition optimization proposition estimate introduce extra element solution easy lead represent row equal I represent zero everywhere one position represent one appear product eq n fact inequality proof bind prove let introduce block block way compute thing eq put together lead bind trajectory instant word mapping stationary vector symmetric bernstein denote continue inequality similar show respect instant assume consecutive state application us stationary statement write define eigenvalue x reasoning assume matrix ab b compute give statement theorem imply last eq probability want satisfied substituting must need far need q impose probability condition look hold notice allow simplify four restriction dominate actually let unit thus prove negative definite block b fact vertex visit generating function form walk move neighboring expression make theorem proposition electrical stanford stanford stanford stanford model dynamic associate freedom tackle learn regularize square set graph direct edge associate take evolve generality give current corrupt represent brownian motion entry pattern system stable
dynamic communication financial classic armed mab offer reward draw distribution unknown player choose total involve exploitation player face objective arm reward play reward statistic case know clear reward play mean essence arm policy sublinear growth achieve know slow effective minimum different reward finite liu logarithmic sublinear order heavy tailed reward evolve process successive play passive markovian reward liu ucb markovian armed classic mab state arm continue evolve play transition arm play accord arbitrary process centralize player decentralize multiple distribute centralized make player choose performance similarly loss compare markovian model arm evolution period markovian segment control bad experience switch potential steady frequency arm balance factor sequence exploitation structure propose exploitation carefully control epoch length player partition contiguous play arm player large e average reward play far grow geometrically tradeoff exploration balanced exploration cardinality epoch accurate rank achieve nontrivial system regret knowledge epoch order case propose reward classic mab weak classic mab markovian optimal stay term long large unfortunately markovian adopt weak regret know reward open discussion arm play exchange occur objective decentralize minimize growth arm perfectly among centralized scheduling perspective player arm evolve accord process play decentralized policy centralized scheduling regret among player global regret indeed know centralized scheduling inherent sharing unknown consider player liu adopt propose knowledge system available algorithm propose ucb arm determine obtain cycle cycle cycle classic mab policy deterministic epoch use offer discard cycle pilot arm reward choose pilot small pilot difficult pilot reward solve govern stochastically identical tractable semi exploit markovian show regret arbitrarily decentralize liu division fair share fair achieve reward decentralized player ucb basic idea sequence exploration exploitation carefully analysis also different furthermore decentralized require highly present develop non bayesian mab treat deterministic quantity value bayesian policy treat probabilistic knowledge prior past bellman mab solve policy classic mab mab markovian dynamic lagrangian certain index numerous example g armed bandit broad cognitive dynamic spectrum secondary search several primary user channel chain dynamic user channel sense subsequently maximize long without primary user secondary user paper apply environment specifically channel choose accordingly reward transmission throughput design policy dynamic problem capital select company year company evolve transition whether company vc long design know integer k rest organize establish logarithmic sec consider performance sec single centralized arm choose play arm amount reward define arm unknown markovian transition irreducible passive arm stationary arm permutation specifie play determined policy measure regret reward play stationary denote expectation order omit regret extension arm play learn markovian steady consecutive bad long enough balance factor illustrate policy partition exploitation epoch geometrically epoch epoch player far play epoch aim learn arm exploration make exploitation epoch accurate epoch every epoch play denote begin end epoch whether exploitation epoch determine fig epoch condition ensures spend exploration necessary frequent exploitation exploration epoch exploitation exploitation epoch statistic give h c exploration exploitation arm arm epoch htbp epoch complete decentralized fashion pre epoch player local arm fashion offset sharing player different rank arm play exploitation epoch detailed description decentralize policy divide fig player start exploration arm go spend epoch go step epoch large sample mean exploitation epoch length player play go epoch divide play th go decentralized logarithmic regret different arm mean policy sharing decentralize epoch achieve order require nontrivial achieve definition appendix detail show global pre agreement eliminate policy join system exploration exploitation epoch difference exploitation play arm player play exploitation random arm efficient sharing among player pre note epoch arm reward affect reward consequence logarithmic exploration epoch logarithmic establish logarithmic absence synchronization agreement exploitation epoch sharing decentralize without global agreement achieve logarithmic reward occur hold access communication condition transmission observe problem reflect reward corrupt rank achieve pre open compare context secondary search assume consist independent state evolve secondary select power channel channel use choose fig short period seem parameter sufficient logarithmic fig logarithmic regret monte run space consider state state arm reward arm make probabilitie arm find appendix avoid pilot state well fig use cycle learn pilot discard armed bandit centralize decentralize develop deterministic geometrically achieve order decentralize appendix rewrite definition show cause arm cause effect lemma irreducible space state time denote reward exist continue play cause switching goes cause logarithmic bind number number exploration epoch player exploration epoch spend epoch bound time spend exploitation epoch total spend spend spend epoch play arm exploration epoch spend epoch denote starting th epoch denote large sample arm arm mistake let condition ii contiguous growth length see loss generality derivation every realization notice occurrence state segment bind chernoff reversible space transition probability pa l arrive show regret cause arm exploitation epoch become arrive point markovian problem require evolve process condition consider multiple epoch segment detail carefully length exploitation epoch b theorem fix three part cause arm cause bad playing exploitation epoch upper bounded cause arm switching arm cause play bad arm cause play bad epoch order caused play bad epoch spend arm lt mt reasoning upper bind cause epoch appendix proof rewrite regret term cause cause cause epoch next spend arm time spend bad arm epoch play bad arm bind bad exploitation combine arrive bind bound follow constant logarithmic term exploitation epoch still consequently cause arm epoch play arm player play arm mistake upper cause exploitation epoch regret player player reward th player player
belong view prop er device center immediate consequence follow expansion claim easily follow tend event since assumption establish complete instrumental proving let let twice partially minimizer minimizer exist otherwise nothing prove twice partially taylor positive obtain eq note normality suppose minimizer belong I satisfied nonnegative specify coincide assumption current noting prove achieve apply neighbourhood positive definite appendix tend proposition subset go rest reasoning tend estimator respectively minimize eq note existence ii tending event coincide precede instrumental go expansion require property unable implicit consequence proof likely direct score normality weak density also alternative normality weak growth condition specify quite comparable difference iii generating mechanism iii p schwarz proof everywhere continuity prove iii trivial belong leave element part implicit suffice latter ii px prop p dp px interior radius interior depend p f nx ax hx derivative compact assumption consequently nx p h proposition follow statement pointwise iv vi vi arbitrary subsequence exist converge converge converge keep mind follow implie imply iv I f consequently follow choice bounded elementary follow central v l cover complete proposition generate close ball separable compact hence p k immediately parameterization view appendix norm empty borel every empty subset additionally empty bound v kx f df nf clearly borel follow make argument analogously sup view proposition appendix convergence parametrization continuity kf subsequent van empty function consider constant satisfied value tr every maximizer j sr k combine imply n k cr inequality outer depend present van constant assumption coordinate projection space proof unnecessary I denote empirical sup f real note integrable hypothesis l c p event entire hold consequence density sup dominate mod inclusion satisfied mod view remark sequence converge compact assumption imply complete proof nd p nd part mod density sup part ball inclusion imply fact sup hold event part mod twice continuously partially differentiable continuously p p note involve mod assumption formulae integral continuity dominate inclusion satisfied integral appear mod lemma large td follow continuity separability assertion precede true probability interpret thm axiom thm thm conjecture thm thm exercise thm thm thm summary em university indirect estimator minimum model parametric maximum density correctly furthermore asymptotic parameter estimator independent I random law correctly identifiable likelihood likelihood method expression density equation imply complicated dynamic concrete alternative well method also efficient simulate equation define simulate respectively specify example maximum estimate objective function auxiliary model model auxiliary simulated indirect estimator consistent asymptotically regularity condition indirect inference estimator efficient sense certainly long suggest asymptotically idea amount choose restrictive maximum likelihood span limit informative auxiliary present long indeed indirect asymptotically fisher information variance simulate show normality originally explicitly certainly possible comment estimator high admit spline asymptotic normality asymptotic efficiency correctly analyze simulated remark indirect topic proof use fisher define indirect q indirect simulated sense extension classical introduce additional proof establish careful weak uniform obtain establish result also derive rate outline existence derive zero estimator indirect estimator establish originally parametric efficient fisher proof technical result f banach real value equip sup empty subset banach space empty measurable associate borel lebesgue measure shall furthermore field banach value equip sup shall thus logarithm space measurable measurable mapping say integral furthermore say weakly borel measure denote say value real write converge outer space fold n ny p r ny n open integer integer uniformly continuous hilbert b b next proof multiplication continuously embed embed compactly compactly empty fx totally minimal ball entropy space norm abuse let non give open I law sample furthermore family simulate synthetic density argument consequently variable value indicate introduction form also arise way application indirect section shall estimator obtain projection measurable remark denote auxiliary repeatedly summarize subsequent proposition proof find statement empty equivalent element iii singleton affine hyperplane endowed topology coincide tt topology exclude singleton natural shall requirement coincide stress proposition norm sup apart maintain assumption frequent representative strict prop interior state interior strict assumption strict inequality interior happen g map mod equivalent e equivalent interior specify density assumption respective mod trivially hold already appendix equivalent ii behaviour imply simulation continuous mechanism satisfy mechanism relate could conversely mechanism principle prefer work assumption assumption introduce auxiliary indirect eq k p kp px view logarithm value give x satisfy similarly estimator element kt eq existence uniqueness uniform central go first lower equal extend consistency norm norm prove consistency estimator even extend appropriate stochastic cf k q kt satisfied view b non empty show radius singleton hilbert subset maintain continuity claim coincide property impossible view np x px p qx px gx px qx ix I apply make proposition potentially argument uniqueness fix lemma compact assumption property appendix mapping nx x v k turn theorem already contain less assumption apply however norm q define restriction value make maximizer c maximizer give theorem already let satisfied k restrict number real law b fix arbitrary event converge element definition maximizer lt go function pointwise integrable monotone conclude equal lp hence analogously part eq q argument assume k k dp k view proposition remark contradiction assumption maximizer monotonicity term k appendix recall h k p l c appendix h p conclude follow theorem tend satisfied tend iii assumption satisfy event idea van norm give analogous generating cf subsequent estimator convergence avoid restriction n trivial result choose remark estimator since kt lemma p ks instrumental assumption subsequent mod assumption mod inclusion establish borel verify k condition satisfy taylor td convexity verify empty denote logarithm imply sup consequently proposition b get display establish condition condition satisfied rate complete recall interpolation inequality mod strict iii coincide follow case assumption theoretic consequence let process result analogous hold h appendix n coincide assumption also k borel measurable kx ff space borel evaluation totally appendix f kx coincide apply establish limit process p part largely mean classical sum true fr turn empirical gaussian limit negligible term ingredient establish uniform previous apart classical major usual together eventually imply maximizer zero relative r conclude maximizer save norm either reasoning necessarily small provide lemma simplify lemma parallel empty h g may hold w differentiable prop conclude k repeatedly prop k conclude last prof contain note make suppose mod subset index bound path process separable range uniformly metric give let empty subtracting use prop lead k k subtract p gd gd every expression supremum display supremum display proposition expression observe p repeatedly prop put thing every real nonempty denote nonempty nonempty subset bound property every cauchy theorem next class view borel nonempty subset already l proof event tend estimator f weakly brownian measurable value function f index k latter property follow correspond bridge isometry view theorem proposition b obtain ff definition space measurable analogue lipschitz outer integral denote lipschitz hypothesis separable brownian bridge denote fix theorem separable constant h display brownian bridge index theorem describe weak discussion respective projection empirical canonical appendix study indirect auxiliary estimator define eq well value separability belong respectively jointly measurable measurable
knowledge indexing map likelihood note invertible equivalently remove correspondingly rewrite normalization calculate use law ps ps uniform distribution calculation agent know perform calculation calculate next estimator agent probability increasingly require place belief al neighboring belief estimator span previously see dimension possible occur converge belief slightly subtle argument convergence diameter current incorporate network option challenge calculation remain tractable leave standard martingale argument area study mechanism neighbor neighbor sufficient calculate even explore feasibility paper certainly convergence happen fast take propagate network star simulation lead believe sample diameter memory span conjecture converge minimal thank network thank al discussion follow helpful suggestion write result section corollary remark theorem open model social computationally rational calculation resource large result extend preserve computational feasibility agent social network member year measurement independently acquire piece deviation belief estimate update converge belief additionally iterative network efficient calculation carry mathematical behavioral economic human utility action world take may social take characterize rational intractable usually rational network clique field receive weak state aggregate majority high enough correct recovery go go known view attempt extension relationship connect goal finding de node perform computationally lead original leave desired maximize unclear hard rational average signal apparent proportional total connection network network matter large constant away setup chain individual set tie model somewhat limited realistic agent act prior network converge receive potentially unbounded furthermore fully rational carry agent begin agent completely aim utility round require know network study paradigm setup show belief agent converge possibility action generality model infinite belief private iteratively pick learn neighbor result prove rational calculation tractable optimal signal finally take place twice agent diameter thank attention briefly place appear main contrast computation theoretically dimension code agent requirement agent structure social justify large network calculation principle derive mean discussion follow variant preserve efficient conclusion part agent state world behavior model network tie social agent chain neighbor neighbor simple certain measurement normal deviation initially belief regard update posterior piece belief prior improper learn improper distribution pick assume current belief optimal want pick carry observe neighbor distribution base formally agent action next observe neighbor know agent involve rather memory computation agent calculation simple algebra involve matrix belief converge access private furthermore converge
gradient scalar de familiar calculus complex diagram vector calculus also combinatorial analog vector de complex triangle edge laplacian analog laplacian finite calculus tool computer graphic engineering mathematic differential form vector decompose potential vector although potential first free homology finite really four idea book subspace slightly prefer terminology algebra refer operator presence always dot identify vector away slightly use middle splitting two subspace orthogonal presence ingredient get splitting come subspace fine piece fundamental idea elementary unique orthogonal obvious complement b ta inner precise write equality identify dual space space early pairwise equation content drop theorems square minimize gradient equation converse sufficient book classical rank positive definite nontrivial kernel function minimize dot ax latter short square square ne tucker equation saddle calculus algebra analogous term purely harmonic existence state least normal tucker easy harmonic harmonic orthogonality also second formulation ease write square square normal notice contrast chain minimize use difference interesting rank fundamental chain relate datum vertex rank part loop clique cell loop length four capture harmonic square component value straight involve vertex orient inconsistent triangle choose part triangle also harmonic would square loop connect equation solver recent explore homology clique arise enyi homology homology extension homology homology harmonic pure globally bound edge enyi homology almost type omit phrase restrict homology relevant enyi homology homology homology et clique almost satisfy column serve new look numerically try mathematical mention apparent bound infinity homology region suppose trivial homology tool ask number measure measure harmonic stacking yield desire find value count number alternative evident experiment picture presence homology transition nonzero homology region homology region graph trend visible appear peak mark vertical line quantification homology investigate much contain homology nontrivial harmonic result column right experiment topology clique scale free graph develop graph plot enyi area heart usually different experiment accuracy system solver square graph experimental special star free perhaps study enyi probability phenomena cluster see shape distribution life graph reaction capture generative process accumulate experience study rank underlie come iterative solver worth variety iterative rectangular aggregation aggregation especially attractive ignore could kernel direct solver nontrivial handle vertex pressure remark rank normal symmetric solve method et optimality however reliable method et available need dominant graph clique six et solve due lack algebraic restriction mesh carry adjacency aggregating strength connection define way vertex belong aggregation center algebraic smoothed enyi random various solver table formulation sa aggregation indicate another variation table homology fail setup mark specify double symbol sparse partitioning list setup problem solution algebraic approach hybrid combine algebraic direct problem direct solve partition dense system hope small form use system approximate inverse guess ia I ix ir solve dense inverse successively refine equation description dense technique sort vertex scheme like reverse approximate order sort general sorting scheme likewise least none yield consider matrix feasible hence report rank enyi smoothed aggregation solver rank two graph result amongst conjugate well perform optimally certain differential would algebraic sometimes poorly algebraic exclude setup solving setup worse last could even reasonably table algebraic take second phase second algebraic experiment expert find contrast box serial currently mean million require computation open explore method result approximate marginally slow alone table system method size triangle detail storage requirement store storage figure visual representation displayed apparent little locality visible mesh er mesh locality whereas researcher formulate rank study present involve take elementary highlight square rank graph area calculus algebraic absence calculus geometry arise information vertex project aim student relate hard comparable elementary state formulate system involve involve decomposition harmonic manifold problem differential equation metric case require matrix star calculus discrete calculus analogous study require stability graph simple code use experimental formulate information field lead algebraic since successful equation poorly graph poisson equation involve operator lack partial equation solver normal low square triangle area small piece minimal one need method method problem create connection solver dominant reliable software implement comparison various surface exploit study surface assignment rank part dms rich wang h harmonic trend graph although comparison section figure gradient estimate laplacian ratio eigenvalue certain condition label enyi graph count obtain bind measure obtain label variety enyi graph cycle star graph graph marker plot next table r homology trivial absolute error enyi graph three experiment algebraic table homology fail phase mark symbol reach number mark double symbol mark symbol dense partition detail enyi graph graph setup phase algebraic table require phase enyi table show coarse grid algebraic useful fail phase enyi graph next nine pattern laplacian four top vertex number reverse label column show result visual figure example middle way ccc er er er er ba ccc ba ba ba ba scale ba ba graph edu http www cs edu rank comparison rank alternative value comparison come impossible residual really remarkable square graph reach area spectral graph laplacian clique connection explore paper easy fundamental elementary algebra aim basic connection topic numerical method work small algebraic generally count calculus theory chain item use lead square way precise usual laplacian square actor study study numerical analysis square find gradient however different field domain conceptual algorithmic implication give direction area seem rank ranking ranking use ranking opinion allow quantity rank formulation link link keyword focus rank independently interest comparison al calculus decomposition motivate rank different ease graph calculus al working calculus albeit never mind modern physics vector calculus manifold calculus make computer graphic partial equation computation try reduce simplify algebra suitable serial result result never appear many experiment need theorem yet numerical hope reader implication exposition aim idea aim mathematic community broad imply stem experience calculus skew tensor object associated orient object simple study discrete calculus display book chapter idea book solely numerical algebra aspect iterative far solver al ranking experiment find time write work performance quite problem reason square laplacian solver mention square rank graph type eigenvector manifold extensively geometry laplacian spectra well operator propose application understanding solver hope payoff go potential paragraph eigenvector detail numerical topology random clique clique augment subgraph consider establish connectivity naturally homology clique lead clique reproduce graph free something analyze elsewhere establish maintain list field development rank graph diverse development need easy old code differential equation away examine problem formulation rank real value pairwise represent comparison opposite skew al oriented precisely pairwise opposite orientation multiple version strength task translate find always exactly vertex impossible sign add procedure close possible whose difference reproduce part edge represent technical sense consistent loop sense rank still inconsistent square particular second equivalent formulate tensor follow introduce pose ranking method differential equation emphasize square graph inconsistent fit equation think al projection viewpoint address web graph theory alternative preference concerned quantification strength reasonable internet win high preferred pairwise exist winner hand square rank rank alternative vote scheme least rule edge weight arrive alternative acyclic graph edge weight originally acyclic produce graph square account remove satisfying requirement ranking future require accuracy rank treat random give mean variance strength variance rule rule account outcome alternative sort mean via interaction least globally optimal pairwise vote give connection ranking connection random propose ranking alternative rank proportional strength preference frobenius eigenvector subsequent rank strength amongst square ranking finds optimally require easy right side create preference preference analyze rank perturbation perturbation particular rank method ordinal perturbation rank page rank preferred ranking social concerned order alternative relative strength completion matrix expensive connection calculus quickly recall square problem describe rank basic idea basic terminology concept need calculus detail call simplex say refer dimension simplex refer abstract equivalence even permutation fall one orientation orientation orient complex orient weighted oriented abstract edge vertex graph augment clique abstract dimension clique yield clique triangle orient triangle refer clique loop valid complex reader may simplex change real orient change orientation algebraic topology integer chain since chain particular
polynomial statistical release release small lower bind corollary fact inefficient additive proceed multiplicative check update omit description remark proof query release call learn implication make prove concept preserve privacy release many point section present need concentration explicitly variable rx dx bounding function strong concentration independently dimension concentrate around crucial submodular satisfy self concentration generalize monotone submodular monotone satisfy string whenever ix roughly element weight rather probability hand string prove work keyword fact conjecture pt know answer query statistical ask query agnostic answer question concern small fraction answer submodular function graph interesting point main application efficiently differentially private answer progress problem preserve datum unconditional differentially potentially statistical query implement hence barrier differentially private element correspond privacy enable rich individual like rigorous privacy outcome nearly indistinguishable set differ sometimes contingency fraction individual open efficiently private datum encode answer output differentially private boolean query privacy imagine analyst wish cut subset cut small number constant complexity give fundamentally polynomial guarantee boolean permit implementation query statement preserve release theory put aside privacy pose question sufficient show operate conjunction apply submodular independent concern implication characterize compute privacy secondly unconditional query release permit regardless privacy knowledge class almost date include recently multiplicative mechanism statistical query make query yet implement statistical summarize develop query release expressive simple class conceptual result privacy theorem submodular submodular runtime fx give differentially private release boolean differential differentially private differentially private fraction may large fraction however strong work bound mechanism euclidean privacy preserve literature ability conjunction release agnostic learning query release exist learn release statistical moreover release make statistical agnostic preserve query complexity neither reduction preserve runtime equivalence characterization factor answer class ability make differentially thus answer simply vice versa release informally submodular submodular satisfie submodular recent concentration self imply query additive submodular correspond query database characterization release multiplicative recently maintain universe query observation datum set much possible query relative provide submodular style learn monotone continuous submodular distribution spirit inspire introduce potentially lipschitz constant submodular function studied introduce gave centralize pac value function class preserve class mechanism universe ask class give theoretic privacy agnostic characterized al conditional mechanism release al query inefficient run universe g universe latter mechanism set encoding mechanism answer mechanism query hardness small know computational set useful unconditional bind monotone interact release monotone lower depend representation almost private hardness private information query bound run differentially private particular average conjunction size recently arbitrarily conjunction query early show give interactive private release mechanism analyst slowly privacy ask make intuitively characterization release implement interactive mechanism statistical query analyst ask step induce agnostic answer preserve privacy database interested privacy privacy differential pair database rd rd query abuse two adjacent database laplacian distribution universe ask query oracle parameter differential distribution statistical early differentially private new example denote algorithm output privacy answer laplacian set predicate release datum interest release statistical query fs concept consider influence reject get claim demonstrate claim suppose must converse suppose sake minimal ps jx imply claim observe mapping property oracle submodular tolerance set function tolerance map property compute tolerance proof oracle always however use q differ establish detail call modification reader step proof size show every vb bs proof claim observe f vb u uv decide whether select inclusion analogous ps also manner place element establish product distribution subset universe product remain collection return bs approximate oracle submodular apply error exact additional lemma equivalent state corollary plugging vb claim follow monotone refined structure replace lemma strong guarantee monotone submodular submodular prove theorem oracle query collection return oracle mapping respect gb vb function guarantee vb c b cs compute item submodular b vb g cs directly bc g item analogous construct appropriate follow analogous already clear mapping tolerance tolerance invoke complete algorithm subset universe sample collection mapping vb tb cs notational throughout detail estimate estimate sufficiently number submodular product oracle tolerance let query tolerance oracle query apply submodular state plug cs combine submodular answer approximate submodular like appendix problem database release result previous monotone describe submodular indeed monotone section element submodular monotone universe element natural corollary differentially boolean completeness product though rely query answer collection mapping vb bs b corollary directly monotone boolean conjunction last release release replace monotone width distribution concentration statement denote uniform string follow lemma throughout less monotone conjunction width thus approximation monotone omit monotone presentation graph discussion multiple edge sensitivity observe cut count database submodular decomposition construct release preserve privacy modification lemma decomposition preserve show query agnostic specific release consider find concept p reduction query release agnostic weak take value agnostic concept satisfying p strong sense agnostic section agnostic sufficient release algorithm agnostic exist distribution query return query correspond proceed induce eq strategy uniform short distribution concept distinguish multiplicative
signal easily however suffer correlation correlation great strength priori assumption impose dataset restriction correlation causal result trivial I restriction correlation since correlate remove correlation find description toy singular assume consider observation time mean datum correlation information correlation spectrum statistically lie low spectrum represent reject exclude dimensionality spurious correlation however uncorrelated variable eigenvalue include find singular spectrum represent theoretical eigenvalue density theory q empirical suggest nontrivial correlation construction check whether describe index publish indicator variable economic indicator explanatory period I standardize carefully optimal internal correlation uncorrelated h cc uncorrelated create rectangular calculate compare result singular benchmark high energy exchange cc eps whether necessary proceed present path necessary conclusion future example correlation redundant significant svd truly correlation investigation stock
countable finitely lebesgue follow make finitely finite finitely vc figure process theory geometry probability inequality law large uniform insight immediate corollary vc vc approximation analogous finite extend vc join consist intersection empty vc dimension large boolean every dual vc introduce vc family use proof collection subset vc vc prove begin specifically boolean boolean pointwise subsequence finite boolean stage splitting mean weak incorporate subsequent stage boolean van private intersection construction critical space essential factor topological section devote family number deduce vc uniform family empirical process envelope f f countable function measurable particular say f need minimal element notion separability regard indicator vc routine argument countable generality cell remove denote follow relation c b c l fx suppose envelope integer finite level define f mx fx let family f uniform law large frequency limit probability give class establish vc class large uniform atomic probability vc appear number x stationary ergodic almost tend infinity corollary number f find complete law vc ergodic early version simple assumption argument follow outline improvement anonymous standard vc countable elementary every countable fail sequence stage wise splitting stage arbitrarily join collection vc boundary great set functional measurable I h mh adopt stage subsequent stage splitting wish nn nc kn kn kn j kn proceeding stage define note fix splitting positive suppose generality eq elementary claim induction note fix cell ensure cell imply cell join join necessary cell imply eq equal inequality hold every let join lemma vc dimension inspection measurable positive define clearly vc equal anonymous comment lead simple general acknowledge helpful van early paper present nsf set corollary family dimension number satisfy law corollary vc major vc space measurable combinatorial complexity say every
bc specify shall explain intuition detail process network actor actor actor community element e bc j z hierarchy likewise receiver use interaction scenario worth hierarchy compatibility hierarchy child position receiver appropriate receiver position place sale theory explicitly interaction community arbitrary severe parameterization decide thorough position share parent element infer path child hierarchy infinitely even guess appropriate bayesian chinese restaurant short extension chinese restaurant recursively integer shall level actor crp draw define hierarchy crp crp top unique top level associate child crp result crp ad newly prior tree crp priors crp share concentration finish describe generative actor display complete model side path imply infinite infinitely compatibility integrate path exact intractable integrate condition I z beta conjugacy value shorthand bc j iterate expectation conditioning actor stick break depth domain ij actor actor actor interact time explore compatibility element actor tend interact within element actor tend interact community element high noise actor actor interact analogously hierarchical denote spectral unclear cluster let pick likelihood calculate tp actor depth false count false total average show quantitative result illustrate result diagonal figure result edge hierarchy actor show illustrate result branch branch branch size correct cluster branch strongly diagonal little level branch imply actor interact community perform good figure figure essentially actor poorly model actor specific rich network web choice reflect two mode see web reflect social could hierarchy branch different organization chart subgraph actor parameter log log likelihood subgraph hold subgraph require tuning parameter experiment likelihood result either dataset model likelihood superior notably role extremely recover complexity rich blockmodel candidate author interpret world early community hierarchy actor via grid importance optimal parameter take lag time iteration sample posterior path generate pair actor share actor share position consensus simply post visual merge bottom actor sampler link augment interaction miss annotation shape specie contain outli community community web species community nd range large separate least six either specie community specie super level identify occur fine grain interaction specie community community level interaction interaction link g community subgraph physics citation construct subsampling paper involve parameter post require hour processor core run sampler burn take sample lag iteration complete hour processor infer community annotate paper frequent title actor learn solely citation sub citation imply paper topic keyword field physics sub community super focused keyword string concept reflect community dense separate specific researcher narrow topic community super surface deeply involve infer community level mixed actor bottom interact community edge head color assume interaction solid represent annotated membership stochastic blockmodel model multiple hierarchical membership actor interaction simultaneously recover mixed membership hierarchy discover role membership expressive non compatibility simulation dataset real recover intuitive membership organization collapse sampler efficiently scale dataset around actor complete processor support nsf gm fellowship technology school science le song school pa school computer science university pa actors realistic play one interact social community analyze interestingly organization mixed blockmodel generation social community membership actor automatically discover gibbs algorithm conduct synthetic real network citation network self organization arise information actor actor formation particularly interested network actor exist actor role single role interact actor social actor play influence actor interact mixed blockmodel specific nature actor structure social network formalism flat role actor structure actor beyond simple simultaneously finance department european american branch european might business orient membership european finance department normally business american finance european branch company american european resource european member specific american motivate yet infer visualize semantic actor link actor hierarchy send day hierarchical membership membership community actor form model structure implication link sequel begin specify branch next relax nest restaurant process direct graph actor adopt convention else edge actor receiver latent variable actor membership short record actor level depth hierarchy actor branch root
non component ac sa choice stochastic however gradient descent incorporate technique nesterov replace closed like fuse solve accelerate apply stochastic word deterministic gradient totally prove convergence combine smoothing gradient determine algorithm independently iteration choose ny hx aspect exact replace length maintain sequence point x ac sa unconstrained ac sa hard depend performance solve true convergence algorithm follow total fx gx technical lemma present algorithm inequality technical characterize q classical see subdifferential x z give follow third one imply updating equation convexity lemma l z fy fy hx z z z z z cauchy schwarz accord side come q imply n n n x convergence sa projection solve similar appear particular fuse lasso iteration could application modify utilize construct smooth control approximation well proved function eq therefore gx tv tv show constant convexity parameter smooth apply parameter get good modify iteration smooth lipschitz z ny tv step algorithm mapping non term close solve small return near fortunately appear dominate observation result find descent algorithm propose apply smoothing dominate obtain word price incorporate technique stochastic develop dominating lipschitz term incorporate strongly stochastic different regularize ac sa algorithm propose matlab implementation run cpu processor gb ram fit function occur equal chance square prove gradient iteration whole whose element interested input influence term force highly relevant regularize level choice eq q problem apply sg ac sa projection sg ac sa apply represent ta parameter generate e numerical represent cpu p n introduce al input subset set induce element group input inducing call apply algorithm sg ac case mapping sg long solution coordinate descent mapping adopt solve sg apply non formulation dual auxiliary associate let ball space sg ac mapping ac sa dataset iteration generate algorithms numerical result though sg ac sa performance sg ac sa reason ac sa length impact stochastic ac approximation sg influence nesterov reflect mapping close apply necessary green curve overlap sg sg sg mapping solution infinitely parameter numerical linear variable eq loss gradient discrete regularization stochastic set ac sa sg performance reflect figure converge fast ac belong minimize likelihood eq regularize choose norm lipschitz lipschitz regularize whole generate apply sg sa follow component probability three generate put decrease logistic ac sa solution projection sg ac rely subroutine smooth another gradient descent smoothing term convergence scalability proposition
finally expression limit variance formula limit way obtain adjacent form term finitely many acknowledgment author thank I helpful concern associate anonymous remark lead considerable improvement manuscript remark la procedure usually either combine approximation however establish posteriori due dependency weighted estimator equation maximization composite strongly convergent consequence overall simulate exist procedure economic analyze analysis datum scientific social science communication molecular biology protein vast book study graph bernoulli edge homogeneous capture feature network lack heterogeneity lead introduction primarily science group characterize relation refer adjacency matrix indicator respect belong exhibit block block intra group connection block behaviour diagonal call structure parsimonious lot model community structure inter intra traffic network consider adjacency financial last concern dense weighted network integrate primary problem vertex cluster widely community analyze graph existence however rely mixture increase stochastic network beyond scope approximations intractable core hide factorize form variational strategy behavior support indeed major procedure variational local maxima posteriori open simple variational focus treating main either moment composite likelihood consist marginal dependency cox weighted show estimate rely binary graph mixture identifiable identifiable moment composite induced establish normality rate distribute graph issue edge order convergence group proportion convergence also limit variance limit result fact variance organize notation limit introduce specific binary first equation proportion rely random model rely composite theoretical dedicated section illustrate datum finally proof notation v sake consider generalization graph without loop mixture first follow use heterogeneity index general independent worth adjacent precisely belong shall assumption particular observe degeneracy specific lemma section later thus proportion motivate single follow weighted assume absolutely namely vast continuous mixture maximum variable usefulness rely whether expectation answer yes good identifiable variate parameter tuple estimator core k kp eq consistency asymptotic estimator asymptotically normal g group proportion us comment expression useful interval parameter simple expression state convergence happen equal prove variance meaning converge limit shall consistency preserve one give insight rate mixture instance model random binary random indicate presence absence node conditional latent bernoulli group attention connect whether belong intra inter bernoulli respect simply display example equation parameter idea moment correspond triplet proportion give rise multiple solution indeed never uniqueness partly identifiability apply proportion know algorithmic procedure proportion use shall triplet finitely completely finite moment three moment induce triplet three characterize triplet namely equation examine proportion phenomenon take recover rational obtain estimate introduce estimator specific theorem able proportion suppose converge surely almost combine shall parameter composite moment method develop preliminary identically mixture component five appear follow q zero soon equation define one relation constrain bernoulli mixture shall already permutation recover mixture identify mixture indeed identical different one appear consider quantity derive obviously accord uniqueness proportion unique let criterion converge identifiability limit attain classical estimator normality define define comment mention model rely comprise graph value result assumption question slightly triplet distribution identify corresponding proportion already bernoulli unconstraine finite variate decade identifiable rigorous elaborate directly hard establish strongly believe simulation seem reasonable restrict parameter space greatly simplify restrictive generalization chapter parameter prove result consistent infinity moreover increase proportion comment result rate rate indicate might beyond group group solution consistent rate least always case approximate estimator assumption converge iterate grow infinity weighted observation either indicate specify parametric dirac dirac accounting sparsity parameter focus structure connectivity identifiability reason constrain parametric admit give parametric could density counting parameter namely poisson would sparsity density concern weight edge htbp weight binary value column truncate use group intra inter community fail find meaningful graph model never propose literature dirac enable proceed rely plug connectivity rely estimate rather section consistently estimate independent variable accord mixture p f directly soon identify mixture composite provide introduce need assumption identifiable range family next deal regularity quadratic twice continuously last compact express log stress quantity derive label issue estimate deal grow infinity least mainly consistency direct criterion assumption consistency note theorem seem indicate phenomenon degeneracy limit estimator composite procedure adapt thus provide implement issue recover graph follow range number triplet let st nu encoding hide triplet observation moreover state coordinate composite dimensional bernoulli distribution optimize iterative compute conditional use step maximizer triplet value constant observed aim recover procedure group plug previous section datum latent proportion simply remove datum simply call latter frequency introduce criterion estimator dependency section label estimator denote latent choose couple estimate weight propose use structure iterate procedure maxima value maxima estimate simply take frequency procedure summarize start z procedure analyze introduce value weight graph estimate section illustrate start triplet transform triplet estimate edge study examine bias proposal propose strategy implement assume distribute describe generate group number create different intra connectivity free proportion model picture three method estimate method triplet method propose result present simulation give density variance magnitude htbp computed group line true estimation three model produce enough asymptotically unbiased agreement bias among estimate number vertex equal group compare deviation simulation show vertex see deviation vertex relate similar converge deviation remain comparable estimation great dispersion rand evaluate agreement latent rand belong consider structure identical rand rand index three size allow well structure structure variational latent since analytical resolution structure miss distribute equal mahalanobis distance model different group proportion empirical deviation function average simulation c computation bias recover parameter bias c high deviation vertex vertice binary slope lying display perfect efficiently consider order economic non symmetric graph
q class reflect rewrite interpret give optimizer nonzero find propose singular globally convergent certain completion problem objective aware work generalize denote matrix satisfy frobenius define follow solved initialize arbitrary stop establish hyperplane w lemma satisfy e onto gradient local outline standard form prove nuclear problem incorporate form partial constructive pick define assume generality small outline kronecker product use replace basis main proof relate bind vc elementary two define subset basis define follow ir b obtain identity support grant fa nsf recommendation express material author view nsf assumption conjecture theorem remark edu alternate observation feature performance optimal hoeffding crucially contribution new extraction cast obtain optimization testing extraction decide favor either assumption uncertainty motivate behavior amount observation simplex hoeffding empirical leibl element kullback hoeffde nonnegative favor hoeffding characterized error exponent statistic summarize give hoeffding observation theorem derive imply drawback hoeffding exponent optimal size observation potentially address hoeffding name enjoy advantage design robust error knowledge knowledge heterogeneous incorrect optimization supremum ratio divergence restrict supremum special case hoeffde expect choose determine divergence exponent critical successful implementation us exponent come maximum different optimization provable convergence context framework paper family pca find estimator kl divergence improve upon motivated exploit limited possibility alternate kl divergence dimension might alternate divergence absolutely respect exist say given require basis fact distribution dimension family mutually absolutely continuous except obviously family characterize divergence neither absolutely continuous error ask pairwise need support parametrize call exponential
strong detection formation vector embed dimension wrong continue point vary strength direction grey te additive suffer obvious system large less measure regardless observe give discrimination perfect measure cause serious implication world presence investigate advantage te noise couple enhance add te allow level te two large sum make weakly drive first system delay absence regard dimension use environment matlab sample large strength future tend increase tend increase opposite direction balance take positive two coupling direction mask rank value direction couple system couple display free legend te value measure give fig b illustrate noise te variance discrimination coupling strength fig give note couple favor te turn sensitive sensitive presence rank te seem presence condition world result te rank useful coupling recently transfer entropy term symbolic te vector modification augment comprise reconstruct state consider show vector joint far suggest rank sample simulation couple assess receiver find couple varied three presence state system fix delay future summarize te mask direction less well te noisy dimension pass correctly justify work couple drive measure wrong measure te measure length strength provide term stable vector requirement future rank te embed length increase couple zero coupling measure couple maintain coupling lack te show large direction complex system complexity rank direction strength coupling discrimination higher particular te te estimate sum showed perform worse especially attribute estimate stable noisy te entropy vector journal system transfer te measure information perform setting define te reconstruct rank system regard version transfer rank propose ahead drive system assess te flow receiver operate roc form coupling realization couple state space observational show te particularly te ahead improve keyword couple causality transfer measure measure causality direction approach extend approach focus synchronization neighborhood reconstruct flow concentrate last measure entropy variant te operate symbolic entropy see comparative give te perform te application mask detail structure treat discretization te response scalar modify give te information mutual grain interact flow ahead regard financial index map well see interact correction measure interaction system reconstruction investigate one ahead measure briefly te compare te interact delay respectively te driving response system cause drive accounting response te shannon mass pmf cell variable appear entropy te estimation estimate directly occurrence dimensional reconstruction estimator demand estimator te near sum follow te discretization variable entropy approximate inter distance suitably estimate sum vector account euclidean symbolic transfer reconstruct rank assign small assign substituting rank vector combination cell combination vector occurrence scalar express rank setup entropy constitute sample horizon propose substitute transfer ahead entropy rank augment rank independently appear time capture time ahead reasoning use te direct te flow joint affect simulation favor te note lag rank vector distortion rank vector large g entropy term involve coupling measure use instead bias remove strength te simulated mean computation te point stable maintain neighborhood preserve distribution insufficient coupling measure bivariate time te good direction coupling coupling quantify discrimination information net assess discrimination set receiver operate characteristic roc coupling direction e g flow detect couple great fig te strength reach reach couple bivariate white te small white noise deviation small te strength
hold would time panel suppose within service panel job server service job service job leave change service job processor job start job leave impossible later job early conditional close unnormalized even compute require new service time affect computational conditional important consequence perspective natural arrival times times fact present severe fortunately though expect small long occur expect typical time therefore queue sufficient develop computationally efficient sampler note queue markovian service exponentially markovian queue chain except queue queue arrival incomplete section difficulty time service time inverting parameter either model em infeasible markov sampler sampling wish accomplish slice sampler leave invariant version iterate sampler sampling uniform slice update leave horizontal update spirit easy slice previous metropolis slice difficulty density normalize unnormalized job slice sampling refer times slice constant pointwise invert set service contain time number fortunately cost updating change factor product nu value mean value select sampler infeasible unnormalized compute change equation service job b e e old arrival job e b change immediately queue propagation service compute update service time change algorithm section relevant job set algorithm section propagation queue arrival queue main service depend previous processor time immediately previous job separate unnecessary queue job time update queue alg change hand none service recall arise job enter service queue penalty job queue become compute queue implement sort queue contain queue arrival compute dt finally time nu index propagation queue algorithm arrival algorithm service times ps function queue structure queue job ps function portion service search intersect implementation variation store initialize nonconvex latent service configuration make service belong cause value bad mix ed service finally service switching distribution time avoid generally processor subset unobserve initialization initialization proceed unobserve task path service time service service step service switching service prevent would service describe analyze propose experimental setup facebook develop web actual implement software profile twitter figure architecture common medium web service show incoming request first web server system image page request content comprise email change time user make request time computed request web server design run request copy store user ability external request purpose assign request copy copy obtain elsewhere run machine copy setup machine per load database involve machine series request generator total cause request range amazon ec utility request record handle request spend library time spend database cause database query bad high practical degradation variety evaluate prediction period average predict interval arrive queue ps networks service single queue arrival datum database baseline regression regression response maximum pt power law processor queue network ps reduction interestingly differently capacity important simple processor difference type poorly inspection suggest cause model outside datum g prediction poorly demonstrate reconstruct miss arrival database time spend measurement intrinsic distinction response add due intrinsic processing log minimize overhead service estimate well use estimate ps delay connection minimal service estimation miss observe axis time second display spend queue visually resemble increase quantitative measure bin bin system root mean square reconstruct service incomplete time infer subset baseline compose service denote job ms ps reconstruction baseline observe linear worse interestingly support limitation final although selection receive relatively attention context performance characteristic software often understand system build external software library whose internal fully code software hardware failure idea behavior pool behavior expect system bottleneck queue add couple consist arrival queue transition bottleneck queue unobserved basis goal miss queue choose determine lar lar aggregate covariate queue choose bottleneck assign second covariate coefficient bottleneck despite poorly relationship covariate response nested service time distribution gamma come arbitrarily close mass zero knowledge bottleneck queue choose augment whether service bottleneck queue statistic alternative selection likelihood demonstrate use bottleneck five ten alternative consider work bottleneck queue technique synthetic generate figure arrival homogeneous service bottleneck queue varied contain service distribution bayesian sampler iteration repeat independently c selection queue decision queue bottleneck queue queue lar well chance technique perfectly hard display roc generate r package curve lar perspective presence miss idea perspective demonstrate web fact network computer distribute application build perspective queue consider setting work focus single rather addition restrict restrict miss queue observe none arrival estimator inference algorithm markov theory algorithm indirect possibly situation single distribution example single processor queue place service arrival metric steady exist goal diagnostic serious distribution asymptotically unstable difficult modify incorporate sophisticated prior issue area network focus estimate delay network inferential link link include request current largely direction model service example run effect service imputation grain computer detailed web book google internet services yahoo amazon request day computer service operate strict great queue computer detailed request heavily bayesian arrival viewpoint viewpoint carlo benchmark web application model internet service google yahoo amazon serve facebook million user retrieve large scale web application thousand request request request request computation web operate strict minimize site web page return response delay cause business http com web concern response example ten web spike poor request handle slow request incorrectly distinguish performance hoc question regression response onto natural server single queue request allow incorporate two qualitative connectivity inherently incorporate maximum question way concern answer question request simply infer parameter arrival second request system much spend queue process request reach queue call request inferential service operate collect overhead overhead number request delay site receive million request per reason incomplete arrive request scheme minimize storage expense introduce inferential incomplete datum idea arrival arrival observation service enough advanced include service sharing arrival approximately monte arrival em present arrival complex arrival time queue necessarily arrival previous consider network model much require request share web service design architecture presentation generate web application storage often request typical service actually consist multiple request web internet randomly make request necessary generate turn request storage friend web page engine queue individual reflect system web service request request assign first necessary third response reality serve request ignore external request terminology request individual series cause request web service external request web typical would order probabilistic arrival job task queue task markov chain call reach potentially individual job time arrive queue job draw independently job queue amount queue job head processing service draw density jacobian vanish jacobian observe knot whenever job job knot write number queue job job precede jacobian triangular bring queue section network develop job queue job job queue job job task whole initial queue queue processor simply service queue notation define denote chain always specify base figure structure define queue structure arrival job service accord service parameter service service distribution solve system network system parameter service service job time use regime jacobian fortunately jacobian triangular time joint likelihood arise path queue likelihood work improper choice prior issue key deterministic service distinction service important statistically service I time dependency queue impose arrival assumption relax complex combination job nonempty queue job enter ps however distribution inferential also make posterior model
equality represent pair x f f argument g pt f f g f f g g vanish moreover f f conclude p x p x pt x x non expression set outcome iff condition need integrate acknowledgement like thank anonymous comment associate greatly improve von binary iv extension way association extension negative independence hold independence normalization take zero measure dependence well simple similarly simple ordinal monotone simulation study random order outcome continuous categorical continuous arise social science ordinal robustness cc b cccc popular variable replication range sign version ordinary expression interpretation firstly see observation see three specifically term reason score continuous aforementioned definition score e obtain implement carry modern computer ordinal test robust zero association alternative coefficient hoeffding function hoeffding henceforth see hoeffde severe drawback consistent page option suitable categorical square applicable suitable categorization chi account alternative test treat nominal rather ordinal although rather decide probabilistic notice complex simple natural call negative independence probability analogously simple independent coefficient main reason popularity pre computer value much hard google currently appear equally binary er von organization state interpretation theorem turn involved related hoeffding new give description independence permutation replication z absolute coefficient result paper distribution mass true equality proof give sign obtain show schwarz normalize exceed define analogously unfortunately possible set suppose distribute uniformly cumulative decrease chapter explore copula probability pair pair jointly pair jointly minus jointly separate pair ccccc convenient four two pair jointly four point notation point permutation logical shorthand straightforward z denote randomly sum two term equal last equal see four say neither tie likely variable monotone ordinal variable hoeffde probabilistic use leave ordinal coefficient simple drawback ordinal consistent characteristic correspond equality definition show hence definite negativity imply show kernel hilbert appear proof simple mathematical formulate square euclidean summation pearson chi square well onto contingency table suitable permutation test value permutation time consume moderately permutation compute th monte evaluation practically infeasible moderately approximated taking well permutation marginal distribution statistic independence usually conditional test b l independence simulation artificial table ordinal category visually monotonic yield chi give association hoeffding consistent ordinal possible like chi hoeffding use hoeffding compare chi yield use hoeffding independence expect alternative gain look value six box represent line maximize correlation function box represent square double respectively box minimize represent sense comparative furthermore choose hoeffding little poorly six marginal copula course give equally increment normal plot increment heavy automatically care use simulate normally distribute value increment increment normal average least hoeffding compare bad walk double box poor section lie zero see likely hoeffde necessarily positive tie bad walk increment significantly cross test monotone
order neighborhood pose stability indicator non cc behave point fold sparse result present problem particular important behave standard structural equation quadratic study clear relationship among difference variable cluster group relationship among clear residual advantage explanation adequate learn necessary nonparametric r lin gps lin fold fold fold fold subsample training pair sign rank test suggest behave mix dependent believe size sample also sparse alternative non decade review classic simple parametric construction interested relate factor misspecification conversely specify discuss factor non none nonparametric structure context graphical real problem often ignore bayesian mcmc work improve mix particularly relevant ordinal mix difficult bottleneck procedure pseudo least open determine graphical promising thank relevant concern detail draw interest number pseudo input latent hasting current variation non variable structural latent indicator mean mixture block latent mutually function pseudo function sample conditionally convention particular explicitly mention fix implicit respective implicit implementation use submatrix update library gamma marginalization done give posterior conditional gibbs parent correspond condition variance j dy j p coefficient constant analogous set conditional structural new value individually reject gaussian proposal center probability graph parent value shorthand point accord cost use submatrix cholesky latent take child parent include implicit drop factor mixture latent respective mean sampling latent gamma random b variable give b pz e ij observe choose aspect plausible child variable target differ application design explicitly account moreover implication distinguish among implication detailed discussion instead serve block spirit determine condition subproblem detect edge establish result exploit identifiability observable assume assume independent easy entail marginally marginal independence variable identifiable scale marginal without loss generality follow result independent estimate latent instrumental residual residual discard justification indeed error identifiable consequence however opposite connect path use technique error input input formulation formulation without pseudo input explicit dictionary space scalar country national product indicator freedom parent instance b dependence essential graphical structure independent acyclic dag simplify presentation likewise treat define cyclic exclude particular latent parent empty distribution parent terminology mixture gaussian measurement parent mutually independent coefficient example square observe circle emphasis direct model sparse study explore exception observe share share indicator less might lead many latent limited condition identifiability latent I identifiable structural identifiable deconvolution element term joint deconvolution extra describe identifiability joint regression regression equivalent x identifiability condition causal additional assumption discuss literature overcomplete brief appendix mcmc gaussian without discuss case imagine latent measure least equivalently mutually input mutual clique instead parent represent index represent variable fix value whole I n equation function sampling introduce representation adapt input reduce choose pseudo notation represent I I x I kk x kk k pseudo motivation alternative mean dd I dm k k conditionally imply context input model overfitte pseudo fact see gaussian rather motivation optimize pseudo variable next pseudo input avoid pseudo choice partially informative easily freedom choose fix structural imply parent cholesky pseudo may cause one way propose jointly cost pseudo metropolis probability latent variable mutually remain accept walk accept ix dx children graph correspond child c x function crucially cost briefly illustrate follow identifiability interpretable study predictive alternative independent gamma model five indicator fix edge intercept identifiable burn burn due prior coefficient inverse mixture gaussians significant intercept purpose observe value able reproduce linear functional relationship pseudo input cccc latent assume variable measurement represent however learn application allow functional analogous illustrate quadratic rotation enforce identifiability study student aim pay indicator variable simplicity latent dag correspond measurement identifiable fix intercept slope describe reference section identifiable dataset
affect affected atom empty slot attain eq write wish minimize yield substitute z j yield collect like j n j atom respect final solve applie notice nearest ise material mean material use determining acknowledge valuable material mm ise institute science technology materials university nj usa show I methodology reproduce solution drawback agent interact ise purpose I tool compare traditional field show case main regardless I approximation case otherwise least justify simple spin individual strong exchange effect material behave determine net difficult second atom net dimension dimension approximation computational difficulty shannon entropy information assigning method method entropy I probability family allow posterior drawback present way assign bayesian method deal expect reproduce purpose updating probability I I tool material lie traditional paper traditional third reproduce fourth traditional I mean atom act atom allow interact neighbor quantum suggest examine atomic spin spin suggest exchange exchange calculate interaction energy spin atom partition write easily strength net energy atom mean hamiltonian hamiltonian atom word average effective hamiltonian hamiltonian part use I justify material I use write canonical I moment must purpose arrive produce field act wish entropy canonical proceed free reduce q I good information provide entropy term I proceed I well find maximize constraint solution energy illustrative purpose atom pose continue similar except solution summation since z yield minimize respect well rewrite eq though atom behave like atom process ising spin examine possible atom
characteristic observer raw return place remove apply change use follow remain decompose table affect rotation random mark ten population guess accurate state everything perform transform component table normalize unity randomly result experiment time use separately length show experiment table test make sense keep mind observer object compare correctly classify difficult different vary proceeding chance albeit linearly observer unity dc band band train ground result identify time instead slightly moment transform mark x band band eight point repeat test show good signature follow transform preprocesse direct angle four moment skew add numerical percentage fold fold set scenario aspect angle object observer experiment enhance experiment result relate technique bayesian popular technique pattern conditional laboratory describe paper aspect increment overall performance bayesian approach process identification increment four vector give cross validation show statistically guess give feature easier transform bin nothing exclusive bayesian machine arrive accurate process result feature comprehensive classifying computation load extraction load main preprocesse extraction least classification operation especially applicable negative release result review proceed grateful thank mark identification signal space identification object datum good fusion band band svm number shape mean sized object little object great interest survival keep geometric shape object determine return cross section signature though geometry approach effective signature different geometric pattern recognition vast pattern signal signature widely recognition signature return representative generally one training separate class simulate shape accuracy percent precision standard run signature decision deal computational burden minimize maintain mention potentially fast enough approach advantage three require database label signature time train input signature label signature quickly weight poor testing evaluation sample pattern evaluation simply consist matrix multiplication input ten one multiplication ten element product ten element adequate application significant neural advantage preprocesse signature automate concerned recognition neural good reference network bayesian massive require technique spectral classification signature spectra computation often quite consume signature advantage nonlinear perhaps consequently global well secondly unlike accept database signature synthesis synthetic incorporate operating location upon intermediate versus object shape ghz band ghz band diameter height diameter edge feature except object sphere adapt optical eq q angle incidence solution derive singular superposition add circular correspond incidence aspect htb aspect b x classic formula utilize center key htb center geometry axis plane center origin axis phase set comprised object approximately broad observe half symmetry observer ghz simulating return object change perspective dynamic comprise object object observe static one five period class vary length result datum second set observer band ghz effectively simulate return ground observe object amount object length goal signature sensor fusion improve identification ready begin process support follow label signature label follow represent signature simplify signature else group depend signature output equation training relation hyperplane hyperplane separation call separate hyperplane problem basically subject constraint label give nc analogous bias compactly representation point weight programming perceptron know basically find support reference several svms basis rbf kernel tangent kernel calculation compute whether support train element phase support result support case object classify belong otherwise take one randomly remain test repeat statistical randomly remain weight train classifier sphere non time select object second test
differentiable gradient show efficient apply suffice proximal may alternatively adapt proximal polynomial slow purpose provably hard nevertheless dedicate cut plus modular min proximal done dedicate derive closed sort element beyond cut cardinality express easily maximize minimum orthogonal subspace greedy submodular synthetic cm vary unique go condition proximal obtain agglomerative example agglomerative cm path variable show prop submodular total thus additional extend extension proximal penalize fw soft minimizer proximal cm matlab replication proximal descent dedicated situation graph chain graph light red penalization level consider analysis design matrix already proximal make typically guarantee first face create see proof supplementary material constant absolutely respect lebesgue one minimizer set define partition lattice prop condition correct one recovery lattice constant minimizer probability well eq design meet interestingly validity imply proximal exactly prop non merging term cm situation increase show presence know issue limit face set intersection prop condition one start practice optimality lattice proximal lattice partition f belong face constant value imply soon extension optimum general maxima candidate prop simply sa fa ii thus satisfied lattice check bf merge lemma moreover proposition fc fc fc fc violate merging set satisfied condition condition equivalent true interval one eight depend possibility neighbor chain trivial fc fc v fc fc fc fc fc bind path agglomerative namely two neighbor get unique dual pointwise maxima distinct union proximal element show full fine grouping order correspond show distinct lebesgue lattice order relationship impose second meet j probability prop j lead bf lemma get fa si p uniquely symmetric fa k moreover j counter gray node complement show connect correct even cm true pt minus pt minus pt minus plus pt pt mm sparsity submodular focus function submodular extension envelope set index whose component lead knowledge level set support predictor unify proximal guarantee allow submodular function statistic cluster noisy cut detection variable solution zero either use match knowledge therein sparsity one lead solution image detection induce automatic design sparsity submodular predictor show similar parallel submodular contribution link extension see submodular specific depend exist norm cut application presence outlier unified operator efficient adapting slow unified predictor always subset indicator subset moreover define index small index denote relevant proof cm throughout power b fa b fa unless usual submodular negative section undirected extension function fw extension moreover indicator positively bf p sa one submodular analysis submodular greedy decrease fw bf g say know tight tight face intersection hyperplane prop say cut set connect cm submodular beyond convexity homogeneity submodular zero immediately invariant addition constant contrary decrease regularizer norm envelope certain combinatorial envelope extension envelope note envelope intersection level relaxation perhaps consider amenable effect next extreme induce cm note need variation fa fw extreme separable subset lattice always topological chain go construct example thus correspondence go back constant subset diagram inclusion order connect represent diagram immediate allow lattice variation ia ii v empty relative face modular I ii interpret certain submodular undirected interior face lattice belong cm section submodular extension cut variation cardinality base submodular ga mutual eigenvalue weight cut extension variation symmetric set justification behind variation allow prop lead one may application direct cut may increase jump along edge piecewise outlier node level good total variation base noisy set detection vs variable
non burn effective monte relative ratio px configuration number indicate rr r expand configuration rr estimate relative efficiency different summarize table sampler px usually inefficient roughly draw posterior table reflect introduce regression large predictor example top middle intuitively failure px offer presence conditionally direct fail conditional matter enter hierarchy autocorrelation conditionally independent indeed seem thing suppose restrict near behave large globally small may study expand big expand handle global parameter default difficult situation posterior mcmc precisely solve simply encourage well option heart conditionally handle slow autocorrelation either depend make poor marginalization family prior beta variance doubly nest sometimes remain area sigma sigma lambda sigma burn sigma save burn beta lambda beta sigma lambda sigma jeffreys lambda sigma sigma rate lambda comment shrinkage sigma lambda burn beta lambda sigma save sigma seed sigma sigma delta lambda p sigma burn matrix sigma save lambda beta sigma lambda sigma jeffreys lambda sigma sigma expansion g z sigma lambda lambda lambda shrinkage sigma delta lambda delta lambda burn burn burn lambda sigma save sigma consider mixture example prior usefulness global conclusion expansion type improvement less especially attempt component practice approach seem parameter matter perhaps view replacement expand keyword mcmc normal expansion simulation history autocorrelation expand expand sampler undirected graph right px indicate replicate many common equivalent parameter expand version identify improve mcmc convergence generalization expand generally equivalence hold cauchy excellent sake illustration despite fact individually identifiable model identical imply easy moreover translate yet mcmc drastically phenomenon widely exploit speed useful sampler particular simulate standard severe px code implement apparent arise fact independently posterior autocorrelation location exchangeable mean look interval exponential outside transform work slice step invertible invertible slice sampling expansion applicable conditionally conjugate prefer slice possible beta beta cauchy ratio two equivalently possible use conjugate note component omit compare px px sampler zero seem perform well default reasoning concentrate meanwhile half observation exhibit modeling goal goal logic order plot expand expand simulation autocorrelation expand
ask extra quasi error square simplicity consider ordinary ol term ol fit ol note choice use easy expectation replace version case close rate limit covariance limit population solution directional give may obtain subspace may lead variability difference latter q equivalently tx tx derivative try avoid estimated level level conditional self adjoint taylor expansion asymptotically tx classical equal level classical importantly tx tx j g tx us attain follow function nuisance approximation define estimator requirement efficient first side negligible certain positively correlate weighted purpose extra negligible tx tx e tx g negligible gain confusion appearance place asymptotically normal achieve fisher information g covariance nonparametric calculation justify interestingly word joint extra automatically extra estimation help achieve reduction compare side limit equal super efficiency root nuisance piece estimation estimate true regard expectation nuisance pseudo link estimation likelihood estimation motivate use nx density direction employ technical handle smoothing argument begin subsection intrinsic propose identifiable idea previous identifiable estimate quasi boundary surface directly parametrization popular parametrization remove yu wang parameter infinitely jacobian pe sp prove invertible inverse parametrization coordinate presentation otherwise th component jacobian matrix component replace brief conclusion clearly inverse see easy invertible whereas rest invertible either latter derivative involve estimator check sign another estimating define estimate solution step positive procedure estimate scalar apply appear limit tend go infinity achievable tend need identically smooth linear derivative sum fan bandwidth confusion classical quasi glm uncorrelated distribution phenomenon may benefit correlation predictor step towards glm exploration glm dispersion interest quasi replace initial estimator likelihood motivation easy similarly previous technical similar structure unknown nonparametric construct conditional estimator obtain classical know asymptotically efficient still nonparametric final equation simply derive case unknown compare simple parametrization variance attain minimum result parametrization issue glm nuisance regard link true automatically estimation positively fisher efficient regardless parametrization small property interest correlation small key extra show glm nuisance paper may extended handle advantage estimation cost expensive nonparametric possible instability quasi give technical compact derivative ii lipschitz condition vx bound vx nh vx derivative smoothing worth unlike use truncation second wang present difference one reference estimation sequence variable variable hold similar main conclusion replace respectively simplicity cauchy schwarz fact wang j nr condition lemma proof omit x I hx f I verify note n condition side imply zero cn separate write note pn pn pn tx tx expansion tx tx tx ie tx tx weak law number use denote derive vx vx iw nj tx g tx tx vx ni tx tx vx ni tx jx vx together imply conclude argument estimator estimating argument also proof theorem easy clear variance elementary presentation ti ta n proof result application distribution variance far j limit theorem imply p tv p last tv q tv derive covariance q vx vx write negative semidefinite definite thus v v v tv tv v v cm corollary china parameter introduce pseudo consider nuisance estimate importantly positively asymptotically fisher asymptotically small fisher fisher information necessarily stage scalar trace limit fisher quasi information extension parameter augmentation readily handle classification word investigate issue develop literature linear blue description gauss limit achievable generalized well asymptotically unbiased information measure variable carry unknown upon variable depend present criterion efficiency probable efficiency linear widely class regression analysis complete result glm dimension write unless suppose method weight square used method exponential variance covariance attain rao er effect linear estimating prove optimum property estimation matrix quasi benchmark comprehensive book rigorous optimality nevertheless seem parameter nuisance model perspective extra estimation large study stage base quasi pseudo nuisance parameter regard trace covariance small fisher particularly reduction relate correlation theorem section reference linear gauss provide distribute shall
particular case mle asymptotically asymptotically subsample problem suggest variation avoid subsampling step method multiscale subject classification accurate physical multiscale one apply coarse grain effectively efficient choose fitting particular coarse grain diffusion apart estimation multiscale coarse average multiscale estimation equation case grain fit grain equation combination example newton question dynamic grain drift process go step path system locally dynamic carefully simulation answer global dynamic possible statistical come full important available full issue horizon either fix time horizon issue separation dynamic scale brownian motion computation reach definite build intuition behavior general tackle goal dynamic address diffusion quadratic commonly go discuss rise review core coarse grain give free grained explicit computation outperform describe heuristic come multiscale describe limit multiscale allow discuss estimation multiscale affect algorithm limit equation completely come multiscale multiscale equation banach space call slow ergodic transpose set eq call average limit condition assume positive prove assumption basic type equation banach space call fast eq step x follow converge eq diffusion similar definite inner theorem replace interested slow dynamic reduce equation simulate much step dynamic simulate review multiscale slow small multiscale multiscale hold differential multiscale limit system case multiscale drift system might pose realistic coefficient precisely limiting depend linearly parameter extend normality come parameter interval relax explain multiscale slowly behave diffusion certain say diffusion demand certain simulate multiscale general answer coming evolve slowly see estimation interpolation concrete describe approximate local path multiscale another starting possibly lead multiscale see particular exactly prove would maximum equation since quadratic lead q fact reason choice square error small significantly investigate behavior intuition follow behave scale brownian length behave like process value go definition process finite total variation non process clearly total variation order behave behave limit real path go function maximize cardinality choose maximize maximize obvious proof satisfy z clearly zero I set equation fact supremum countable set countable give k e random finite section law grow prove estimate define properly study error need py expectation simplify already z z note explicitly p p dt theorem find behave thus finite need get know proceed write laplace behave laplace transform transform substitute back get inside operator put everything consequently variation error proceed moment pz w e come p p
control begin control intensive use euclidean hamming define distance minus mean case exhibit positive control exhibit snp quantify pathway derive pathway conceptually clear significant drawback snp analyze test drawback distance issue control employ allele frequency major minor frequency minor allele individual first term quantify different term quantify relative minor allele frequency centroid along close standardize across obtain quantify across consideration across provide mean whether consideration inspection numerator distance centroid sense centroid denominator distance across snps consistently close snps consideration assign control quantify relative distance systematically remove individual leave manner distribution hypothesis control close manner pathway select pathway remove needed control quantify control significance systematically step pathway identify pathway homogeneity control indicate play eqs magnitude influence difference distant centroid correlation penalization denominator application pathway yield influence pathway snps yet subtle pathway influence snps gene wish snp correlation gene rather end snp gene detect snp effect practice pathway pathway database contain annotation associate gene retrieve pathway exclude minor allele necessarily pathway gene snps select gene snp association snp significant snps significantly filter snp marker interest remove pathway snps yield another control case quantify computing rank amongst thus control close snps accumulate yield group contrast nan set control yield control figure snps snp pathway snps across minor snps relative snps none snps show association snp share pattern snps pathway comprise pathway snps snp neither snp directly truly evaluate snps gene pathway pathway distinction procedure informative snp pre pathway gene likely highly significant concern snp proxy toward pathway contain abundance gene account permutation describe select compute classification normalize distinction pathway effectively preserve correlation q permutation practice permutation snp snps pathway distinction score adjust quantify close high control pathway difference simply whether pathway permit distinction significance distinction pathway assess choose number present snp random pathway snp pathway snps pathway yield particular choose snps address pathway consideration permit separation control snp pathway expect chance author concern impact select snp proxy gene snps chance pathway appear snps kolmogorov assess whether pathway contain gene chance pathway conjecture pathway genomic biological retain pathway result report kolmogorov interpretability comprise breast match control participant participant european descent array snps genome smaller comprise sample snp array snp approximately one million result genomic marker comprised breast number control participant participant american array snps across comprise control population snp one million breast breast figure expect turn snp comprise snps rs rs rs distribution frequently control peak great obtain systematically pathway represent make use build pathway snps pathway snps snp pathway gene mean pathway distinction pathway pathway great score sample relative regard membership metric odd disease status odd adjust pathway adjustment significant ratio plot increase range odd allele snps closeness pathway carry first compute snp pathway remove pathway excess pathway greedy algorithm pathway order frequent output keep share pathway un supplementary pair pathway assess statistic one pathway correlation suggest biological playing process perturb novel cell connection element significant pathway breast cancer genetic difference mechanism disease pathway biological playing role process pathway observe cell majority pathway directly cell gap pathway activation addition associate breast cancer finding interestingly link breast breast sequence study breast risk functional could influence cancer modify production contrast wide variation contribute snps pathway four cancer contain gene associate significant randomly may reasonably ask pathway snp pathway resample base significant suggest difference kolmogorov whether pathway pathway large continue pathway pathway gene genomic relevant cancer pathway trivially cover snps unique keep snp interest pathway mean differential computed pathway pathway odd hypothesis fdr adjustment pathway breast cancer cover complete overlap pathway give examine remain pathway maximum pathway breast form various type include breast play protein several neural pathway drive cancer drive pathway genetic irrespective breast cancer many pathway cancer datum well know tumor associate far support mechanism contribute pathway breast cancer appearance interaction pathway suggest exist genetic irrespective disease site along breast pathway pathway notion contribute appearance pathway contain drive increase cancer driving pathway pathway overlap complement pathway il dependent il pathway nk interesting suggest activation play study research cell response connect human finding nuclear although human connection human appearance significant pathway suggest may variation contribute cancer pathway result observe even pathway odd unit typical approximately case pathway pathway overlap pathway snps take union sequentially pathway yield considerably combine significant illustration case comprise top pathway breast give finding spread mechanism rather single pathway distinction identify pathway closeness permit infer across snps snps gene test pathway breast cancer suggest disease dna find pathway link role complement snp unlike gene approach gene find significant subtle consistent marker permit pathway joint phenotype possible many pathway breast detect snp include highly pathway relevant role dna rna show pathway along statistically effectively closeness individual control conceptual relationship neighbor centroid classifier base derivative must goal genomic heterogeneous near centroid derive marker classify case control complementary application g f pathway significant snps drive near near neighbor pathway heterogeneity significance snps snp place snps likewise fashion importantly yield regard control snps benefit phenotype minor allele interact gene slight consistent eqs pathway geometrically situation centroid sample linearly separable exclusive two I one control plain yield pairwise permit interact snps package possible pathway identify cancer suggest generally study even phenotype different snp level pathway pathway examine pathway pathway set conduct run statistic analogous pathway cancer alternatively pathway pathway rich pathway one pathway potential use pathway snps case pathway advantage rely marker genomic cause disease would snp biological effect rich system level availability request future acknowledgment research institute md cancer fellowship cancer institute national md like helpful simulate snp status minor red yield fold snp pathway comprise snps pathway htb snps snp comprise snps close four snps value htb pathway breast cancer scatter plot pathway leave panel high distribution toward odd ratio fdr adjusted pathway scatter pathway sample case control shift toward see odd ratio fdr adjust union top pathway snps breast cancer cancer control shift toward pt wide study increasingly advance nucleotide disease typical marker complex diabetes amongst unlikely snp reveal case control snp pathways analysis pathway snp identify pathway ideally distinction technique upon pathway disease appear snps systematically apply potential hypothesis pathway snps exhibit great class importantly snp snp analysis require pathway independent permit pathway detail method cancer suggest exist pathway wide genomic contribute analytical complementary system novel snp genome study motivate snps associate case exhibit within across snps pathways pathway gene gene snp identify pathway distinction systematically pathway potential pathway snps control distinguished pathway disease demonstrate system genome wide association variant disease modern yield million nucleotide human genome yield insight basis disease diabetes list publish find national national genome institute typically produce analyze
extremely resource integral distinct discuss augmentation scheme td specialize td algorithms selection da td bernoulli interest status derive show additional effort care focus modeling seem fit individual one york compare three list death medical trend year multiple material class many order describe abundance accept loss change abundance outline simple alternative nsf grant rgb circle rgb cycle rgb rgb rgb rgb rgb rgb rgb cycle rgb model thin central credible vertical span symbol thin central correspond quantile model black gray rectangle rectangle rgb rgb rgb cycle et horizontal span central credible thick horizontal median area space ess ess per min model top six ess min model parameterize standard effect thereby alternative integrate monte model complete complete proceed monte eliminate sampler explicitly account move state product avoid move td algorithm subsequently augmentation specify refer approach complete phrase da refer miss auxiliary status consequence make modeling purpose fold firstly issue underlie various augment representation capture heterogeneous integrated likelihood super population within bayesian framework abundance implement generic extension software offer comment vary likelihood obvious difference approach ii absence iii presence absence conditional unless specify capture capture capture study align compatible individual history history integrate integrate individual parametric distribution hierarchical integrate across detail supplementary material term idea replace vector label exchangeable write probability observation subsequent two distinct step involve equivalent replace usual da specifie individual population integrate sum summing permutation multiply induce conditionally bernoulli random parameter specify binomial include material instead partition augment zero show material modeling eliminate provide integrate carlo fitting implement gibbs specify detail include difficult binomial induce unlikely single across wish specify outline fit note define overcome implement reversible procedure implement necessary upper value specification de specify likelihood purpose pilot explicit pilot de k induce realization integrate obtain satisfie requirement use unable specify zero material abundance year count count across closure ill manuscript suggest treat fit extension exchangeable logit specification
subsection estimator size combine spectral norm eigenvalue norm find family satisfy regularity ny log fitting maximize mle another n particular th result rewrite normal let eq nb follow establish hellinger exponential argument hellinger density exponential define satisfie chi inequality repeatedly normal basically accuracy terminology formally paper indicate usual change paragraph next constant bound approximation need quantity kk assume regularity gaussian proof initially emphasize serve essential estimate component estimate maximize try position invoke modify allow suitably regularity difficulty asymptotic true misspecification choose maximize cutoff differ argument three set x universal ensure ready truncate conditional sequence know x nc therefore total step lie invoke notation subsection respectively give q n independent cutoff define state conditional lie high st w iw taylor eq behavior l follow imply deduce q must unit establish bind mle decompose n cauchy cauchy schwarz n moreover may f r ensure x x dx matrix taylor invoke absolute value state generality virtue e preliminary lemma preliminary section eigenvalue definition proof sum function save lot argue pa use range contribute contribute covariance inequality prove covariance lemma rewrite uniformly c use iii iii lem adjoint cf crucial perturbation theoretic result extend q j k k prove six five x complicated expect appear kb p expect x assumption hold lemma equality together nc n f c n assertion eq form assertion hold inequality know bound section theorem definition nd triangular uniformly inequality plug n dms nsf grant dms functional rate parameter change measure eliminate cause extensive exploratory decade comprehensive discussion application among many problem functional regression receive minimax truncation show depend smoothness slope theory know device problem nonlinearity function analyze estimator establish estimator exponential parametrize change inspire le eliminate nonlinearity consist pair process index parameter exponential conditional measure fr section precise specification control decay eigenvalue characterize smoothness result bind assumption bind functional generalize provide optimality establish due issue overcome measure theoretical advance regression prediction prediction establish minimax consider generalize regression economic predict curve minimax result minimax proof family size aid reader stage kernel key idea invoke theoretic family pt otherwise yy quasi provide difficulty cause nonlinearity quasi specify achieving cover broadly regression unknown assume due model model become difficult interesting generalize eq exponential regularity set c observe pair like I satisfy decrease offer estimation procedure provide theory assumption justify regression widely application component zero curve functional activity analysis basis wavelet fourier basis regularity component regression applicable wavelet basis basis basis slope case theorem degree ill setting need low explain page eigenfunction lie denote depend regard strong assumption remove help regularity case exponentially decay logarithmic case eigenvalue slope decay exponential essentially much methodology rate state estimation trade mle misspecification
optimize optimize handling enable subsection ready optimize weight formulation regularization objective monotone function speaking correspond fit example severe h regularizer norm mkl consider consider sample task task dependent common propose optimization hyperparameter learn simultaneously norm regularizer generalize presentation early paper regularizer let variable kernel weight x h employ regularizer special mkl elastic mkl let x discuss early elastic net mkl block mkl combination connect base block norm base regularizer analytically eliminate weight regularizer origin satisfy monotonicity mh conjugate divide concavity straightforward composition convex line use monotonicity infimum exist formulation correspond original note convexity strong concavity regularizer separable regularizer suppose kernel regularizer separable function slight abuse follow convex regularizer separable straightforward omit regularizer summarize regularizer example mkl regularizer separable therefore accordingly formulation regularization norm regularizer example minimum minimization lagrangian taking set mx expression obtain structure task case mkl q jensen jensen note discuss joint different problem may task regularizer example convenience lagrangian back converse denote conjugate converse statement conjugate base elastic kernel elastic net regularizer function weight since concave square regularizer component take q substitute back equation therefore example norm mkl tb concave derivative q follow follow rapidly numerically compare mkl discuss take dataset sift pyramid kernel precisely combination four type sift sift sift sift px image choose local visual point center local histogram obtain partition partitioning whole level cell measure similarity histogram pyramid decay four pyramid see linearly space computed sift result block mkl elastic mkl mkl mkl loss mkl mkl logit include make comparison mkl possible difference square elastic net fix model logit implement toolbox update toolbox cross except norm tb tb mkl elastic mkl uniformly weight favorable mkl block mkl tend perform especially elastic net mkl mkl perform almost mkl component specifically vs dataset elastic mkl rbf agree vision literature elastic mkl band kernel never choose structure mkl elastic task see kernel regularization two formulation systematically map transformation likelihood employ furthermore derive block regularizer bayesian mkl test classification mkl categorization accuracy uniform roughly kernel usefulness mkl sparsity contrast mkl tend probably obtain extremely one avoid prior relax fine characterization regularizer bayesian theoretical concern sparse mkl mkl mkl non mkl thank helpful discussion partially point completely specify function mx mx n mx easy mix side form mkl receive attention paper understand weight systematically map separable naturally generative probabilistic numerical mkl comparable kernel might preferable mkl elastic net usefulness kernel elastic weight representation kernel represent web page bag might classify discuss also page useful classify whether page support b categorization descriptor classify car representation manner source framework kernel combine nonnegative weight base approach mkl recently mkl understand constraint regularization kernel weight meanwhile structured assumption employ regularization ask well base map manner conjugate far contribution
phase integral go configuration weight rewrite bayes language field theory partition actually negative logarithm joint permit transfer interest norm minimize often field pixel discretize signal fs sx integral multimodal density determine many functional integral path integral functional expand around integral reader theory treat analytically start depth wiener parameter use get interesting corresponding want reconstruct distribution covariance subscript value indicate read xy sx sx ps far signal process linear device response general depend theory case focus discussion concrete covariance eq function give source response fashion finally constant ignore depend quantity signal partition permit calculate call map apply filter signal work p identity connect correlation function obtain classical treatment hamiltonian classical field map signal characterize theoretical classical acceptable much easy approximation follow specify thereby abstract either determined determine joint give expression contain parameter fix permit construct partition parameter information estimator simply multiply additionally average compatible get recognize hamiltonian hamiltonian energy obey enter calculation analytical calculation hamiltonian theoretical representation hamiltonian around reference hamiltonian dp provide signal repeat thought interaction permutation permutation leave implicit tensor notation ax jx rank tensor uncertainty variance introduce signal x long definite relevant invariant fully characterize mean hilbert identity transform spectrum deal simply scalar product fourier adopt since applicable space like abstract basis also unknown spectrum combination support band positive band cover domain therefore variances verify distribution law amplitude cutoff jeffreys limit provide e reconstruct full field turn assume purely improper jeffreys interesting hamiltonian five cast determine equation discuss filter wiener spectrum assume iterate specific freedom follow assume filter decide principle wiener spectrum ask hyperplane pure approach first traditional map principle joint jeffreys filter jeffreys improper prior class would appropriate informative decide concrete inference individually subsection summarize fig display section joint filter pure filter sec critical filter line estimator spectra wiener realization gaussian verify j p p p rhs account lose wiener rich individual drop last jeffreys obviously information logarithmic critical scheme sec filter cast critical mark phase critical apply successfully sparse noisy transformation latter identical without normalization jeffreys hamiltonian symbol mark respectively stationary hamiltonian thin dotted hamiltonian hamiltonian eq spectrum hamiltonian sorting provide respect filter critical logarithmic map express hamiltonian calculate hamiltonian regard poor critical lose filter single signal hamiltonian dimensional show map provide acceptable skewed reconstruction goal calculate average effective optimize construct eqs expect case theory expansion truncate low order practical uncertainty localize hamiltonian repeatedly add uncertainty source accumulate thereby entropy parameter follow flow parameter decompose narrow mutually prior auxiliary onto mutual auxiliary mostly strict sum convenient value full effective increase accumulate uncertainty analytically hamiltonian assign measure accumulate far suitable accumulate dispersion auxiliary find time hamiltonian cast back spectrum hamiltonian construct posterior sense maximize let individual infinitely hamiltonian hamiltonian hamiltonian especially effective localization typically factor various coefficient effective interaction perform step uncertainty let specific hamiltonian virtue might correction hamiltonian eq vanishing therefore ignore introduce four vertex correction theory dominant correction account drop subscript rule inverse hamiltonian belong high uncertainty due contrast free uncertainty correction field lead free hamiltonian subscript uncertainty arbitrarily derive flow pseudo accumulate dispersion prior transform compact operator partial spectral sort closure hamiltonian original structure affect uncertainty ideally change map onto effective amount uncertainty distribution limit q apply problem sect need additive decompose moment concentrate parameter auxiliary convenience properly dispersion result normal take jeffreys logarithmic permit pure receive uncertainty rate center appendix expand free shifted flow dynamic flow latter differential equation pure quite expensive simplify filter equation original evolution equation need dt split evolution rhs signal bands evolution orthogonal contribute evolution obtain part evolution yield fast evolution strength whereas inverse evolve long evolve direction band get project reverse value accurate determine evolution evolution equation original pure uncertain project parametrization onto infinite result jeffreys prior trivial limit jeffreys arbitrary probable basically infinite however likelihood datum must finite amount stay stay either unlikely amplitude thus probable pure averaged imbalance dominate remove jeffreys imposing look alone take would remain small jeffreys spectrum formal future filter jeffrey project plane representation display jeffreys spectral coefficient filter band variance band low band map fluctuation appear calculate filter exhibit power thus investigate extreme algebra since mark vanishing filter jeffreys scheme filter exhibit vanishing lie threshold datum criterion position datum noise live spatial fully characterize spectral band band spectra instrumental kernel spectrum fully power spectrum band generic eqs separate independent individual fidelity far significantly narrow permit determine band coefficient drop normalize fidelity case jeffreys simplify formula solution three simultaneous solution take decision ignore trivial amplitude increase decrease branch root asymptotically datum work exhibit follow approach spectrum filter however filter bin filter threshold exhibit negligible spectral amplitude since might imply assumption significant filter spectrum measurement band filter regard response separate pass band critical power spectrum wiener map iteratively filter spectra filter power procedure name kl spectra filter exhibit mode expect exhibit mode application literature measurement usually kl iterate system initial correctly estimator reflect roughly grey guide double logarithmic spectra dataset top panel spectra display bottom many frequency trend spectra reflect spectral filter want filter performance test wiener basically small difference clearly adopt reason decade frequency mode convolution fourier coupling unknown spectral homogeneous observational signal response pixel display spectral band low since mode take negative one band identical band reconstruction five filter show roughly way threshold analysis filter band noise generality filter prior signal spectrum adopt member spectra exhibit incorporate want specify specific length double logarithmic spectrum limit additional smoothness logarithm spectrum derivative collect quadratic line actually combine log repeat use physical evolution force jeffreys specify reach neighboring band thereby produce smooth spectra without gap see fig regularize filter smoothness think fidelity pure derivation uncertainty lead trivial amount uncertainty spectral influence result desirable since guess might available abstract want include fair comparison filter inclusion non wiener correction filter seem well determined unchanged understand roughly poorly map one evolve similar speed evolution map large region uncertainty large thus observational remove fast pure filter aware assessment filter principle critical obviously fidelity smoothness regularize filter one probably lack gap comparable wiener power also wiener finally full fidelity comparable without spectral show deal uncertainty theory effective hamiltonian joint signal parameter beyond treatment uncertainty successively feed parameter uncertainty pure many calibration uncertainty pure provide full maximal consideration concrete noisy model uncertainty various four classical filter regard wiener operation spectra single filter prefer significantly informative jeffreys filter suffer threshold band power threshold filter four investigate threshold variance fourth filter try logarithmic mode noise soon successfully depth frequently estimate spectra galaxy threshold truncation full iterative scheme hide band signal adopt typically large exhibit large slightly asymmetric spectrum much bad know margin degree band datum determine result parameter pure filter principle tree poorly perform correction include pure essential improve threshold smoothness fourier may show pure smoothness approach wiener since identical critical already filter spectra keep spectrum critical logarithm spectral wiener correction evaluation flow equation filter conclude general uncertainty calibration implication assumption commonly improvement pseudo time flow amount dispersion feed knowledge circumstance real measurement device calibration noise evolution offer dispersion per equation permit control way pure calibration acknowledgment acknowledge comment manuscript signal covariance expect fluctuation free special invertible expansion logarithm general second normalization linear shift gaussian unity different combine
eigenvalue compatibility universal distortion explicit get prediction code deterministic construction construction sense knowledge use construction selector construction code code time unbalanced graph degree side chernoff bound inequality proposition unbalanced vertex signal selector polynomial red indicate entry target panel code draw bi design gaussian moreover gaussian performance latter establish dimensional design compare performance design figure simulation agree theoretical cm author would like anonymous adjacency unbalanced prove selector multiplicative estimate dimensional situation know explicit eigenvalue satisfy compatibility unbalanced deterministic time design processing soon choose find rely target recently emphasis put selector identically center random variable variance first impossible recover noisy ask tuning estimator suit sparse soon satisfy coherence property prove design challenge adjacency unbalanced great refer property selector design inspire frame article matrix I ball distortion design interestingly show design matrix compatibility universal property constant moreover four matrix unbalanced constant remove coordinate great optimal multiplicative less risk side bound show coordinate root code soft phenomenon coordinate error moreover ordinary support eventually sense value selector unbalanced constant left design construct follow positive deterministic euler number selector exist universal time sensible construct selector recover article fold part unbalanced third satisfy satisfy compatibility distortion explicit involve design unbalanced vertex denote regular vertex neighbor adjacency unbalanced unbalanced bipartite graph eq subsequently property small property art unbalanced theory let adjacency unbalanced degree adjacency matrix begin loss assume consist large coordinate neighbor finish ns ss ps give event choose finish unbalanced great lemma finish instance form begin restrict isometry property unbalanced graph short conversely binary satisfie proper one unbalanced satisfy property restrict isometry high vector property property deal context adjacency check unbalanced expansion q frame code eigenvalue cone compatibility analysis carry assumption
ks e notice continuous uniformly point weak continuous dominate case general decomposition j k large numerator equal converge theorem simplify eq e j virtue eq us equal vanish sum term vanish q term yield complete modification weight treat tu ks tr replace notice display virtue establish result suffice substitution detail treat brevity fx ts proof make lipschitz small arbitrary boundedness dominate ts open ball arrive finite coupling allow one paper value real coefficient notice otherwise assertion trivially cf consistency hand large finite sum fourth consequence dependent check moment recall bound estimating result consider moment ts us block schwarz coupling follow derivation consecutive partial sum sum partial sum odd independent putting conclude expression argument replace denote field put replace allow mix variable complete definition height cross sequential mind application aim prediction detection task law large consistency cross validate extension discuss error mix cross validate series illustrate class nonparametric regression attractive area science devote study selector subtle treatment prediction detect address setup classic especially choice thus classic next sequentially maximum design form result namely epoch issue order external time internet traffic sample application preferable issue matter nonlinear medical response explanatory pressure science study influence political opinion regressor practice ii process evidence mean differ usually nonparametric kernel smoothing spline wavelet amongst concern detect kind proposal chart type latter amongst frequently asymptotic classic invariance dependent asymptotic play well literature see separate change drawback course detector certain optimality restrictive sequentially prohibitive ease interpretation applicability base reasoning fit possess class smooth select smoothing extensively classic regression practitioner sequential validation treat yet bandwidth version arrive mind really quite time engineering consistency observe converge available aim present cross focus time point time grow also bandwidth consistency result epoch plan discuss assumption detail interest sequential error section extension data extension general ensure result appendix discuss study power module cell mathematical process distribution error would allow clearly validation mind possible aim detect large assume recall weakly dependent let semi specification sensitive mean function bandwidth sequential classic note classic control chart I kernel cover canonical sided detector monitor indeed stop use illustration assumption smooth kernel satisfy kernel number even weak discuss bandwidth concern choice kernel detector average result extent change cross criterion validate estimate bandwidth shall provide certain remark sequential classic estimation cv since mind past current present ts similar sided cross propose classic interested computational feasible exposition fix finite small appropriate would prefer large value nice relax allow increase devoted careful cross validate bandwidth consistency identify asymptotic q equivalent minimizing identify weak bound uniform usual integer number conduct uniform law put eq worth remain condition combine additionally q extend weak bandwidth weak mind parameter follow write abuse eq cross validate bandwidth minimizer sequential objective uniformly uniformly validation provide many functional functional continuous measurable formulate behavior validate sequential selector apply technique minimizer separate exactly validate large formula sufficient estimator cf perhaps suited criterion requirement separate check analytically numerically ensure sufficient criterion easy require differentiable derivative series communication collect e g longitudinal clinical trial social survey assume ask carry since subject scientific qualitative assumption let weakly discrete mix mix instance mixing provide cover exist eq dependence bridge generally definition approximate l coefficient satisfy
consider power subset modular norm cardinality indicator define resp give submodular section theory preserve separable optimization associate section submodular minimization detail result present subset cardinality value infinite submodular submodular subset simple I e multiplication scalar submodular discrete analog behave duality theory link extension equivalent difference submodular fa fa fc fc fc condition necessary opposite satisfied sum fa k difference submodular fa k condition weak simply inequality fa prop lead modular indicator submodular prove submodular pf sa fa bf sa pf empty unbounded empty interior let submodular interior show empty true constant pf cm function extension link submodular convex transfer extension prove actually definition unique ordering integration summation equivalence equal irrelevant integration fw w w w letting tend modular classical extension indeed set extension fw fw fw v v fa symmetric w f w fw regular cut co reference submodular def basis many result submodular show maximize extension otherwise programming require greedy submodular maximizer fw apply prop introduce multiplier constraint sa fa w fa w feasible I loss generality assume decompose interval sa fu fu f v sa non notably finally value draw link extension prop equivalent submodular convex equal fa fa fa fa proposition complete prop showing equivalent minimize minimize submodular submodular function submodular fa property fa p ia one consider large fw fa fa fa fa next proposition complete prop full support function behavior submodular let submodular exist proof greedy thus fw fw optimality prop algorithm characterize maximizer otherwise order unique convex combination solution exactly submodular w si fa ib optimization v I cm v objective equal equality result optimality condition fa solution let submodular take set sa last sufficient characterize face characterize interior submodular separable say let submodular base must intersection hyperplane interior factorize contraction sa bf bf ab fa fa fa separable detail base e contain hyperplane face intersection support hyperplane support hyperplane hyperplane face potentially empty interior sa together prop face interior face prop non empty interior prop submodular proposition conjugate note prop regular indicator conjugacy function conjugate submodular function extension fa sa submodular proposition desire assertion fact relevant submodular present minimizer submodular sufficient fa fa desire moreover strong equality one moreover w present several submodular exist one submodular affected operation simple restriction moreover extension moreover submodular projection component obtain notice conjugate contraction submodular function submodular contraction function extension pf pf sa tb pf pf bf sc fc fa fa pf build similarity submodular prop prop minimum submodular submodular extension fa ga b ff pf aa minimize get conjugate one sa ga fw q fw fw let fw v proposition give submodular set modular ga vb ga ga b ga ga ga bb pf pf sa let submodular ga ga ga ga ga ga obviously get take extreme definition inclusion consider separable penalize extension separable function problem reference therein simplify statement make equivalent dual continuously problem eq maximization prop w conjugate strictly separable optimal function also minimizer minimizer sufficiently contradiction optimality separable exchangeable q immediate prop relate tangent differentiable separable optimization define set check belong unique become prop another interpretation prop ps great maximize order derivative cast order base minimizer maximize vector pair prop moreover unchanged contradiction imply exchangeable imply prop base greater equal prop show one strict tb j base previous prop prop k ks prop leave zero exchangeable maximization prop equivalent submodular prop satisfie specific cut e maximize submodular general maximize cardinality review submodular first briefly consider function call present submodular know equivalently minimizer negative old norm base possible maximize function solution prop complexity essentially order know form value know prop identity combinatorial usually subset base tight output high complexity submodular f attain function minimize e subset regular include well sum particular cut general submodular direction submodular thus value soon assume fa affine obtain imply find minimum point minimization fw w fw base strategy prop large enough minimum recursively f procedure quadratic fa piecewise otherwise find f k adapt algorithm slightly decrease submodular minimizer minimize base associate associated contraction stop part let submodular aa aa tc th prop thus tight tight minimizer submodular ab fc fa tc decrease bf optimal exchangeable imply tight stem k kt come tight prop optimal prop show finally apply restrict g submodular decrease refer decrease truncate submodular decrease maximizer may fw prop constraint simply replace monotonicity also base include positive consequence prop include compact unbounded prop prop support prop face support take sa fa special non decrease submodular strict partition sa f f relative interior prop submodular positive base first non non decrease separable let primal pair maximizer statement consequence optimality correspond strictly denote prop hence set prop prop involve pair define view view apply prop positive submodular convex pair maximizer prop ks ks kf non separately non decrease fw w submodular associate submodular depend cardinality proposition concave submodular prop cardinality concave submodular result concave base extension extension function disjoint subset fa fa fa f fa da db c b da c fa denote fa b b fa fa fa f f b fa fa b weight submodular also symmetric prop cut extension play role image process compose total therein partial g cut weight fw fa submodular dedicated plus modular flow add node graph non weight cut lead minimizer proximal section define fw obtain use linearity subset always decrease induce norm fa k jj w replace cardinality prove type weight cover q show I
program dl boolean resp boolean correspond conjunction nm simple partition answering depend boolean semantic answering follow case query apply case database schema partly tuple view cast building stand name tuple fact belong integrated false induce hypothesis concept occur form mm mm contain database kb adapt concern build p predicate form usually assume guarantee target belong target kind tuple individual description theory incomplete inherent set appropriate answering reformulate kb rule satisfy axiom model consequently note precisely every thus force every force cover three support induction need equip subsection devoted hypothesis employ original problem induce hypothesis must issue language regarded clause logic program directly carry normal logic program treatment adapt result relie reasoning r boolean boolean difference nonetheless boolean answer aforementioned link answer check reformulate kb reformulate way solve apply consider mm example want immediately hold say r r r nothing else ground answer refinement operator refinement language hypothesis build rr refinement l since nm semantic add apply rule intuitively reduce rule mm rule mm mm mm discover induce nm kb main start contain element queue include deal induction database consider terminate specify redundant pruning post processing pass nm theory process queue become report satisfied framework far hybrid dl cl less expressive propose focus discriminant interpretation hypothesis recursive play coverage usual learning interpretation relation extension procedure provide pre processing enable represent clause framework mean induction generalize quasi check constrain setting interpretation query answer opposite partially implement support variant frequent pattern conceptual close problem nm illustrate case learn aim rule head head kind rule kb whereas extend principle powerful coverage generalize instead scope point notably relative interpretation deal formalism language hypothesis apply engineering task interactive rise call inductive functional dependency valid almost constraint decompose restrict framework reformulate nm notably evidence go beyond illustrate traditional nm dl hypothesis far accommodate viewpoint expressive treatment define generality refinement turn induction framework towards plan analyze produce adopt expressive may turn dl definition refinement operator ideal refinement operator theoretical inefficient constructive improper preferred refinement promise reason deal expressive mention rise currently expressive incomplete knowledge feature third deal rule rule group concern core likely inspire dl language use case final shift extension mining name onto relational systematic report relational integrate onto grateful valid universit di mail di community semantic web solve database well issue database ii schema reformulate expressive framework inductive database knowledge reasoning concern example classification purpose relation logic relational interesting extension attract little effort make able pose relational scalability respect far still remain prior conceptually rule organize conceptual seem ignore conceptual artificial intelligence refer precisely principle certain plus explicit regard intend word form logical vocabulary appear unary respectively concept formal explicit specification among thing semantic mean propose semantic dl language dl expressive semantic combine limited form framework argue difficulty address web help database understanding perspective consider view database schema reformulate problem benefit expressive mainly nm integration brief introduce reasoning induce view within survey conclude remark resp resp role n r relations atomic concept dl construct dl form atomic restriction limited restriction call concept restriction whereas dl add dl role restriction role expression contain inversion role r equivalence concept axiom role equivalence axiom role axiom assertion assertion assertion resp concept resp axiom also inclusion axiom role role axiom dl adopt nothing else possibly together dl kb directly theoretic show map default individual equality dl kb interpretation kb satisfy axiom kb coherent semantic dl kb least kb write reasoning dl kb try check decision mostly calculus another reasoning service check assertion implication dl sophisticated instance dl individual possibly expression individual entail intensive application retrieval aspect complex expression query relational express arbitrary cyclic possibility name name alphabet conjunction atom predicate whose tuple kb entail write check boolean follow boolean alphabet expressive finally ground clause aim induce require thorough discussion induction affect explain confirm explanatory setting observation clause hypothesis observation observation interpretation clause cover observation learn aim none example validity call drop due absence case interpretation traditionally partially order inductive theory single ii clause notion clause clause general define quasi clause clause substitution order clause substitution definite clause apart definite program say iff substitution appear substitution lattice partially order least infimum great clause yet generalize structure accord generality ii refinement operator generalization accord search kind refinement operator refinement desirable illustrate hold practical use condition application operator clause ever satisfy simultaneously refinement language implication language usually locally improper full retain possibility refinement operator couple bounding clause concern specify syntactic clause space connect definite link link term link link chain link length integration kb accord formula upon mutually name alphabet predicate call either alphabet expression tuple atom atom logic rp ir
course optimize simplicity type close likely mostly alternate likelihood hierarchical integral graph make bayesian integrate since posterior fitting goes determine although set tie type topology hide way active learn study type laboratory resource vertex vertice type vertice query mi type entropy average expected decrease result mutual good vertex entropy assignment accord gibbs heat vertex proportional assume type fix explore collect distribution offer theoretical time family exponentially r enyi vertex connect switch state type bottleneck state real world try far chain improve average mi algorithm say already vertex gibbs query large move next query another call give type agree correctly really maximize assignment draw gibbs project expect assignment draw condition imagine vertex independently give assignment agree agree thus three vertex vertex well sampler except assignment chain two independently initial denominator give keep track average draw increment numerator numerator denominator unchanged alternate estimate type vertex correct query large mutual mi vertex large agreement aa text label algorithm accuracy converge vertex test mi aa consist member split around around vertex form highly correctly identify vertex belong aa course community identify perfect review choose sort central lie boundary understand lie aa well experiment seem place examine take attribute attribute half vertex correctly attribute less although aa perform measure decrease easy vertex hard form fraction arrive time remain type attribute aa specie label correctly label word accuracy continue get end learning find extent attribute attribute species large include type attribute block half misclassifie block likely specie confidence know member friend close expect regard mistake cut aware belong confirm misclassifie modify artificial make block iterate follow species value equal accord block iteration change specie reach fig perfectly predict specie web difficult order suggest agree extent relative positively pearson begin type specie fill available mutual varie condition specie largely due topology adjust mutual perspective heuristic vertex high centrality heuristic popular centrality reflect heuristic mi aa although mi however situation resemble surprisingly mi early stage achieve poorly actually predict rest leave
weight random consistency open analyze base gradient optimization conjunction hour ahead wind problem classic management demand lose constraint constraint observable generate demand mixture variable include mixture generate bi width pdf width new pdf bandwidth accord package range kernel online period decision posterior gibbs collapse sampler run take iteration base dirichlet weight base sampling decision optimal dispersion measure sample use collapse method location wind ignore wind dp ignore display algorithm wind function type outperform ignore possible dirichlet still substantial wind simply free optimization optimization rely weight give kernel complex weight dirichlet show dirichlet super work statistic solution problem search space element aside decision observable machine tool avoid proposition conjecture david problem noisy search behavior variable shape currently density state outcome distribution query problem function examine two scheme weight method hour ahead wind result dirichlet substantial solution class uncertain hard compute instead problem decision classic stock uncertain demand distribution rich theory important variation search allow belief state belief decision challenge outcome something use current decision say conditioning often frequently hour ahead wind wind company hour future little difference lose depend market contain solve stochastic sharing use outcome observation decision weighted ensure summary mind formal search problem become q technique distribution treat state independently nonparametric affect similar new observable motivated realization depend reason subscript nonparametric approximate weight response state outcome gradient z solve wind entire resource allocation involve problem base restrict weight observation currently exist almost sure convergence global optimization popular variable explain extended try behavior original optimum previous weighted accord linear approximation around optimum separable heavily weight give easy implement good develop give unstable situation propose mixture dirichlet nonparametric distribution derive place current monte carlo clustering method kernel problem hour ahead wind synthetic weighting hour ahead synthetic wind north american assimilation computational solution scheme dirichlet produce contribute base dirichlet process dirichlet empirical promise paper review treatment variable propose incorporate state prove base state section weight scheme analysis synthetic hour ahead wind operation optimization study form search problem diverse within entire outline problem consider individually case construct wind select variable wind create require problem portfolio bandit literature theory community study portfolio handle treat separate bandit establish area sample study parametric study however mechanism parametric view problem sensitivity control community deterministic machine problem theory reinforcement state state arise way dynamic might state determine influence next visit choice propose problem example major close value learning deal challenge decision variable information portfolio reference reference close online player select convex propose decision value update give turn method use decision hour wind wind value sampling deterministic complex hold almost surely ratio problem study program average event gradient variable fairly straightforward create joint convex like respect observe change produce average close particularly solve deterministic solver decision would surely pointwise subset set compact subset surely convex every fix minimizer decision pointwise section discussion assumption decision per nx ns almost surely pointwise heavily every compact minimizer proof uniformly compact converge pointwise uniform satisfy turn corollary function however present optimization take stochastic use gradient general iterate back another construction piecewise function variable straightforward difficulty state base iterative state construct approximate give piecewise separable aim approximate well easy convex interpolation dimension component univariate piecewise convexity restriction ks weight outline place decision eq sufficient k minimization nx ns nx nx affect nx nx piecewise reconstruct fx v df kx ks step optimization perform set discuss extension subsection heavy part order number observation find easy format fine evenly parameter equation never grid second break converge optimum query care say sufficiently neighborhood accumulation however define assumption finite write separability convexity gradient parameter continuously observation unbiased equation cone every decision place net center partition convexity variable optimum eq often non phase infinitely subsequence every infinitely combine fact sufficiently therefore hold choice determine importantly well approximate discuss weighting procedure response calculate object density approximate either compose observational observational weighting kernel implement however dirichlet rich weighting material rely conditional common bandwidth weight bandwidth observation estimate q continuous every ease guarantee highly sensitive validation overcome propose dirichlet alternative weighting kernel non weight observation similarity dirichlet partition introduce gain popularity powerful computer computation partition model countable simple number dirichlet place lead trivial proportion dirichlet issue determine like derive cluster mixture query state average response state proportion produce method clustering discuss mixture satisfy mixture parameterize eq wish determine find review find good number dirichlet dp parameter proportion location dirichlet effectively place give probability call space prior place reason actually measure construct draw model condition mix give dirichlet base concentration produce dirichlet explain later dirichlet produces surely demonstrate dirac know also take control example approximation model conjugate normal dirichlet equation associate place place partition induce give partition wish place partition
size throughout definition transpose matrix way mode analogue fixing tensor rest robust gaussian perturbation conclude problem non convex minimize iterate always take respect converge stationary mm stationary jointly continuously full meet guarantee stationary point global simplification iterate ii unique minimize r fashion hold mode subproblem separable problem toolbox tensor draw dense variable generate tensor randomly draw shape parameter dense entry I add value initial mode toolbox goodness match factor perfect provide two recover score decrease increase exception find minima compare true median error factor capture solution base comparison lp found perform slightly fast similar factor show size tensor indeed consist independence parallelization simulation demonstrate suffer insensitive presence noise perturbation cause degradation conversely suit tensor finding tensor array cp typically propose perturbation alternate fitting cp cp factorization generalization decomposition fit cp tensor
employ leave analytic give become core suboptimal start theorem give partial yield translate argument bind yield claim formula optimize space fix build primal derivation kkt formulate macro mkl optimality compute accord solve subsequently compute analytical stop gap iteration disadvantage full propose thereby solution solely operate standard terminate fall precision normalize constraint mkl variable accord mi old old optimize accuracy subproblem exploit method useful gauss prop continuously define gauss recursively let proposition establish mkl hinge let positive globally p svm ignore speed hinge suffice show transform requirement prop fulfil expand thereby extend variable may loss prop fortunately prop substitute constraint maintain closeness gauss method solution block objective continuously differentiable latter minima uniquely attain solve svm definite analytical solution limit globally hilbert imply violate either due long strictly computed equation achieve one exist want improve little bit indicate cf sect lot optimization suboptimal overcome ensure svm try cf like primal svm section cut plane newton toolbox two class task decide svms algorithms optimization scheme outer offer list semi use violate thus single sequence constrain program alternatively analytic update perform modification core currently implement objective kernel step couple trivially typically perform svm positively access solver implement cache become mkl scale sense reach cache track partial objective individual cache size regularization optimal imply magnitude modify regularizer feature regularizers imply kernel sensible uninformative experiment fundamentally generalize spirit contrast nevertheless beneficial normalize formally positive rescaling rescale corresponding feature satisfy equation equivalently express function eq kernel empirical frequently kernel rescale unit also multiplicative spherical substitute point length whether far section show norm extension aim mkl mkl translate mixed formulation penalization equivalent use op prop yield close look parameter real range perspective extend sophisticated norm mixing explain straight q prop analytical update derive block mkl mkl unweighted study toy light sparse mkl localization find rna bind gene genomic study intuitively expect directly sparsity apart suboptimal set true go level sparsity scenario step study mkl illustration toy balanced covariance opposite mean encode true sparsity imply latter carry fraction set fix hence noise work classification norm mkl grid optimal attain interior grid relative gap optimize test pure I latter error reflect corresponding explain theory unstable result observe agreement test scenario kernel carry scenario perform mkl variant truth I select datum imply towards contrast unweighted kernel informative bayes block sect truly limitation shall keep artificial balanced scenario level mkl achieve less tuning sparsity mkl low increase nevertheless observe mkl prediction scenario variant scenario contrast reasoning discrepancy explain minimizer stable become test appropriately fix mkl summary size increase expect scenario mkl uniform unweighted svm well appropriately mkl rare norm mkl define capture protein mkl unweighted improve establish problem gram section investigate matrix set predefine split norm rest prediction r coefficient proper affect norm equally split yield optimize averaging split data split table non sparse regularizer hand mkl arbitrarily norm intermediate norm r std std std briefly mention mkl four kernel pick differ place gap part protein sequence consider vector carry overlap information gap principle high degree polynomial svms individually case similarly highly redundant exclude non norm model detect rna bind genomic site regard key determine start site start genomic detector appeal mkl contain gene utilize translate positive instance extract window around negative size represent angle energy describe instance use pool soft grid attain test size pool training bar indicate size contrary mkl high auc mkl suit norm mkl yield superior remarkable application domain unweighte recently confirm comparison program area roc test indicate error repetition mixture weight spectrum good explanation optimality sparse show moderately informative recall report auc individually energie drop remain discriminative investigate compute ai frobenius dot observe kernel show slight little auc carry complementary include mixture precisely mkl fig reason right energy expect auc performance auc auc surprisingly kernel auc originally database functional unknown employ experimental setup phrase edge capture localization profile additionally use integration kernel match unweighted sum validation fold however omit guarantee problem setting table slightly svms result observe unweighted svm perform approximated mkl improve result include maximal mkl unweighted kernel moderate show suggest hence orthogonal kernel experiment explain mkl experiment mkl mkl mkl mkl mkl mkl product mkl mkl exp one suggest beneficial non mkl task odd digit lead example analytical mkl experiment analytical convergence compare infinite program mkl svms unweighted mkl kernel train computation expect one counterpart optimize outer gap stop criterion duality gap norm counterpart trade method execution svm analytical training number kernel figure display result gaussian bar standard unweighted fast mkl slow compute matrix notably requirement gb double number time slow evaluation kernel optimization allow kernel gb mkl optimization naturally slightly slow taylor rank non figure vary sample unweighted taking time mkl increase fast mkl thereby somewhat fast sparse analytical might bad contrast conjecture part solution hardware storing gb allow effective cope gb considerably fast slow method increase memory propose analytic cut plane order limit impose store future improve mkl pay translate multiple minimization problem regularizer arbitrary penalty mix regularization dual kernel mkl analytic base allow application algorithmic approach norm implementation toolbox execution reveal commonly approach scalability world learn empirically validate mkl data computational control various mkl achieve scenario wise real bar either unweighted closely approximated mkl hence seem natural worse unweighted along suggest mkl bad despite appeal first understand appendix scenario mkl fast sparse well ground suggest mkl mkl inconsistent exist advantageous conjecture bound improvement serve conjecture bound slight advance cross case call non analysis world discriminative carry play distinction exploration beyond scope remark redundant uncorrelated variance sect speak observe redundant keep put zero strong preference scientific already connectivity causal flow social true zero interpretability correlate arbitrary predictive demonstrate wish discussion manuscript acknowledge contribution suggestion project fp european exchange service theoretical norm mkl rademacher rademacher generalization mkl mkl remarkably effort line mkl valid match want hypothesis risk class bound w rademacher eq rademacher value remove dependency random rademacher show technique rademacher main rademacher exponent old norm technique extend norm convert rademacher thing rademacher illustrate bounding let rademacher substantially improve loose favorable whenever plug obtain new rademacher exponent compare one p coincide differ polynomial asymptotic considerably infinity contrast approach precisely p om bound vc rademacher literature employ aim assume large w loss v hx iv functions loss lipschitz bound immediately bind mkl fix margin assume n w reason remark extend norm different example block sum norm norm seem beneficial somewhat contradict norm refined theoretical support intuitive claim uniformly priori promise beneficial block norm prop norm generalize introduce parametrization straight pc q exploit h illustration leave word bayes mean advantageous attain latter bind yield interestingly obtain norm mkl variant generalization theoretical priori optimal see fig simply precisely sparse mkl truth expect ground truth vice tight ingredient paper let convex f exist rise duality satisfied constraint vice versa op equivalent op vice suppose exist translate vice versa duality optimal point max removing show corollary multiple berkeley university california computer division berkeley usa yahoo com yahoo research laboratory com life learn combination appeal choice support scalability rarely outperform trivial practical mixture generalize mkl norm devise new mkl like demonstrate strategy compare control artificial mkl empirical biology sparse achieve accuracy kernel convex conjugate coordinate bioinformatics generalization allow find appropriate representation kernel immediately machine wide domain appropriate even good hand validation probably prominent right alignment aim vector support machine svms alternative select dc mixture problem relaxation obtain focus scenario guarantee seminal combination semi subject require combine learning mkl automatic version mkl namely sdp however sdp expensive application conceptual develop mkl utility simply constrain negative constraint restriction transform problem constrain drastically burden optimize dual formulation decompose problem mirror prox min independently mkl training generation amount repeatedly mixture coefficient immediately convenient algorithm benefit svm allow large scale training svms still prohibitive reason svm drastically scheme extension exist major coefficient regularize optimization control mention learn solution mix mixture first sparse combination irrelevant evaluate appear additional simplex simplify constrain beneficial mkl outperform svm unweighted consequently progress field mkl need useful model improve plain since compete formulation set achievable outline side cast loss function regularizer norm penalty mkl variant dual objective optimization state mkl well incorporation knowledge isotropic include sparse plain regard introduce form step infinite unnecessary utilize work svms mkl current mkl implementation employ enable ten thousand thousand completely small set machine toolbox claim experiment artificial world represent diverse application bioinformatics negativity unbounde partial allow optimality discard problem hinge supremum threshold recover solution apply kkt start composition side conjugate dual term dual recover regularizers point highlight arise theorem optimize employ depend dual term depend dual present duality limit strictly positive duality learner optimization unweighte recover special regularize risk minimization regularizer
tail attribute level transfer bank take equal bank word increase coverage comparative risk bank conversely figure bank attribute completely view bank similarly transfer condition bank risk receive bank understand must motivation behind price bank bank cross var mit bank consequently region region bank price risk deviation bank hence retain bank bank portion transaction region solid line exceed bank risk dot bank mit bank conversely var mit solid bank less risk measure optimum bank advantageous bank relationship distribution optimum point measure decrease portion entire exposure bank prefer criterion us bank give bank policy advanced tail index frequency set ht bank risk dot exposure receive bank dash dot unit unlike previous low frequency unit attribute infer coincide observe attribute coincide discount grow optimum infer discount apply bank light increase tailed linear cover limit linear bank bank exposure dash bank bank appear dot line discount event exposure demonstrate payment transfer perspective create line peak illustrate risk band moment less tailed shift bank appear risk band policy unlike dot consequently yield region bank fair range maximum uncertainty bank expect risk exposure light tail study context capital reduction multi period risk modelling investigate examine ii capital extensively demonstrate flexible capturing process process provide form claim lda cm remark operational advanced reduction capital policy capital extreme distributional lda modelling involve compound heavy comprise analysis capital question bank financial heavy tail question ii address quantification capital risk consider fundamental policy several two extensive simulation study policy var fair provide analytic solution loss multiple operational distributional capital reduction mathematic email mathematics north modelling impact risk business challenge operational management capital partially attribute limited impact fair policy ii advanced financial bank strong capital impact scenario allow accurate pricing study transfer bank aid model capital develop international ii specify resource hold capital phase distinct capital requirement level capital also naturally allow capital estimate measure differ particular unlike capital measure capital correction concern treat since contrary capital hard define management company explicit analog ii particular es claim product loss bank capital percentile claim allow level relate accounting business closed form analytic basic transfer capital policy capital lda claim understanding policy understand extent capital offset informed analysis scenario heavy rare extreme consequence introduce model claim family flexible incorporate tail infinite pose bank policy capital behave perspective claim build policy address structure cell business event model variable discrete year distribute bank year calculate requirement seven time eight business financial section limit select block structure linearly duration year second stochastic propose trade bank capital attribute loss risk begin individual follow policy applicable practice exploit risk see simple concept via year loss exceed bank incur black lda model accord process claim process lda provide loss complete limit bank loss bring bank bank loss bank simplify policy function claim period accord provide maximum basis maximum year illustrate claim exceed bank loss claim hence exposure incur bank express conversely period multi bi loss denote provide limit simplify result claim generate period express require modelling condition outline payment advance base particular aspect model policy year bank section model explicit arrival time consider scenario single discount sensitive factor simplicity year claim describe payment payment discuss generally bank financial loss realize perspective claim account uncertainty account severe loss likely payment delay severe discount payment arise counter party claim uncertainty stochastic across reflect coverage loss demonstrate influence measure loss equation result claim period quantify segmentation define identify band locate beta payment claim process unlikely band limit equal convert lose truncate model one family scale dt result chapter ht function lemma allow compound jt process mixture density comprise exact form lemma corollary compound mention compound jt express comprise poisson loss random furthermore loss analytic directly jt copula aggregated loss close capital var quantitie meaningful exercise report chapter furthermore measure var es jt poisson tail quantile allow unique compound maximum lda jt median poisson loss compound state theorem relate simulate loss loss lda fundamental extensive basic consideration meaningful fashion process consider quantification capital bank define var refer year expect loss exceed consider quantification policy deviation context correspond claim loss define demonstrate lda stable exact analytic purpose es potential second claim process ii jt loss accord closed form analytic derive process policy risk process quantify j claim finally es var depend whether risk example capital economic capital es bank infinite nc proof th second process es stable var nc numerical losses capital result capital capital capital var however although context represent report capital typically simulate require uncertainty capital estimate uncertainty sufficiently large restrictive report accurately estimate simulated loss experiment frequency poisson intensity rare modelling systematically simulate parameterized index stable well apply capture policy capital requirement exposure exposure capital generate level purpose multiply percentile purely individual range indicate provide study intel ghz simulated loss provide study capital measure ii risk measure detailed capital loss realize via allow question policy detail stable model case policy advance study statement type scenario comparative var exposure process comparative capital var bank question denote claim make quantify claim comparative present eq bank capital var reduction loss present single risk bi simulation study identify exposure
direct straight line elsewhere necessary unit decrease represent lie body pair body may group contribution bin decrease length equal straight straight affect different bin separate lie inside divided normalize sum sign equal quasi construct length produce limit line aggregate body consider measure straight body early express ratio cf straight cf devoted integral particle move along straight line medium fraction body straight trajectory possible describe amount intersection ray analogue angular limit direction projection body surface sphere yet calculation could general adapt develop integral b particle double integral eqs many body intersect modify due discuss separate calculation however may simpler disjoint almost straightforward advantage straight integral integral different due hundred carlo critical possibility take variation calculation boundary surface omit integration discuss distance expression write monte technical may ref analogue write express cauchy due ray q dirac finally application sign measure calculation integral may useful many area physics fairly technical theory elsewhere ray essential property extension necessity sometimes negativity certain delay application algorithm ray distribution quantum physics call function reasonable physics distribution conceptual challenge appearance illustrate ray outside body describe measure know pc pr pa pr overlap affect removal another bin effort simple distribution reason appearance ray complicated sign formula correspond set dirac quasi dirac suitable six integral distribution homogeneous body analogue dirac call function nonconvex relate interpretation sum integral two object dimensional calculation body apply calculation double pair surface integration due obvious length may analytical calculation six initially dirac ref analytical expression shape reasonable integral method dirac analogue get carlo neighbor even nonconvex straight intersect body widely construction nonconvex interval inside nonconvex multi distribution introduce divide justified body definition nonconvex range argument construction sum present analogue calculation ray dirac introduce version tool sec collect discussion body sec carlo mainly analog ray instead ray segment draw isotropic write expression intermediate ref explanation inside straight distance leave side six inside track ray proportional useful explanation appearance alternate nonconvex intersection ray nonconvex body possible introduce distance maximal intersection distribution rewrite produce analogue rigorous treatment sign quasi use ref description picture may consider essential condition possibility use integral depend merely absence inside medium indistinguishable ensure possibility generalization expression call factor isotropic distance introduce coordinate second consideration body precede together ray angle angle brevity ray direction early may take multiplier body area surface explain discussion emphasize ray particle trajectory formula disjoint nonconvex arbitrary isotropic medium inside environment term hand distance straight track inside fall outside boundaries monte calculation advantage integral many give calculate integral numerically analytically carlo visually maximal divide bin simply monte fairly useful application different create origin odd index bin interval index bin carlo generation reasonable relevant alternative definition via derivative relate ray length autocorrelation far convenient dd body length find widely autocorrelation define sphere radius multiplier comparison easily nonconvex system explanation relations q explain convenient hand multipli measure volume six space division correspondence mathematical expectation cumulative distribution function random distance dd rewrite agreement prove analogue detailed proof elsewhere briefly five product body use multiply surface area analogue ray equivalent derivative work integral space integrable function define consider body topology test simplify notation instead rewritten b b derivative body ensures rewrite derivative appropriate dd integration integral formal four straight rewrite produce straight body analogue derivation dirac ref double along integration source intersection
error measurement survey portion response space relevant solve play useful extend current section discuss condition identify sense possibly mis identification fourier transform fourier continuously grow fast topology function topology function class include lead discuss generalized density function derivative belong equation define ordinary positive eq g polynomial choice generalize discuss example regressor transform continuity characteristic characteristic hold set include interior necessary interior characteristic equal point shift continuously differentiable continuously differentiable continuous proof density proof univariate density though still b identification necessarily restrict include equivalently identification restrict factor exclude unbounded gaussian weak topology therefore b b n nx ne ne far lead function arbitrarily nonparametric support wider fact parametric pose al closeness support n function theorem converge solution pose exponent imply slight two gaussian differ gaussian belong class separation gaussian specification weight convolution super transform multiplication growth bound function cube wiener complex exponential long theorem axiom conjecture theorem theorem exercise proposition remark proof em la la la university abstract identification problem convolution nonparametric measurement panel examine issue receive attention well far give period let observe nonparametric period distributional
effect symbolic calculus analytic function program implement calculation gradient etc ad outside symbolic list ad possibility derive gradient use ad price code ad long might affect overall analytic ad iy j iy I j I iy j j j iy iy I iy model iy iy iy j c grant research science thank mathematical sciences research proposition physical phenomenon whose estimate sde intrinsic randomness identify drift interest model dynamic effect distinction source variability effect method like root automatic close transition equation cox class become theoretical make applicable field dynamical ode pde repeat take individual role form experimental subject parameter vary mixed model difficulty arise deal model distribute term extend kalman one dimensional growth curve reference demand dimension handle contaminate measurement likelihood normally sufficiently behave distribution develop density give integrate effect quadrature result quadrature application extend multidimensional automatic result obtain work methodology flexible accommodate complex normal employ satisfactory subject observation per often drawback measurement situation section introduce function consider method tool dataset summarize result limitation method bold character sde evolve different population unit pp dimensional fix effect effect unit dimensional standard mutually condition diffusion assume ensure state denote unit evolution unit realization brownian path random express sde among unit explanation variability unit distribution give r lebesgue unit time let unit di scheme goal datum parameter also estimate observable density density behave function effect result possible random integral computed effect case explicit estimating explicit integral analytically solve ii unknown evaluate numerically solve direct equation expansion mention comprehensive propose approximate show approximated transition homogeneous sde suggest review dimensional focus extension expansion reference drop sde observation infinitely drift growth expand approximate taylor exist transformation solution sde ease interpretation scalar sde namely transformation give reasonable reader mean many diffusion see software symbolic algebra homogeneous obtain obtaining obtain general integral numerical dimensional integral irrespective effect implement though grows suppose give approximated series q tf special laplace exact likelihood denote mle laplace inference still valid intra variation variation happen general either ii symbolic iii automatic recommend costly grow choice symbolic package become however package help specific situation see ad tool comprehensive example assume random name rand rand create automatically file name contain return precision hessian rand hessian rand b hessian gradient b rand returns rand trust derive strongly recommend ad tool speed estimation random symbolic toolbox hessian much effort remainder drop necessary follow internal sometimes symbolic calculus value limited precision gradient hessian trust external plugging especially internal variation external free namely internal estimate internal might however author latter inefficient consume iteration external efficacy e toy growth model allow growth rate growth growth datum growth tree effect balanced consist seven measurement tree reference age age mm day infinity logistic quadrature method solve consider ignore ode ij growth often observe proportional root state effect pdfs zero estimate dataset thus obtain trajectory scheme size extract interpolation trajectory linearly equally spaced sampling transition expansion technique close euler discretization symmetry report skewness mean ci ci skewness ci ci skewness ci skewness produce report datum simulate trajectory scheme band trajectory trajectory realization draw order panel solid dash estimate see figure report fit relation find histogram population fit density fit equal methodology expansion approximation euler exposition sde start sde approximate draw lead close euler simulate obtain approximation considerable frequency contain true result surprising sde model conduct although worth numerous physics engineering finance application anti density multiplication mutually effect external internal total internal external step strictly positive reduce superposition transition realization effect dimension parameter euler approximation transition use expansion skewness ci skewness ci skewness ci skewness ci skewness figure coordinate empirical smooth solid five report simulate euler scheme use band five trajectory draw estimate population estimate strong estimate sum equal sum determine occur individual deviation sum moderate return internal round plug explicit hessian plug pool equal distributional thus equal effect plug exact maximize converge computationally costly huge expression exact expansion polynomial root process ergodic many application finance interest integrate neuron emission electrical see reference call population p standard symmetric constant quantity determine moment determined plug variance p value observation interval setup respectively satisfactory drift parameter problem especially calculate order length interval observation
lee writing update drop index simplicity analogous lee case exactly eqn learn view complete hope encode guess guarantee encode remove eliminate remove particular generate term attribute generic imputation keep decomposition place finally estimate attribute decomposition done learn classification engine additional convert single test base multiple write obtain converge attribute error follow write write get increase eqn extension claim square q increase define analogous result dot close repository represent vector ar rest point random mark mark replace entire test comparison baseline experiment substitute substitute substitute dataset base accuracy random lemma nmf datum interpret factorization miss attribute provide joint miss classification miss factorization nmf recently environment recognition mining dna extensively lee field various extension recently nmf point generally basis researcher decomposition basic need limited dimensionality similarity introduce motivate classification would miss motivate due inherent optimal nature
computational goodness theoretical simulation snr use distance digital priori characteristic class base kolmogorov ks statistic classification accurate letter distance simplification classifier classifier sample symbol perturb receive loss generality propose method testing compute small cdf concatenation quadrature cdf give method obtain theoretically distribution respectively point order j equal evaluate get thresholding counting sort respectively although brevity address approach metric metric letter set point sort order notational partition conclude fall distribute pmf individual sample class classifier replace snr store snr value far store complexity sort sort complexity use cm short requirement application memory requirement smooth worth generate cdf observation classifier l multiply distinguish ml snr fix confirm small size region detection snr exceed converge snr ks perform constraint analytical match perfectly simulation result db set evaluate present observe cm classification upper perform size slightly ks third classification phase snr fig actual
last implication kkt strong rule j slope condition drop inactive bind kkt drop derive rule penalize logistic coordinate piecewise enter leave slope absolute change move start product plot predictor simplicity rise interval intuitively solve kkt exclude buffer account may move strong kkt rule kkt slope violate condition section counter slope rule believe counter yet somewhat exceed short took independently center standardized eliminate nd nd continuity interval break sequential rule random hence maximal solution line horizontal line plot red slope segment value slope red segment reference sufficient slope basic guarantee dominant result hence never general show problem dual slope lebesgue measure cone weak simple hold outside model suppose inverse one dominant haar haar triangular matrix arise dimensional transform design possess condition work express slope sample concept mutual nonzero condition extremely optimization put distribution design mutually incoherent subset condition unfortunately argument become complicated seem translate need hold tool check kkt condition fails generate sequential logistic global safe rule gaussian inner residual vast example regression discard predictor least association exceed million say solution snps check verify find k scalar suppose form variable satisfy norm globally sequentially easy check reduce rule covariance covariance penalize likelihood sum diagonal penalize would useful graphical optimize entire rule discard zero retain occur rule problem lasso group strong sequential q value predictor predictor kkt predictor predictor start active predictor maintain non coefficient solution kkt check repeat strategy ever ever non value current broken unit slope add plot contain ever active occur ever due sign also occur light advantageous value check kkt go warm start kkt predictor go step warm c fairly common step strategy table see offer case never seem slow sequential rule inner offline intensive subset paper strong sequential discard kkt condition offer speed yield plan version use discard predictor apply discard unable correctness time thank co share publication helpful work author support national grant national sparse seq strong seq seq discard safe univariate predictor outcome zero paper rule rarely fail practice tucker kkt convex rule optimization focus statistical fit start observation denote write omit solve tune considerable past derive fast give zero therefore perform kind solve problem substantial discard author derive safe penalize logistic set similar construction fit tucker kkt condition repeat screen related optimization screening rule author argue performance estimation propose rule discard predictor problem involve penalty discard safe rule exclude nonzero rely end effective sequentially generally rule stems fact discard rarely practice net penalize look counter example large counter although global extremely safe rule rarely mistake condition discard need check elastic net penalize logistic section general problem strong speed convex answer detail penalize linear safe discard I author lasso sketch find scalar represents bind nothing predictor certainly imply safe bind somewhat recursive safe safe advantage discard truly nonzero discard fewer strong global rule predictor hand always small safe weak illustrate safe basic rule safe keep order rule discard practice predictor useful much
term joint approximation expression eqs q integral eqs analytically conclude sufficient component computing eqs mean eqs analytic kalman predict kalman covariance many gp covariances eqs recover equation filter sufficient smoothing distribution measurement mean state terminal step equivalent distribution tt recursively integrate measurement multiply integrate tp integral p step rule conditional joint approximation joint close gaussian approximation first second filtering fully determine measurement cross covariance matrix filter computation rule obtain shorthand define invertible unnecessary obtain multiply gaussian state p tc integral eq integrate unnormalized constant constant eq smoothed conclude p tt approximation filter previous step smoothing recursion filter approximation consecutive smoothing mean covariance consecutive state joint smoothing implication filter distinguish covariance algorithm filtering smoothing implication covariance smooth kalman smooth x ig w x I overview cross notation kalman measurement respectively sigma use mapping desire computing computation slight modification coordinate none compute joint implicitly mean filter smoothing eqs paper recover presentation smooth gibbs infer gibbs infer mean filter sec alg priors time prior infer I filter fp wishart estimate covariance burn similarly generate map mapping conjugate inverse wishart prior gibbs posterior unbiased alg joint tb conjugate x generate datum iw parameter j posterior sample b estimate covariance mean prior parameter moment p require pass show per give rmse solely filter measure filter unlikely experiment choose gibbs system px unbiased filter error gibbs sampler tb cc filter filter nonlinear dynamic system exactly setup filter quadratic measurement nonlinear run start expect rmse c cccc filter filter rmse rmse system error coherent smooth I high consider consistently improve coherent gibbs distribution area confidence area area latent variance filter preserve lead differ infer infer system part unlike particle degeneracy due filter computationally efficient sufficiently approximation preserve infer infer covariance joint increase infer remove time due introduce relative avoid error gibbs able evaluation requirement elliptical slice potentially combine gps model publicly gaussian gp paper smoothing extension gp general perspective filtering probability filtering distinguished use determine filter kalman filtering smoothing compare filter robustness acknowledgement thank valuable suggestion support intel partially foundation research computer general filter smoothing allow show filtering distinguish solely approximate covariance novel filter straightforwardly insight smoother robust extracting respective model play signal decade context filter control smoothing efficiency filter appear filter smoothing implementation concept require get lose implementation derivation kalman nonlinear kalman kalman filter
update proportional magnitude motivated compressive sense author result framework algorithm cost sparse empirically superior performance rate steady state compare system furthermore penalty penalty addition time vary result regularize provably dominate conventional counterpart proper formula white signal introduce furthermore group adaptive useful enforce regularization propose close expression guarantee provable white demonstrate advantage simulation regularize mis base organize section iii filter identification iv simulation summarize proof denote upper letter transpose denote briefly derivation vector goal identify impulse input desire independent instantaneous error cost instantaneous size control steady yield scenario knowledge study subscript allow constraint time instantaneous coefficient counterpart term always regularization manner approach identically mutually vector filter bind regularize provably conventional upper bound true show regularize robust superiority simulation correlate provable suitably negligible application transmission acoustic divide sparse identification system location non bind impulse non know suited adopt surrogate regularization equation regularization alternative small yield wise many application often group zero originally whole index mixed encourage among coefficient inside reduce ensure denominator knowledge follow generalization q weighting zero coefficient assign plotted fig initially gaussian input variance zero filter implement filter refer project vector onto ball use tuning time estimate upon steady behavior well outperform operate memory begin perform well however storage computation memory computation investigate fig highly exhibit unstable converge behavior local minima indicate iteration phenomenon ball fashion projection mis specification rather correlate ar process system noise benchmark filter set employ calculate curve plot white input empirically confirm severe degradation signal independent frequent signal correlate b study initialize change active shift signal unknown repeat achieve conventional filter section system start group response input treat benchmark divide coefficient set number coefficient number average db steady white coefficient update next parameter filter value white average curve fig outperform improvement correlate filter system initialize response coefficient inside block iteration block active propose general filter penalty close provable filter sparse identification demonstrate simulation regularize simplicity memory requirement likely steady extension order filter extension filter study accord note
cluster scheme determine scheme r alternative way map require existence f use condition exist informally suggest finite define metric space finite metric space produce partition say whenever say metric space finitely existence generative let endowed fix write remark x bp depict diagram conversely z ki diagram refinement refine partition prove pick point f z regard bb endowed restrict hence bx bp finitely metric definition finitely represent fix metric px ii z z z xx iy k z p reflect follow assume recall uniqueness assign block piece piece proof r px xx xx claim indeed exist x conclude half block belong px qx x piece lie scale axiom cluster c assign metric block let metric consist piece piece piece fix finite write xx piece assign block space scale piece pick pick subset easy belong assign partition follow every partition proof find behavior also kk cx assign metric pick pick distinct block partition arbitrarily force produce partition piece prove partition rich cluster scheme include already scheme practical cluster meanwhile cluster metric admit interpretation produce degree point reflect sensitive point counterpart consideration cluster implement two together consecutive inter exceed proposal provide metric clique shorthand allow could densely connect cluster metric consist vertex together persistent equivalence relation object act regard easy implement hierarchical linkage dataset preserve cluster mean output name agglomerative hierarchical begin cloud construct dendrogram describe merge lack come correspond force order root output simply root tree amongst impose larger demand pass powerful unique satisfie natural condition complete linkage linkage agglomerative clustering explain metric metric length refine counter linkage htb uniqueness prove instead hierarchical xx persistent mean persistent two persistent element block partition point block block equal another characterization linkage book proof hx r effect lie lie skip follow lie claim lie lie lie persistence preserve hence block conclude skip b assume imply belong skip equivalence class equivalence equivalence distinct would imply contradiction write x block relation x rx x depict r conclude b observe condition point cloud multiply example rx trivially partition replace persistent exist block impose persistent dendrogram check persistent obtain evaluate indeed finitely belong minimal belong belong block let xx xx permit restrict positive define equivalence relation require cardinality class singleton persistent check xx rx regard lie cluster represent arise definition construction spirit thin chain point additional increase hierarchical clustering manner write paper graph fit case determine could produce classification desirable ingredient topology vary construct scheme diagram reflect stability independent believe conceptual cluster algorithm qualitative quite notion many argue valuable well claim open proposition stanford optimize partition framework happen impose refer set one analogous obtain uniqueness theorem vary notion space require rich family sensitivity technique hypothesis apply deal interest regard exploratory equip input might optimize choice clustering family define desirable property cluster practitioner notion satisfactory however one thing intuition fact incorporate linkage nice method average linkage sort sensitivity unstable theoretically support theory sound incorporate persistence little regarded consist collection ad method generate picture one demonstrate construct plausible map metric axiom cx xx cx upon reduce distance simultaneously consistency choosing hierarchical prove slightly present understanding theoretical develop rich version familiar include familiar regard recall path component topological continuous point path crucial one set key topological decade reader familiar map interpret notion symmetry one construct useful conceptually basis space value circle construct spherical harmonic function regard invertible rise understand organization topological algebraic homology powerful key homology determine homology group equation field rational observation associate symmetry theory day study transformation law modular form continue powerful present classify cluster density domain moreover construction similar set rule assign induce replacement topological space consider nest increase map point hierarchy give respect quite restrictive single linkage requirement restrictive enough permit specification isometry space implicitly property require existence mathematical powerful axiom odd bipartite generative interpretation parametrize connect criterion point permit respect find collection account include effect account whereas scheme find scheme construction rely study finally scheme admit generative composition linkage change input believe incorporate cluster operate space isolate space construction application topological method connectivity paper demonstrate notion induce operation category discuss construction output standard scheme whose set category finite partition b p fx constitute category scheme persistent set say persistence preserve refinement object persistent persistence preserve easily persistence preserve map persistence preserve persistence preserve three range value hence one proceed similarly collection object consist map easy composition clear therefore another map category consist isometry finite whose eq proper special finite space yx x next concept formal construction fx clarity refer pair letter diagram equip carry without multiplication respect inverse set x ff xy regard define framework study define object object hc input finite regard category object procedure assign follow case partition persistent concept refer cluster object happen let stand accord map input output diagram order view specify map object category diagram eq discuss section study standard idea order inclusion study category demand require algorithm category uniqueness category category order real
length useful understand imply almost surely mass sometimes purpose suffice moment case dirichlet suitably rescale pd sequence self average limit random extra assessment cluster ultimately govern whole random beta atomic converge detailed moment exchangeable partition self finite big big cluster average confirm size accordingly group cluster relatively dp mass beta autocorrelation priori decrease atom great consideration guide prior obtain weight latter specification lead characterization identity true evident determination problem sensitivity dirichlet process cluster example short asymptotic relationship choose priori low matter suggestion present consideration hand long base application cluster otherwise second moment autocorrelation possibility work end form initially suggest stick break characterization dirichlet stick breaking characterize dp sequence generate dp beta special matter stress stick analogy stick break new tag stick evident consider dp piece unitary stick define one prior straightforward dependency exchangeable series datum realization measure random beta define marginally center dp mixture model parameter interest hierarchical beta conclude section exchangeable conditionally identically distribute statement see entail effect assignment observation describe e indicator retrieve analyze refer th draw base datum assignment label turn scheme move space mix easy assign usual indicator set sequence block cluster analogously indicator rule appropriately adapt believe merge move problematic imply exchangeability cluster configuration fast integrate however interested possible unique gibbs observation conjugate otherwise hasting addition metropolis equation adaptation well beta thorough adequate elsewhere provide specification example develop popular simulation range detail inference beta generate point use assess typical analyze section variability model default well basic specification fitting gamma hyper mean large fit dp model concentration basis compatible parameter dp fit package framework dirichlet stress underlie exchangeability dependency report aim provide synthetic goodness fit cluster assignment together follow terminology metric correct assignment predictive bias output parameter cluster original nearly compare bias incorporate intrinsic generating process typical nonparametric ability ground relative magnitude hyper material specification choice remark beta mis specification exchangeable exchangeable dp sensible study generate normal component deviation either robustness level weight short priori rescaled choice gamma relatively simulation beta robust specification assignment correspondingly parameter close process accordance finding around simulation inference beta affect specification confirm represent guide choose estimate mixture material result specification overall remark beta specification beta cc truth assignment table compare specification generate process gaussian variability bias hide model specification generating level ccc c c yu study process markov process hide spend state markov refer duration variable duration generate dataset hide state binomial term negative poisson simulation present robustness beta beta hyper assume behavior beta hide state result report implement package alternative hmm markov affect hyper induce beta generate middle column allocation beta attain hmm avoid overall plot induce hmm fit seem shorter contiguous identical hand beta hmm fail fair representation practical experience default informative beta formulation accordance long contribute wide range generating mechanism result note beta classic link breast medical raw copy gain loss array comparative genomic precisely intensity cancer microarray presence dna repeat fit tumor region high contiguous typical replicate frequency genome copy gain loss genome location copy lowest accordingly low copy gain say minimum say current mean plus two consider level iteration report bottom expect tend localized genome increasingly reinforcement embed correspond live adjacent point shape different adapt compute rate fdr neutral value minimum fdr determine correspond copy describe paragraph region account string consecutive call constitute subsequent fdr alternatively specify information available literature optimality procedure discuss also value previous location platform kb fdr end rp rp rp rp rp rp rp rp rp rp rp rp apply methodology publish match location queue take hour also fit negative length parameter estimate technique length share lead deviation purpose genomic location breast cancer tumor file genome project consensus genomic size list list likely list particular reference comparative beta semi detecting consensus genomic give consensus genomic location span beta correctly location versus hide hidden course take single illustrative study suggest detect array account improved compete approach characterization specie beta random defines exchangeable discuss beta process specification latent beta use hierarchical finally study outline illustrate beta useful array complement tumor suggest clinical study currently speech flexible state model exchangeable formalism heterogeneity exchangeable sequentially order enable unknown single transition state persistence since beta convenient underlie g complicated issue framework dependence cluster arguably major obstacle wider rely hyper distribution suggestion case enough differently experience dp mixture suffice inferential latent conduct inference mcmc discuss specific beta limitation method inference scalable facilitate believe possibility cluster imply directly complex development substitute probit specification observation record covariate curve imagine sort paper model base call asset across market liu acknowledgement support national foundation dms national institute health grants research office research european community fp grant agreement author associate anonymous ex definition proposition example primary c
three example ex moreover simulate increasingly realistic scenario span kb project ex correlation useful column indicate pairwise follow firstly induce x x z jx j introduce relationship hereafter complex placing effect exception vary finally range difficulty six example dimension intercept intercept analyse idea create firstly involve strong covariate secondly since simulate interested select belong motivation ability combine version secondly side copy design covariate high complicated challenge covariate different challenging simulate contaminate sample size model respectively retain r know account without negligible easily jump mode simulated project population originally contain snps nucleotide redundant example linkage naturally force correlation snps test ability ess report ex replicate ex change across ess example visit burn tune temperature exception example fix value remain equal correspond uninformative section fact ess panel replicate ex overall ess variability albeit reach respect thank tune temperature distribution overlap allow exchange panel replicate chain mix gap tune drastically exchange information overall move implement acceptance delay stay stable deviation ex variability model across example average chain believe performance operator tune small simulate relate ability ess remarkable superiority find explanation ess indeed ess number actual drastically ess ex explore conclusion rich parallel ess space competitive art york genome selection use bayesian predictor implementation discussion ed uncertainty vs fully bayes prior bayesian em approach em j uk em liu evolutionary em green delay reversible jump search gene underlie nonparametric regression combination basis function reversible chain em variable application liu new york york j em proper drive wang em accurate assess regression north ratio de f example ess move panel mc use retain replicate mc move plot region uniform accumulate ess block top panel vertical correlation grey leave triangle ess vertical vertical dash replicate ess upper green triangle replicate probability inclusion right triangle ess ccccc gp min ess ess ess determination well visit unique visit chain specific dr exchange stand rejection exchange move cc ess ess ess mc ess l ess l ess ess adaptive ess ess ess ess ess ex ex mm mm swap exchange c c c p ex ex ex l stability l min mm cm theorem college uk model datum many field make operational upon monte design thus datum example demonstrate extensive simulation evolutionary fast scheme model selection discuss recent advance make different coefficient propose algorithm huge algorithm implement idea evolutionary monte difficulty sampler face high become covariate approach operational indicator illustrate prior far hierarchy present specification namely devote sampler portfolio include good algorithm variety algorithm contain conclude remark response dimension describe tp want variable selection interesting large predictor interpretability bayesian perspective place latent overall exponentially predict gaussian coefficient column correspond intercept contains recommend intercept separately assign gamma q value jeffreys specification joint far write main neutral lead prior scalar replicate likelihood rise first conditionally specification unified tx clear alternative prior binomial illustrate eq choice ep pn centre prior attractive firstly possess posterior standard recommend skewed measurement feature appeal constant become tx reason next qr regression despite prior complex comprehensive place back shrinkage z hereafter lead analyse prior point prior likelihood close something advantageous even though laplace derive appendix never define tx x demand extra determinant fix different predictor order place value illustrate place regardless specification use qr know difficulty multimodal unified mass space know tackle parallel monte hereafter green hereafter inspired extend algorithm finally neighbourhood predictor propose issue related dependence strategy hereafter add temperature swap every non try genetic scheme integrate conditional population temperature chain population retain population update move genetic ordinary metropolis hasting selection swap probabilistic measure operator different exchange move crucial allow propose several novel aspect two fast scan hasting particularly bold pattern predictor new adaptive proposal update sketch useful implement discuss furth new paradigm complex predictor space believe move use indexing indicate indicator population local fast sampler mc york add move require indicator select try rather affect hereafter affect chain update indicate chain l l main relate gibbs evaluate full cycle implement least consume rely however notice cycle much likely sample large thus mc scan hasting scheme hereafter pt demand idea index gibbs step sampler probability use base size temperature th chain therefore aim save simulated finding save computational consuming l p l variable rejection step temperature indicate chain l update approximately l l indicator higher become indicator give mc embed move select pair chain firstly boltzmann tf log assume use boltzmann chance select chain configuration probability refer suppose new latent chain chain different operator select random adaptive move uniform indicator chain essentially try swap variable correlate practice block calculate retain j consideration specific move exchange extreme operator receive chain exchange adjacent chain temperature limit capacity mix idea delay move exchange chain full detail bold well simulated temperature liu ingredient preserve accuracy population consideration limit recommend example relatively subsequently monitor acceptance delay rejection exchange operator strategy laplace quadrature interval strategy follow integrate implicitly coefficient situation arise product reach response chain marginal likelihood see chain unstable make hard reach increasingly place close temperature balance acceptance move instead lp whole condition allow fast come extra relation sample useful apply adaptive within asymptotic enforce go I benefit amongst gibb proper give upper exploration tail proposal evolutionary ess denote ess prior without generality independent matrix rescale conditional global move ess population state em point select sampling scheme independently moreover complete gibbs perform delay rejection exchange operator delay view update qr qr restrict help burn discover genetic variation quantitative datum type quantitative trait find parsimonious set covariate illustration report gene ess table proposal reach acceptance deviation quickly inside see mix panel ess case reach inspection reach automatic tuning distribution exchange chain confirm without reach exchange rejection exchange operator acceptance experience detail burn implementation ess size uncertainty support knowledge mean variable small ess couple visit ess chain visit drive discriminate compete relate snps select genome gene predict divide bin finding bin analysis whole bin mix note influence figure clearly control univariate example large around burn index believe visit big gain would ideally suit superiority ess scheme illustrate wide portfolio enable search complicate I ess mc move ess respectively comparison fair version ess burn finding ess reach visit ess model prior rank datum provide indirect great superiority explain compare figure exchange ess rather mc contrast ess retain chain mode see look right tail kernel record chain regard second comparison ess operator probability high acceptance already notice efficiency briefly comprehensive ess simulate fair assume prior simulation set appendix behind example g realistic include effective exchange show collection move implement remarkably specifically move temperature ess far swap equal overlap parallel hyperparameter size inclusion posterior overall ess algorithm ess explore find true goodness ess ability towards compete explanatory ess contaminate ess pick ess remarkable superiority especially estimate large design bayesian sampler competitive spirit example also version visit rank ess effective extra enable local mode balanced gain chain develop good chain bold conservative automatic burn ess range situation multimodal tackle important index update move scan metropolis local gibbs large simulation superiority move model variable update particular difficulty contain consume illustrate spirit present far response response identification regressor carry multidimensional acknowledgement adaptation grant detail omit related sampling indicator l j th normalise version gibbs binomial crucially l derive scan metropolis hasting evolutionary define l j moreover introduce metropolis gibbs similar proposition omit check first calculation marginal j l l l scan chain requirement acceptance since normalise work follow chain propose value bernoulli one similarly otherwise select covariate efficient scheme iteration gibbs low gibbs expensive become operation move exchange extreme second chain receive chain acceptance adjacent chain temperature capacity mix obtain implement related delay rejection green gibbs possible pair rejection exchange try swap usually apart temperature reject traditional chain drastically rate flexibility extra evaluation maintain balance report suppose swap since random difference move delay rejection acceptance pseudo detailed balance alternatively attempt exchange operator far
nonlinear get back concept nonsmooth nonconvex accord f v nonsmooth generalize point nonlinear condition useful euler positively homogeneous characterize critical condition necessary nonlinear critical sf sf prop continuously critical nonsmooth thus single value everywhere small eigenvalue building block iterative eigenvector transforming try solve fail problem otherwise unbounded convenience whereas yield rescale inner formulate completeness convergence suggest multiple eigenvector small htb initialization sf sf rf sf initialization g sf sf kf sg rf solve speed quickly suggest consider propose tailor case generalize omit constraint produce alg sequence terminate produce sense differentiable point throughout proof notation u objective problem lemma note homogeneous minimum attain set feasible otherwise rf sf sf ff k nonnegative limit every converge sequence subsequence towards definition subdifferential homogeneity contradiction converge fact everywhere minimizer argument convergent positively homogeneous homogeneity q imply generalize positively homogeneous functional subdifferential hold q subdifferential q sum yield limit become substitution f p rf necessary imply homogeneity eq note replace even lemma minimize eq attain p ff kt rearrange type proposition give minimizer problem terminate attain minimum sphere sequence bound convergent subsequence clearly ff ff nu ff pf contradict fact minimizer imply argument subsequence subsequence towards optimal inner ff k important convergence vector satisfy instead early one limit accurately initialization spectral graph method overview undirected graph find relaxation cut ratio normalize cut weighted vertex weight partition connect normalize motivation c compute cardinality subgradient condition inner invariant opposite initialization htb weight accuracy g produce terminate conclude terminate minimizer invariant verify divide side converge eigenvector f k satisfy negative terminate case use low converge limit contain sequence subtract proceed analogously though spectral lead spectral c else f f invariant generality assume c distinction sign follow fact q easily contradiction cc dc second f second eigenvector graph result cc follow solve nonsmooth subgradient exploit dual w bound note rewrite non empty standard min corollary unit give euclidean unit transform regard lipschitz primal solve use fista convergence fista lipschitz fista good descent make modify fast alg tb input initialization rs rs rs reduction subspace maximal compute covariance pca want like component human enforce component trade standard constraint cardinality simple thresholding component focus good candidate sufficient globally approach et al simultaneous component penalization enforce sparsity control recover whereas yield trivial easily fit function inner become closed analytical objective positively optimum iterate attain derive noting objective concave equality thus optimize get principal component subtle difference thresholde current whereas fix lead iteration htb control initialization total eigenvector laplacian construct cut error point initialization initialize unnormalized initialize laplacian propose tv yield clearly spectral avg avg unnormalize spectral resp point successively number reach minimized scheme initialize thresholded eigenvector laplacian normalize random initialization show eigenvector outperform effort run well however want bi cut least spectral thresholded gene compare power plot method coincide observe art b cs statistic problem linear critical nonlinear apply achieve quality beyond useful inverse eigenvalue semi machine view
monotonicity key n monotonic denote zero define full rank concave concave suppose hence definite ordering agree except relate follow algebra identity monotonicity strict monotonicity entail entail view calculation form contradict hold contradiction rank right hand positive hold first show limit inspection monotonic strictly monotonic strict monotonicity theorem let generate ii yu present yu example even inspection zero point eliminate yu omit analyze convergence let assume slightly weak starting pattern practical assumption tractable convergence section present notion algorithms context mapping convergence eq global rate spectral modulus eigenvalue restrict linear implication impose notion eigenvalue lead yu situation shannon interval surprising exceed nonnegative show precise iteration define speed appeal toward converge secondly slow design iteration reach increase converge note imply intuitively explicit q resp mapping corollary imply hence prove algorithm fast reasonably practical quite slow explain et version monotonic discuss iteration behave dynamic twice practical assumption early usually reasonable measure eq q similar equivalently guess mathematically employ dynamic design side first column interpret speed equal accurate ratio become concern dynamically agree ratio believe situation iteration design space dynamic acknowledgment author li david valuable pt lemma multiplicative compute strict monotonicity establish variant formula explain al keyword computational aspect approximate seek determinant closure linear et iterate iterate resemble closely read design intercept broadly applicable intercept refer ii interest example et al yu recently yu work aim extend monotonicity thereby monotonically
recursive dealing instability solution parameter tool g choice explanation portfolio zero portfolio optimization use statistical estimate reason become f later online moment reason find substantial discrepancy financial standard recursive could version et naive mean expect stable may update section core development allocation efficient nature algorithmic us notation lagrange often approximation qr q recursion q equivalence filter understand similar filter robust statistic choose datum window efficient compute recursive deviation consider exponentially median correction huber replace slide preliminary investigation term financial inherently ability accurately dependence pointed instability portfolio weight cause adopt low eliminate datum optimally lower find frobenius see denote return update less straight time objective n equal one eigen decomposition yu operation inversion specifically yu calculate eigen dimension q incremental free numerically initial perturbation portfolio datum initialize balance allocation return period generate large portfolio base recommendation find extremely scenario balance incur eigen procedure filtering criterion portfolio selection balance approach filter complementary firstly deal separate var secondly account outlier quantity embed account outlier var adaptive account need advance place every shift environment rank set advance give example subject cycle would significantly expect may expectation allocation link theory asset allocation version technique dataset rate perform exchange fx see american cover approximately year daily www com approximately year yahoo http uk yahoo com adjust financial daily market http edu pages ba daily yahoo data observe allocation financial trading subject account looking develop could distribution huber significantly naive compare idea way avoid look ahead bias var strategy portfolio balance instant go balance window balance datum balance period initialize day actual portfolio balance balance time trading transaction cost strategy use transaction cost assumption since transaction cost often substantially employ financial application day calculate daily gain win draw perhaps need explanation percentage constitute peak cumulative percentage term measurement frequency trade absolute portfolio balancing times fx equally spaced value exploratory depict contour ratio evident exhibit become evident structure suggest good performance basis plot approximation extent vice versa computational issue memory storage year substantially imply purpose decay great run balance day first third fourth var initial converge balance period speed batch code version dimension compare batch increase batch second strategy table clearly ratio volatility underlie maximum strategy volatility balancing period partly infer truncated portfolio contract var whose literature indicate imply naive return exhibit bad strategy growth necessarily price familiar pattern report display ratio perform marginally satisfactory financial performance tend increase success var link naive figure remark section secondly var method fair train year insufficient algorithm although case smoothness beneficial outcome well discussion final economic business wise many perform poorly financial trading return encourage extent suggest useful first procedure drift computation noise potentially derive table drawback potential naive naive assumption direction position ann var var naive var naive var var naive var var var var var r gain ann ann var var var detail method need technique benchmark measure et extend transaction bid ask portfolio well balance explicitly incorporate lead addition work make adaptive require statistic finance signal computer drawback allocation remove online portfolio require standard wise left exception zhang li little methodology currently g investigate portfolio algorithms portfolio management stream stock return technical fast recursive portfolio r update singular good dynamic portfolio inefficient v r portfolio finance appear optimization h portfolio program new line use multiplicative huber p york wrong dynamic portfolio algorithmic http www return portfolio li ng portfolio period portfolio capital market gaussian expect market exploratory datum mining trading recursive presence tail portfolio early history decomposition shrinkage via lasso stanford university zhang zhang online quadratic motivation context trade recursive portfolio exponentially regularize var simple statistic empirical financial allocation dataset benchmark allocation term demand portfolio management portfolio capital fraction capital asset fraction capital asset know weight portfolio portfolio maximize fix maximization small portfolio prefer portfolio small return portfolio small desirable capital allocation portfolio return capital little interest visit wish mean portfolio unstable difficulty substantial improving include al b et impose portfolio portfolio allocation concerned require historical oppose recursive mechanism procedure computationally efficient stream nature handle asset trading perspective regarding take automatically allocation statistic
small zero different hence overlap work image strategy calculate yu rewrite require q equivalent eq clear explicit component component make sum denote whether observation treat derive one treat indicator expectation check tucker condition maximize determined iteration em ensure efficiently yu less informative entire collection slight extension fix monotonic yu yu rate work within convergence comparison automatically combine comparison upper possible hence recommend helpful optimal choice convenience current eq uniquely suffice inspection show potentially transfer mass section neighbor em overlap fast scheme exist component severe apply example nonparametric assume draw normal mix distribution put denote mean increase adjacent density conventional em may nearby component hold somewhat pair element number support perform e use eq choose naturally implement tool composite near find add considerably speed usually start interior receive subsequent near bad support may eliminate one easily vertex direction derivative maximize choose choose add em near exchange convergent actually show add globally convergent near yu design report performance near output conventional iterate one overlap density room use combine hope near complement focus purely modification purely one conventional poor many guarantee start global maxima yu denote output step subsequence converge step index equivalence reveal uniquely say derive framework define observation interval inspection inspection failure censor may censor unit censor inspection variable failure observe leave censor right censor censor observe fall within random notational convenience open close though trivial set maximizer jump em type define maximize implementation em section censor take implementation substantial order bootstrap resample repeat briefly give em zhang doubly interval censor show admit implementation censor effectiveness algorithm doubly censor propose em decide evaluate include bivariate censor work progress simulation random censor degree censor start criterion experience benefit insensitive neighbor conventional em replication second censor heavy censor em mean count either situation reduce conventional large reduction significant number remarkable direct slow much bad moderately censor heavily censor somewhat moderately heavily censor time censor significantly higher another fewer censor moderately table superiority augmentation design type maximize likelihood advantage overlap conventional near exchange censor work conventional natural ordering censor bivariate censor variable censor present inferential challenge extend encouraging extension accommodate truncation censoring facilitate semi parametric adopt van type maximization one strategy former maximization investigate potential would helpful censor conventional em contain special fast zero zero note equivalently algorithm cumulative initialize obviously valid output output agree algorithm compute induction compute clearly conventional implementation apply track algorithm near exchange affect limited implement belong satisfy entire algorithm input sort slightly set censor em strategy efficient van proportion ease monotonic improve slow overlap component speed stability overlap censor improve adjacent consideration carefully improvement realistic augmentation doubly censor mixture density multinomial miss partially al example mix distribution support closely van among well censor al computing exchange wang em em widely use experimental yu emission shannon ar density mention motivation fast censor partly overlap em monotonic strategy overlap augmentation em observe mixture collection step dramatically expectation scheme van effectiveness wu liu van liu nearby component pair strategy maximize log overlap explain datum illustration censor effectiveness simulation bivariate censor implementation em type take
appendix add subtract break infimum property expectation nest expression part arrive loose indeed pick supremum due bad hand inside expectation condition identically independently draw identically history argument rademacher proceed identically proceed fashion way arrive b arrive pass supremum tree start immediate successive expansion coordinate zero argument assume q ball jensen inequality conclude lemma proposition jensen eq desire proposition specify verify smooth see definition tree martingale come smoothness tree tree far integrate trivial depth let path element sense pair value q successive let j trivially verify hold affine argument sequentially irrelevant specialized response put ensure claim smooth fix mapping fix cover q interval interval enough tree precisely ti I belong plain word partitioning define q simplex function homogeneous need prove close consist hence supremum value shorthand tc q pick appropriately sum integral fix later find player subgaussian tail augment restriction player belong pack choice observe simply equal response mass reasoning quantity supremum martingale choose third bound put upper get adversary almost sure similar divide episode episode length player play subgaussian episode step adversary infinite regret incur union episode choose ensure q prove section separable banach property value define sum follow embed everything work replace key define scalar function q derivative case g x g g plug immediate previous whenever first note second eq valid satisfy whenever control style union family divide optimize plug give expect recover choose slightly take outside proof game eq application minimax subtracting infimum nest expression three arrive mention replacement appear loose pick supremum random draw study wide notion well beyond framework simultaneously notion internal additive calibration perform know study focus algorithmic online external learning present extend wide appear framework give external internal recover extend improve quantity sequential rademacher play derivation decade reveal somewhat biased former focus primarily understand supervise learnable risk minimization serve learnable approach dominate decade base unified tool develop situation unify manner provably learnable feasible yet show scope precise characterize external within circle well know base rather banach ever tool show utilize successful situation section discuss result greater break contribution payoff formulation might appear show framework game upper various natural tool allow performance include smooth non payoff generalize cumulative consider abstract payoff rich whose complexity average number combinatorial usual regret regret see several equilibria infinite banach specifically space shoot calibrate forecasting improve upon rate calibration outcome e global al provide notion regret suited environment tool example great preserving study game unlike research online general problem proceed long study able even hope believe bound arise inefficient recover minimax often inherent describe organization serve whether progress paper whether specific decide specific risk potential first page avoid define framework appear establish general framework derive overview detail skip sake defer paper however basically notation omit abstract problem player adversary course generality say minimal assumption meaningful let move learner adversary generalize round end information set player adversary move separable banach player randomize learner minimize measure eq value payoff additive form cumulative sequence mapping adversary maximize game concern identifying general play payoff class formulation work likely one seem summary mainly payoff payoff transformation emphasize payoff mapping transformation payoff payoff mapping function act modify choice payoff say transformation write payoff transformation mapping shall abuse use mapping transformation metric probability adversary formally strategy mapping respectively adversary sequence mapping learnable game statement measure discuss sure might abstract mapping framework nothing illustrate external regret z mapping eq cover notion swap banach close z easy ensure vector vertex calibration detail global game let scenario consider capital ease already shorthand expectation distribution p f xy bx ba kk supremum infimum write quantify range understand denote tt notation payoff singleton sequence transformation banach let denote dual corresponding ball upper beyond section version rademacher progress additional assumption refine upper definition generalize complexity global payoff payoff function payoff mapping supremum take value depth I omit sequence transformation base write adversary appear definition rademacher external turn act player choice section notion study external sequential expand version online ability perform convexity condition third last twice first singleton consist let mention decompose game interpretable decomposition yield well constant nonetheless inequality essence treatment fact bind third inequality distinct achieve theorem quantity former choice draw mixed complexity draw crucially easy sequential oppose since random sign mathematical arise easy control martingale typically provide specific response adversary similar condition arguably set payoff assumption term proof might appear loose nevertheless substitute pass presentation decide give replace z z z z transformation instance conversely new x become exactly analysis detail smoothness cover existence banach crucial term increment yield informally promise appear consider say say exist finite smooth argument linearize term increment establish inequality argument true assumption lemma bind third lemma smooth bound complexity sum smooth q taking reduce gradient reader familiar notice resemble sequential rademacher studying ask finite question finite transformation finite sequential complexity gradient normalize account cover average power calibration basic power norm separately smooth unfortunately bind value smooth z concrete smooth calibration example statement assumption finite payoff z general case average turn get good payoff transformation z f result exponent previous smoothness smooth probably g f absolute payoff result expense amount resolution proceed however notion generalization notion introduce provide payoff whenever payoff invariant mapping complexity identical section detail comparable tree consider section particular assumption generalization eq particular drop soon repeat statement hold first number adapt coordinate sequential take familiar value tree simplification notion invariant transformation generalize dimension subject next section assume invariant definition dimension tree class exist dd define version generalize tree large tree invariant payoff base control instance proof tree role tree number bound value tree generality evident cover sequence transformation lead appropriately number bound transformation little hope non trivial good news variability cover external time obvious external decision appear dynamic path general one would assumption naturally capture notion change transformation cover comprehensive list payoff obtain piecewise change assumption proposition payoff norm transformation think segment accumulation point payoff transformation vary scale cardinality simplicity though tight payoff sequence payoff vary much sequence payoff transformation sequence claim cover tree increase payoff let large close clearly eq control solely move define notion repeat whenever proposition immediate strategy many since action strategy check face function low term convergence set minimizer expect singleton minimizer specific literature establish framework decide major unified simple inherent comprehensive randomized expectation randomization sufficient sure game randomize least randomization natural expectation measure exceed value non random inequality integrate well integrate respect fix existence player adversary exceed suffice want prove existence consistent strategy round formal round game calibration game existence strategy call trick rest devoted probability version range think conditional distribution decomposition bound roughly speak typically response adversary third third term apply bit mild draw tangent variable
image phase couple factorial sigmoid impose space phase couple generate unit hide unit square interaction restrict call build reference couple produce unit unit example unit generate dependency implicitly activation implicitly conversely implicitly unit ignore signal datum gradient ascent likelihood express energy visible bias possible contrast unit condition exact integrate hidden statistic update denote indicate expectation calculate straightforward however distribution equilibrium approximate summarize divergence approximate run run hybrid monte produce momentum run rejection weight function panel order model berkeley color patch preserve unit initialize random variance bias rate update length vector length vector grow match subspace set column normalize unit batch various part sequentially adapt parameter fix add hold adapt examine localize oriented band roughly quadrature fig learn pattern rigorous analysis need verify position weight hide learn hard visualize remove view filter high column eq coupling subspace matrix find sort pair filter map plot direction angle couple separate suggest explore couple hide varied matrix choice rather factorize factorize additional motivated model subspace structure phase like thank would thank document nsf grant nsf c minus ex minus plus minus ex ex plus minus minus ex plus plus minus minus plus minus minus minus pt minus pc pc plus plus minus plus minus pt minus pt pt pt pt ex center institute california berkeley berkeley describe capture amplitude phase factorize third boltzmann capture structure model dependency output subspace local norm subspace quadrature additional unit dependencie subspace phase form coupling difference spatial image recent year e decomposition utilize infinite mixture gaussian image focus radial step order phase across contour shape image machine phase restrict quadrature pair subspace couple condition phase model couple view within attempt orient filter art natural mathematical factor negative negative model learn dependency scalar way correlate noise angle pair filter coefficient high level high order filter related pairwise logic multivariate describe boltzmann local phase amplitude dependency variable dependencie combinatorial phase couple pair neighbor connect coupling unit cosine connect couple stroke bias omit review name covariance angle pair dependency phase coupling phase define energy term define entry bias filter sum hide however deterministic image unit combine produce covariance representation describe mean contribution connect unit form contribution rbm hide visible unit mean specify unit unit invariant local well model subspace
operator version measure result conditioning operation computable even set construct variable measure computable every individual conditional least construction encoding time joint computable central infinitely relative explanation phenomenon circumstance condition computable conditioning computable discrete corollary situation condition noisy capture computable absolutely corruption independent additive contribute give computable enable theory notion develop computable build recall potentially correspondence finitely say computer program output element eventually c precisely basic notion see sometimes real computable approximate e give report rational real lower rational low computable computable theory convenient largely use metric enumeration dense uniformly denote center bs enumeration metric characterize computable dense eventually string standard enumeration string metric enumeration denote borel algebra ball measurable computable sequence point computable space familiar computable computable every open set enumeration ideal note open intersection modulus subset domain computable enumeration computable sequence computable image set set continuity let computable program restriction equal consider canonical representative computable intuitively map input randomness induce independent coin randomness formalize infinite basic extend henceforth basic relation instead say drop hold font variable measurable measurable p neighborhood open ball computable also say characterize class computable measure term computable metric computable computable metric computable measure let real hand open computable empty whole computable borel open immediate almost computable ideal also ideal enumeration computable form almost union enumeration ideal ball computable e k enumeration almost construction almost characterization almost computable real let c theorem e compute sequence almost union computable real supremum desire computable computable computable computable computable informally event occur occur measurable write conditioning precisely insufficient define interested measurable set atom notation modern kolmogorov solution space function b uniquely set e surely refer generic equivalence version define conditional measurable variable natural proposition version natural want individual conditional order distribution detail g call probability measurable show measurable sx sx measure everywhere agree random variable version conditional map b g image conditional build elementary probability hand measurable b demonstrate almost computable measure corollary computable almost numerator computable real denominator likewise finally computable conditioning notion distribution metric probability give computable computable correspondence allow computable index ball make space computable rational subset topology metric let rational subset computable uniformly ball center write c suffice ideal characterize center constraint bt rational decide whether show uniformly computable computable uniformly interpret computable computable computable space uniformly rational c open weak open computable uniformly e computable function computable space density lebesgue measure therefore immediately lebesgue density rescaling admit lebesgue admit lebesgue measure positive version probability q x version unit eq result index I conditional let numerator k r computable open u rv q u u u exactly distinguish case decide equivalently uniformly computable follow computable computable sense although encode infinite computable compute set map computable obtain pair variable pair continuous nan conditional f computable relation let dense consist rational coefficient computable metric computable probability computable borel computable computable measurable equivalently discussion let consider derivative measurable computable conditional n express numerator one denominator along line proposition low decide rule ask conditional despite conditioning restrict begin address fundamental obstacle admit nonetheless computable possible think bit stage occur long simply stage variable rough cause set infinitely lebesgue everywhere dx n one lemma random surely variable py dx integral computable early uniformly k q kx k straightforward admit differentiable result result measurable let distribution disjoint open contain remainder similar replace replace k computable carry show simplicity computable value random value pair everywhere many setting understand situation inference possible begin common conditioning elementary event condition discrete variable computable metric countable let characterize almost numerator take computable note could proof sampling several formalize unbounded computable finite countable metric space characterize version conditional computable metric space exist computable computable result easy distribution discrete metric let computable discrete distribution distribution way calculate say conditional first definition integrable density complete dx parametric family gamma etc admit case rule give result bayes variable version satisfy dy dy denominator finite borel dy dx dy dy dy dx imply denominator set point denominator characterize dx definition bayes rule dy hypothesis ct respect probability low rx rx hypothesis integration continuous complete integration computable bayes rule computable density computable computable computable conditional computable immediate situation observe corrupted let computable computable computable variable absolutely computable pour show continuously computable computable derivative continuous computable computable corrupt computable even mean introduce theory capacity yet cause channel capacity drop prevent encode bit real much noise couple noise uniformly computable absolutely density magnitude nice reduce time despite raise question show classic exchangeable conditionally I sequence condition measure given version cover nonparametric statistic acknowledgment preliminary version appear partially support nsf dms dms publication grant publication reflect view foundation national mit newton international fellowship college thank comment regard computable comment em rgb claim proposition theorem font theory mathematics subject primary inductive inference success limit probabilistic probability fundamental notion bayesian inference inefficient computable variable unit first second encodes problem nevertheless broadly condition computable computable corrupt probability modern science inductive directly raise phenomenon exceed researcher propose language describe joint automate reasoning inference computable demonstrate generic exist possibility amenable automate challenge theory explain characterize circumstance broadly computable intelligence ai language describe answer language language abstraction modularity practitioner infinite produce effort logic probabilistic language research model formal language high order e structure essential modern statistic expressive language describe arbitrary traditionally contrast system introduce compute conditional toward hope eventually support computable progress towards random hoc explain probabilistic operation build classical notion arbitrary sufficiently programming basis precisely describe operation probabilistic programming capable perform describe computable computable study metric g distribution probability experiment outcome compute straightforward ratio modern probabilistic place high situation notion theoretic notion kolmogorov characterization conditional
tree inference inference go target application character recognition probability encourage show use real justify reasonably work coherent fairly coherence need call write special partial ts ts non ms real map complicate variable joint gamble formally singleton real number element denote similarly also mention tuple corresponding element tree independent meaning simplify device identify example sx device gamble x sa spirit gamble correspond gamble throughout conditional model may systematic gamble condition unconditional unconditional every subject supremum conjugate sc I gain super I homogeneity set follow intuitively similarly sc sc sc sc notation behaviour hereafter frequently thing interpret belief keep mind coherence uniquely consider low kk satisfy consistency coherent sense coherence concept describe unconditional easily recover sure conditioning disjoint subject cx ix subject separately irrelevant infer coherent sx sx ms unconditional uncertainty generic local conditioning show letter unconditional tree compatible ss reflect type list need explanation condition encode turn classical take parent irrelevant follow turn assessment child irrelevant tree infer empty interpretation focus help go tell irrelevant child child even tell non child irrelevant conditional child terminal ci irrelevant grow child child equivalently publish elsewhere refer detail result mention coherent low belief set construct low reflect structural disjoint proper word value affect belief generally speak unconditional disjoint proper assessment conditional independence family lead definition notion model coherent coincide domain low product necessary make distinction term implicitly version marginal small coherent coherent play tree separately proper non nf ng important separate right side equality explain coherent natural extension trivial empty independent coincide want separately joint proper subset jointly coherent low calculate consider eq meaning generally speak yield conditional n regular lie detail extension interestingly extension assessment independence shall course conditionally independent empty finite construct coincide marginal follow proper irrelevant variable belief speak conditional turn lower alone assessment conditional eq assessment family separately coherent domain family marginal conditionally coherent conditionally product denote conditionally extension n indeed independent imply global model conservative extend express encode justify crucial lie tree recursively leave root basic building block dot grow child construct recursively heuristic manner justification child already cx sx cc ci marginals cc blue child child c need combine sx sx see coincide conservative small conditional low blue level distance also account ss start recursion lead joint alternatively eq reference follow collection end discuss property global derive ss positive empty c ec ec singleton separate arguably tree family compatible requirement satisfy requirement coherence coherent reflect leave hand involve model one would model strong generally prefer condition positivity restriction uniquely reasonably coherent approach coincide uniquely coherence condition requirement coherent requirement inference model consideration encode tree dominate family turn model construct satisfy requirement ss eqs constitute low satisfy unique family finally empty derive kk coherent family inference discussion coherent inference go much detail property one net bayesian net tree message namely interested sign sign next thing tree tc indicate ht pi x x node node x node node x node node pi south constitute message node message reach independent arrive represent wise product eqs message child move stop low send leaf send remove overhead create x te left node leave left message along fig calculate recursion formula q eqs influence calculate find maximal principle concavity character drastically fig exp exp exp exp exp exp exp exp exp exp circle exp exp pt node yshift circle yshift exp exp circle exp exp exp node xshift exp pt ht ct sa pc mt intersection axis respectively next evaluation interval become become stop iterate soon interval exactly zero briefly complexity algorithm make iterate multiplying make employ fix tolerance linear represent base algorithm focus discuss step property strong tree observe markov chain thick dot blue grow child node x thing like mass restriction notational inference variable want section message message qx send transform lead obvious notation separate soon long additional observation probability include conservative computation unclear whether independence address availability fast preliminary different separation particularly leave root reason compare interval root binary interval chain interval former wide irrespective least ten chain seven extreme point strong similar independence summary difference inference notion outer safe light could still make tool right pt smooth paper expressive enough model interesting section character illustrate difference traditional notably arise come available reliably character one sequence hide generate refer processing speech text observable sequence hmms correspond tree informally every generative generative grow child child node child model precise version state bayesian technique learn multinomial identify might realistic unconditional sample positive inference local identification tree generally speak computing inference hmms likely observational sequence precise probable adopt overview state text observable text perfectly reliable process device count single transition character another generative two sequence model generative might character th albeit completely resort gram character quantification exponentially transition depict recognition might ed child grow observe child grow right grow observe child grow target child I percentage prediction alone hmm indicator contain percentage correct prediction return htp lrr precise accuracy output quantification character include character hmm take one report value accurate precise output happen rarely remarkable return accuracy display basically return character shape allow notion stochastic focus notion limited net develop efficient algorithm update belief precise distribute fashion pass message expectation remarkable feature fact never practical net moreover clear behave axiom separation net would result encourage open net requirement coherence establish requirement inference update tree able reliable counterpart go would extend expressive tree net graph might affect net infer could achieve global rather many tree lead assessment condition observation indicate lack certain might application way necessarily together separation necessarily require match net de project partially nsf p de comment anonymous net tree strong markov rgb circle draw fill cm width pt nd fill nd nd fill nd width draw line mark draw line mark lemma c focus net replace independence commonly net weak arguably suited theory focus combine local uncertainty justify compute update belief entirely coherent satisfy requirement character approach comment availability truly year intelligence address variety finance name still net create inference close convex notion net vast majority speak independent set stochastically strong precise net mathematical consist bayesian graph ideal net consideration net precisely consider net cause existence partial knowledge disagreement amongst another case
constraint propose know propose encourage combinatorial thus tractable group either entirely solve solve side note similar original understand whether discuss cast find direction assume orthogonal achieve take two problem consider row standard obtain optimal k far illustrate constraint impose toy plot plot x black square vertical matrix square dotted direction represent green dotted line differ correspond error example fact next precise pca additional pca red row sparsity constraint compare constraint dotted method produce v f high v directly implication set eq last strict cast propose produce row two observation choose base row express entirely recall view encourage matrix penalty lasso group entirely importantly convex solve follow form column row equal zero consist successful attain reconstruction correctly optimize compare reconstruction comparison case ii underlie I except always even tune iii notice performance expect vector term low rank pca type wise encourage entire microarray soft set observe x since measure precision plot randomize every practical light alternative plot compare specifically enforce gene exclude c gene microarray ps ps minimize consider explain unconstrained subgradient subgradient equation write claim explain subgradient equation b rearrange take norm side b b connection adopt highlight difference general modify intractable term encourage optimize operate randomness resource thereby understand concept acknowledgment would thank art support stanford fellowship stanford fellowship lemma definition department stanford stanford xu stanford stanford department mathematics stanford stanford provide reconstruction span appear take randomized whereas pca convex optimization viewpoint implicitly sparse pca observe attain possess method recently often svd decomposition interpretable motivation interpretable procedure recently machine learn start far approach purpose decomposition viewpoint connection put combinatorial optimize highlight interesting practical viewpoint second implicitly contribution formulate randomized problem relaxation original pca implicit objective third propose provide paper alternative help notation column contain respectively column brief background particular emphasis subsequent section seek hyperplane write datum column think factor equivalent nonzero provide number randomize variant variant nystr om variant consider set combination ask intractable randomness submatrix least randomly leverage sample singular norm relevant onto generalize interpretation attempt pca find
application mc real mc computational limit optimal decide neighbor differ dimension expect difference large spread dimension small spread separately skewed use result mc study need verify carry mini mc investigate question situation good kolmogorov generate integral hold case ks repeat time show nominal nominal slowly increase especially inferior encouraging size event event going indicate mc true type nominal agree statistical suggest equal preferable multivariate dimension go illustrate normal time figure reject ks fail routine carry available http edu approximation search sophisticated could combination code equality two dataset conceptually suffer curse study show high computational capable detect visible high hour ne ne ne ne compatibility abstract head chapter head head head head head head compatibility lemma corollary theorem ex ex false compatibility true em mu mu mu false mu mu mu mu mu align align end end align environment style style environment array allow tag tag pr box email usa could might dimension neighbor sciences rely monte simulation stage detector simulation mc experimental number higher belong consistent test concentrate simulate mc testing concentrate observation correct generate near neighbor bernoulli binomial slight ignore instead consider
institute statistic virtual center use patient visit retain per say visit standardized independently binomial particular enter spatial term adopt star link structured structured account particularly suit fix purely frequentist use north france public region disease model parameter latent prior inference mcmc various explain latent field strongly dependent development instance poor remain end marginal recently give accurate explain introduce justify assumption obtain comparison finish public north east france adjacent dense centroid km region record record unit unit median per population potential practitioner patient practitioner patient unit region statistical autocorrelation improve explanation reflect access influence specifically focused measure policy notion proximity show patient big specific age status herein patient prefer practitioner us star model adopt follow response binomial logit spatially component use proxy environmental autoregressive process assume adjacent normal distribution proportional unit area write adjacent adjacent unit accord common adjacent two share common ask spatial unit association residual heterogeneity traditionally disease mapping risk convolution autoregressive process heterogeneity gaussian hyperparameter structure prior posterior present approximate three approximate marginal approximation compute laplace laplace previous posterior approximate task sufficient integration mode quasi newton compute let gaussian assess compare penalize indicator measure fit measuring pc core processor compare hour former parameter indicate enough risk different lrr alone proximity density summarize indicate spatial effect prior contrary improve prior remain major patient live less greatly use covariate fix summarize figure adjust north south east h fix introduction mind nothing gain covariate remain furthermore powerful walk use combination b useful offer flexibility coefficient linear spline walk spline spaced walk prior flat distribution assume spline reformulate latent model spline knot example devote model prior rational round several spatial spline centroid population population population concern e population diabetes logical try adjust percentage example technique simplify laplace approximation integral use spline distance vary binomial logit assume distribution application manuscript enough smoothing walk retrieve easily aggregated observation application interested account patient number combination category poisson krige access time hence example proximity model walk third support university thank frank careful review recent spatio model environmental disease propose include potentially ratio patient population within response variable link relative additive account various spatial effect covariate close form integrate nest approximation recently model give fast model comparison assess lemma spatio temporal field environmental disease model potentially distribution patient unit framework response
thousand gene simultaneously decide differentially express gene response absence presence covariate whether nan hypothesis expression setup differentially test exposition thus concerned design mean wish depend standard rank decision whether practitioner software package concern global nan detection plain like know datum noise zero compare nest science nucleotide form position genome population reference allele common snps record copy allele quantitative trait decide trait scan need trait genetic thousand hundred thousand approach test hypothesis fit regression global adjusting see problem denote impulse user noise whether assume user detector max detectors concern detection white popular consist model sparse superposition element multi one would employ detect superposition time signal dictionary problem material arise compressive theory signal fix dictionary projection projection reconstruct interested place motivate study alternative absolute regression nonzero various detect sign need familiar alternative versus type alarm detection dimension asymptotically asymptotically understood bad risk define chi simple asymptotically powerful quantity fix alternative mild sparsity mean identity suboptimal require absolute sparsity max review literature mean asymptotically powerful see analyze procedure higher show within detection established explore matrix multiply triangular decay polynomially fast prove unchanged achieve theoretical literature locally setting would test resemble suggest level sparsity manuscript vector become comment paper signal salient end research focus entirely focused whether datum white paper match simulation match term sharp coin mean result cite applicable situation number greatly exceed simple hold assumption convenience since simplifies exposition essential majority condition mean unchanged interestingly number observation negligible sharp sparse soon threshold optimal max asymptotically powerful ratio good give design correlation role orthogonal low clinical trial compare balanced replicate treatment remove correspond model q block dimensional design even concatenation orthonormal basis result apply mutually incoherent incoherent basis e compressive communication application subsequently normalize column unit design fact know rademacher discussion simple straightforward bring moderately general match elaborate carry important moderately amplitude covariate correlate model put differently linear small may signal correspond way false yet full though design triangular rapidly decay result go much sense design far triangular include decay pattern simplify discuss whereas setting obviously involve nuisance nest nuisance balanced interaction nuisance represent clutter noise nuisance result concern apply provide far application space space also column low detect mechanism space obey organize design state assumption pair correlate contain design embed numerical section file provide brief summary subset bold resp resp letter bold represent column vector likewise denote sup distinguish f survival brevity say stochastically denote introduce column orthonormal viewpoint serve warm determine alternative resp early whenever content second orthogonal detection need define definition follow compare max exponent obey conversely sequence asymptotically pt absolutely clear statement understand simply identity high situation concern design clear asymptotically whereas begin matrix weakly correlate depend obey correlation relax main fix probability belong mean quantity computable alternative identical orthogonal design study lower understood sense generating suppose resp sequence asymptotically resp interpret least sharp increase exponent obey op resp asymptotically essentially multi linear reader appear nonempty instance bind aside special impose slightly weak lb turn bound follow random coefficient sign sign occur make potentially vanish study accordance low although obey asymptotically resp powerful design contrast say method moderately set lb say achieve however importantly negligible proposition turn ht multiply define obeys suppose op p op without conclusion state theorem say condition weak ever close related max fact discretization establish basis discretization leverage higher detailed relatively fully grid statistic strong adaptive exponent obey op cover combine correction operating mention high coefficient restriction dynamic amplitude large test assume design first similar equivalent proposition fourth moment reference consider apply design standardized entry effectively design weak implicit randomness carry discretization threshold high design course generate fashion turn orthogonal remain op asymptotically powerful max powerful assertion relationship response covariate mostly correlate whereas achieve maximal magnitude support use fine comment situation denote must accurate slight significance suppose apply methodology biased consider apply amplitude equal contribution complement argument mention case orthogonal design color unknown note treat matrix standard strategy construct possibility discuss signal near chi freedom say obeys draws apply obey valid obeys condition impose remark replace concentrated around correlation portion depend fine give apply impose impose balanced eq design exponent obey conclusion valid divide balanced way linear easily coordinate balanced replicate set balance
jacobian maximizing require gradient source u consider independent write jacobian mix write equation implicit ignore relation follow gradient entirely write must replace expression replace universit de france fr occur paper correct article maximum separating log mixed
satisfy satisfy k ix moment term moment therefore parameter invoke straight since c k minimum covariance cluster covariance take ij clustering recover leaf corollary university department computer university department distribute pattern several identify activity trace statistic wide aggregate consider arise activation contribution activation pattern method learn independent network activity weak corrupted node observation pattern strength gaussian baseline operating signal activation arbitrary activation physical structure signal arise practice consider pattern structure node exist pathway bandwidth embedding structure efficient social weak activation network trace internet chemical hazard differentially microarray analysis social detector aggregate location contain interested arbitrary detection global aggregation testing average node reliably signal strength location I drive signal strength reliably detect irrespective iid random probability limit activation detect measurement global aggregate adaptive fusion node aggregate fashion approach processing consider statistic pattern intractable weak study subtle test adaptive various range unknown investigate test propose level problem assume activation independent result strength must activation tend locate sensor environmental phenomenon aim detect weak leverage dependency activation closely offer also establish fundamental limit structure paper nature pattern paper pattern reflect present world lead method relatively add practical pattern support structure even orthonormal adapt transform basis coefficient canonical scale early method attain activation pattern exploit extremely activation three fold transform hierarchical structure propose motivate arbitrary pattern organize node focus exploit variable pattern effectively fusion summarize snr activation quantify relative method constructive procedure third necessarily priori dependency structure learn measurement rest organize structure characterize learn discuss introduction internet sensor structure exploit enable pattern activity node structure pattern agglomerative similarity hierarchical cluster group denote figure must satisfy agglomerative recover hierarchical several see agglomerative agglomerative produce binary case straightforward omit save collection similarity pair agglomerative figure cluster pattern cluster merge agglomerative basis denote project pattern onto compute result vector possess vanish moment haar wavelet activation merge cluster yield augment compute orthonormal unbalanced haar transform basis construction spirit lee al balanced haar wavelet dendrogram transform correlate result difficult analyze procedure unbalanced haar wavelet sub thus vector set similarity initialize merge unbalanced end difficult transform support contain multi activation constant basis ising structure since increase level pattern group high pattern approximately sparsity govern interaction mind situation activation network naive model scale pattern tree strength sufficiently zero canonical result determined canonical domain small strength propose sparsity pattern group enhance snr additive denote strength unknown activation project yield pattern energy concentrate thus pattern simple maximum bind max pattern draw tree ise model pattern ising strength activation alarm condition draw signal strength polynomial significant improvement canonical structure pattern detect signal pairwise similarity covariance hierarchical cluster constructing propose recovery multi estimate covariance similarity concentration measurement variable possess specifically satisfy condition let denote difference gap covariance leaf kronecker delta true variable affect behave purpose realization agglomerative network measurement need model add pattern imply signal strength vary compare domain global aggregate false false discovery method exploit algorithmic complexity detection unbalanced haar except summary vanish one contain activation agree parent tree edge basis node different value specify pd chernoff dl dl q dl dl sparsity sparsity inactive level level essentially argue canonical govern activate number condition root inactive consider parent active understanding expect canonical follow repeatedly apply similar fact hold enough proceed e l repeatedly apply similar get l l bound canonical invoke derive recursively binomial chernoff q pa chernoff p notice p n p conditioning probability pd dd canonical sparsity p l fact enough establish canonical upper recall chernoff q enough
average vc ergodic borel sigma collection vc denote small vc vc role machine let union intersect complement positive formally depend interest existence uniformly possess vc finite eq q corollary next immediate corollary least need note define minimal element countable remark routine countable weak function pointwise without positive remove clearly give ergodic ergodic ensure vc uniform show ergodic process set ergodic large number property argument uniform law major vc law dimension early work preserve mix mix countable vc countable mixing transformation nn nd approximation cell measurable satisfy follow triangle previous arbitrary countable vc mixing establish uniform vc sampling several follow definition exist countable elementary countable atom atomic countable atomic lebesgue interval equip borel existence preserve ensure generality may let countable finite proof assumption construct fashion sequence th stage sequential splitting set use arbitrarily collection join vc dimension proof contrary k nc great continue split set remark splitting stage let select join measure continue continuous define exist integer generality intersection lb lb lb induction let interval display subsequence set sufficiently lr cell contain definition boundary ii set hold join j let subsequence interval previous j l c inductive complete give dyadic disjoint intersect dyadic remainder contain acknowledgement author
denote instantaneous security simulate market immediately w p definition maker loss rewrite substitute bind market maker show converse market maker must quickly pricing function parameter use equation equivalent use majority mention apply weight state market market market crucial later finance originally different towards risk market return outcome preferred function property convex monotonicity negative translation financial interpretation important provide low allow result informally include completeness semi continuous quantity vector differentiable function decrease monotonicity translation theorem imply convex immediately concave constraint respect kkt necessary optimal q price precisely envelope ensure convex represent strictly differentiable price maximize ability represent form loss maker function correspond convex risk market maker n market scoring market maker scoring cost base regard show section market scoring differentiable satisfy mild market scoring score base market market guarantee receive every change quantity long achievable achievable correspondence certain market scoring function base market scoring market however provide guarantee circumstance satisfied market strong give function market score q class market proper market rule mapping equation market equation market class regular market rule differentiable scoring pair mapping hold market equivalent price market price price every price market strictly rule regular regret another market instead market price quantity share price share solve convex order share share share market price share function limit imply follow expert risk suggest penalty function function market maker bad market maker contrary regularizer necessary market scoring regularize discuss score equivalently majority expert relationship market write equivalently market market correspond descent setting give quadratic market scoring score market base payoff whenever price nonzero cost cost market uniform previously price nonzero maker market market regularizer price apply bn descent match demonstrate elegant connection market market think interpretation market mechanism function assumption way market scoring infinite accept permutation finish ahead either google run naive exploit majority price permutation connection market grow algorithm infinite set price type outcome space fix place integrate calculus dx I iy dx iy dy rearrange price play derivative n price differentiable iy dy dx iy dx dy absolute value every j b j p dropping equivalent kkt maximize price equal lagrange satisfy kkt condition price case eq price outcome outcome positive price outcome outcome denote bn em price price definition derivative price price derivative outcome change price derivative positive price everywhere rgb exactly choice market regularizer show equivalence market market cost function scoring follow commonly study market score aggregate imagine interested united states month fall hour news read informed could potentially save appeal design prediction market outcome event offer security state neutral p security collective accurate practice wide example rational expectation equilibrium offer insight market converge accurate say nothing market mechanism logarithmic scoring might accurate estimate practice insight mechanism deep prediction market come connection market market build proper loss bregman knowledge weight majority regret marker maker majority efficiently combinatorial space show converse maker however connection deep market interpret maker make observed think maker treat cost market time minimize combination cost market regularizer furthermore another market convex base market algorithm insight prediction market accurate describe review market regret variety mechanism market dynamic market broad scoring function market sequential mechanism section probabilistic forecast market rule encourage individual belief scoring rule mutually exclusive exhaustive rule score outcome take line intuitively reward forecaster receive predict relative scoring rule say outcome equality strictly score scoring rule rule extensive nice survey scoring many scoring market maintain enter may rule distribution allow pay infinite amount happen outcome turn possibly payoff score receive payoff payoff proper scoring belief market converge collective scoring essentially responsible thus maker responsible score bad market maker bad market uniform parameter logarithmic scoring affect payoff maker discuss mutually exclusive exhaustive outcome event market security outcome different mechanism specify market function quantity share security currently share must pay market instantaneous security share say price price security price guarantee second ensure price something respectively great quantity security ensure price valid outcome price market current occur function function elsewhere important completeness property monotonicity positive sufficient require monotonic require require equivalent price fix da da translation invariance invariance fix set translation invariance combine bad market maker maker might pay price formulation receive formulation briefly review loss goal almost cumulative expert assumption make adversary every receive loss cumulative maintain weight receive instantaneous receive always cumulative none say randomize weight know trial eq advance yield bind weight choose specifically minimize entropy broader name time algorithm simply place algorithm highly unstable dramatically change weight time next suffer alternate overcome instability suggest random perturbation perturb instead minimize regularize guarantee wide variety regularizer l quantify trade dramatically round range big choose minimizer unique compute efficiently strictly regret optimize appropriately regret foundation describe loss market maker expert market maker treat formally market
nominal understand effect calculation present fit calculate analytically statistic use goodness fit also bias observe value distribution histogram probability purpose illustration present bins histogram carlo true give histogram reconstruct reconstruct pdf histogram pdf divide take true pdf reconstruct pdf accord define simulation initial parametrization pdf reconstruct gx reconstruct monte carlo generate event bin histogram equal histogram reconstruct pdf weight equal gx px histogram convenience histogram px cc result reconstruct monte reconstruct pdf cc cm result monte reconstruct reconstruct pdf monte size calculation reconstruct carlo reconstruct present calculation part measure detector compare histogram unnormalize previous method new experimental permit decrease flexible restriction domain investigate goodness fit describe histogram various bin numerical goodness fitting model detector finite model find minimization experimental simulate example present validate monte experimental simulate homogeneity unfold density reconstruct detector resolution acceptance represent true acceptance characteristic experimental resolution variable pdf unfold problem true parametric reconstruct reconstruct possibility acceptance resolution must majority monte use unfold inverse ill pose solve statistic systematic priori influence preferable disadvantage approach matrix rather element case monte carlo author goodness fit distribution reconstructed present rather use illustration calculate author state degree impossible statistic alternative apply fitting acceptance fit goodness fit error practically run validate event correspond characteristic call weight oppose unnormalized carlo reconstruct weight avoid monte reconstruct pdfs weights histogram eq bin weight quantity estimator comparison reconstruct pdf carlo reconstruct histogram homogeneity represent distribution assume belong bin experiment monte equal histogram weight sum convenient normalization problematic statistic sum extend bin bin unknown find minimization substitution homogeneity valid substitute parametric formula weight dependent estimator find l fit example validate procedure take true reconstruct domain pdf describe
move limit different cumulative sum zero begin prove restriction except correction improvement point represent mass substitute special second consist point feature allow represent continue mean remainder notation appear identically similarly special positivity apply ergodicity positive case region neither recurrent split irreducible positive recurrent recurrent hence irreducible recurrent recurrent form recurrent irreducible recurrent argument proposition irreducible recurrent class period recurrent periodic contain recurrent thus period must reach odd even exception periodic evident give policy enough hide boundary unclear equivalent interval transformation write denominator exactly fractional transformation fractional interior inspection fix least three order fractional point contradiction thus must apply monotonically lemma exception threshold policy policy threshold since contradict information chain periodic period sensible depend appropriate policy without write closed limit give probability exist suffice ps obvious ps q mass transition multiply require exactly denominator exclude generalised case long nature policy limit good calculate limit expect hide natural place approach formula product infinite hence truncation specifically ic update limit entropy c c h c function maxima occur follow requirement vanish series series th combine inequality easily calculate calculate expect unbounded reach desire precision denominator replace calculate later simulation prescribe within computational arithmetic operation call call approach expect simulate drawback work state infinite possibility atom width variation regardless iteration greatly limit list contain location start list least either require calculation iteration computation correspond represent differ method grow indicate limit set threshold policy represent represent threshold simple collection threshold test threshold policy extremely inefficient countable move threshold change figure tend apart naive test accumulation space even subset next previous choose give policy policy equivalence avoid however depend iteration number determine point create circular depend policy tested adapt threshold policy prove order require bind exception policy ingredient bind depend uniform suffice estimate search space finite policy policy check simulation number policy feasible whether include four possibility inclusion round strategy simplify policy policy orientation lie mass policy thus past every threshold terminate finitely policy step require since policy policy generality contain threshold find small application respectively error empty right run next great encounter record minimum repeat primary content optimality analytically description valuable insight six orientation diagram figure demonstrate threshold qualitatively equivalence assign region accommodate datum partitioning set orientation inclusion define precisely depend space policy interval present extend interval policy equivalent illustrate precise definition six figure cm iii vi red accumulation equivalent mass region finitely belong blue equivalent h region vi strict non calculation generate significant extreme value occur half diagonal blue prove autocorrelation expect symmetric policy convex eq equality occur order prove optimal explicit lemma limit illustrate inequality concave symmetry inequality cm increase write identically vanish prove require fact equality attain empirical description provide perform notice unimodal circle prove simplification threshold test two policy policy entropy determine policy uniquely view hypercube infinite space finite hypercube justify decay geometrically tail contribute truncation policy similarly possible look sense change hope since hypercube connectivity region tend find locally truncate pick policy digit cycle give policy entropy leave previous otherwise find pick integer spaced figure hypercube locally policy indicate policy locally strongly globally policy infimum general locally globally new method find pick probability observation suggest slow significantly slow parameter entropy far uniform likely optimum unlikely boundary region deterministic simulation policy previous section consider criterion limit term attempt maximal gain strict strict current method newton allow computational result greedy idea appear greedy easy range discount suffice show close distribute greedy error tolerance average greedy maximum point greedy suboptimal policy h grey policy error boundary region make final globally optimal closeness greedy threshold figure exhibit behaviour suggest indeed qualitatively study thesis find strong policy aim able ergodicity positivity policy expect general policy find author aware paper date incorporate thesis algorithm policy algorithm computationally prescribe error thus room pt process requirement master department mathematics statistics department electrical university cm abstract consider hidden process policy deterministic focus derive calculation computationally almost thesis comprise original towards use thesis acknowledgement hour thesis together observation case model consider observation choose observation information sufficient past accord arise information state special definition limit entropy explicitly rational series limit optimal general finding also greedy reasonably suboptimal series paper name soon speech calculation influential consist observation observation vector simultaneous sometimes prevent observation evident operate mode must another example study must location limited device even simultaneous may restrict availability processor much sensor consist sensor insufficient analyse processor choose sensor receive multiple limited communication decide bandwidth virtual target move possible site special correspond special file channel give yet choose sensor calculation adapt replace multiple observation hide exist state underlie explicitly information choose measure additional choice infinite cost theory model observation markov exist sensor scheduling autoregressive horizon thereby write optimal suggest work dynamic programming markov dirac piecewise norm piecewise function could fine mean state information state observation path consider infinite horizon primary tradeoff use usage cost aim uncertainty measurement work function function prove policy restrictive assumption use monotonic choice observation still problem cost discount sum cost differ verify underlie chain oppose current simplification transform proceed far search move partially prescribe certain observation transformation appear optimal policy extensive state argument provide full however mistake recurrence formula correct precisely define convenience end list model notation consistently symbol uncertainty many choice good naturally real vector structure finite sensible choice natural countable measure accordance fact particular policy converge among policy limit interesting measure region observation observation set variable chain actual variable already ergodicity incorrectly en recurrence certain chain mostly elementary limiting begin find dirac th markov observation satisfie recurrence py definition simplification give satisfie recurrence observation ps nz ty ty ty ty give chain analyse ps ps recurrence z give criterion recurrent discrete irreducible positive limit state chain ergodicity observation process latter whenever observation reach probability recurrent transition many much recurrence anchor pair coefficient definition fix finitely exclude reach model measure outside bound rewrite definition notation integral integral measure integral union map q give since always large
function moreover know normal statement sequel property detail functional eq function cdf characteristic f f infinitely iff infinitely sense r relation allow obtain stable clearly f provide strictly usual sense moreover analogue analogue iid family f straightforwardly transfer paragraph stable distribution cm stability pz sense scheme direct exponential analogue strictly example branching scheme generate nz another family v nz n actually consider probably neither work familiar year aim involve summation purpose balance equality exactly pair approach however straightforward certain typical outline rational iteration fractional subset complex plane call invariant invariant conjugate conjugate ne see e number point condition union dense plane behave precisely cycle shape lie cycle neutral complement cardinality real left behave part generate exist eq polynomial power negative converge convergent non negative chebyshev function take directly property chebyshev odd consider let n generate clearly p chebyshev state eq q n plugging give well actually fact well clearly pdf form cdf needs apply relation represent somewhat property wiener consider see laplace equal coincide cdf let via reformulate kn attain constant therefore standard normal deviation certain analogy representation r scheme summation consider family e way turn iid pz z laplace analytic book example family laplace transform clearly monotone mention book page negative sufficient pair rational investigate family obviously geometric mention hence check generating pn pm omit notation already analogous applicable strictly follow give distribution normal similar transform standard wiener ex plus minus ex plus plus department email grant university school college research centre mathematic provide number add variable also call stable term chebyshev polynomial result class particular stability characteristic certain class useful
smoothed proximal fast shrinkage smoothed carefully summarize scalable qp thereby applicable convex structured regression enjoy convergence subgradient implement smoothing optimize function adopt mm quadratic surrogate fuse penalty detail connection organize formulation group lasso present along connection extend algorithm simulate overlap lasso penalty parallel outline high penalty feature denote lie univariate obtain solve g encourage regularization parameter lasso scenario many pairwise similarity employ induce penalty joint among specific structured kind penalty norm total variation penalty sparsity induce recently regularizer optimization encourage coefficient jointly composite power group group mix play widely hierarchical hierarchy achieve grouping effect comparison alone group precisely estimate use regularizer variable detail prior structural pairwise edge denote proper measure desirable encourage share similar magnitude fuse fusion effect feature monotonically increase correlation positively would tend whereas opposite effect calibrate graph fuse lasso encourage densely select q fuse globally difficulty arise nonsmooth although overlap lasso penalty type penalty nesterov induce penalty use technique nesterov easily calculate utilize induce reformulate problem norm write variable associate overlap single nonzero store memory first rewrite highly nonzero rewrite fuse auxiliary entry generalization penalty optimization norm make optimization use construct view one dd fuse verify know maximum control gap result continuously lipschitz smoothness nesterov pt plot onto suggest show nonsmooth dimensional project surface see compose sharp figure similarly axis sharp remove close equation group fuse proposition project fuse fuse lasso operator define simply approximation nonsmooth induce fast iterative shrinkage reformulate gradient smooth optimization q q large involve nonsmooth part penalty fista algorithm determine overlap penalty fuse penalty compute proximal q w closed soft operation coefficient soft thresholding objective w l step obtain zero e parameter could far accelerate backtracking could dynamically assign operator eq rate ensure iteration satisfy particular constant find predefine optimize rate next solution f dt upper pt theorem part involve bound balance three detail present much subgradient convergence term monotonically objective decrease f objective value estimator eventually depend convexity learning solution solution speed recognize open community main computational iteration calculate share iteration subgradient descent store computation x j per iteration order table time pre storage significantly store insight draw early composite widely problem loss simple nonsmooth achieve consider operator challenge smoothing work entire nonsmooth approach separate nonsmooth norm structure induce penalty enforce fuse lasso smooth complex leave benefit irrelevant feature avoid processing truncation condition predefine impractical optimal induce full rank obtain problem tree expense time boost idea construct smoothing adopt widely mm maximization maximization problem minimize replace difficult objective surrogate bind gap objective iteratively construct construct apply proximal optimize surrogate penalty mm matrix surrogate structure fuse gradient quasi newton smooth convergence seem derive convergence develop various overlap fuse lasso develop handle penalty focus group solve ascent quadratic min flow min flow min min flow subgradient summarize applicability give solver proximal sake computation proximal although propose worst importantly superior addition method active propose involve active alternating involve expensive global rate proximal guarantee proximal accumulate fuse ty coordinate proximal due compute two solve great advantage method update qr propose original version algorithm column add objective especially balance trade author extend deal extend require schmidt initialization extra avoid heavy pseudo practice column solution entire efficient require time consume entire column entire column parameter sparsity prior association goal genetic nucleotide input influence phenotype measurement phenotype naturally guide graph relevant phenotype regression penalty previous application briefly induce illustration matrix output b j coefficient task structure naturally wise sparsity induce multi coefficient output tree structure output output fuse task reformulate tc replace output auxiliary f constant per qp complexity evaluate scalability efficiency smoothing overlap lasso real genetic overlap lasso qp qp graph fuse qp proximal solve function g nonsmooth slow pc gb ram software matlab fair criterion dual gap stop criterion widely objective stop addition maximum purpose overlap lasso receive group incorporate view singleton ease singleton receive constrain smoothing reasonably univariate simulate univariate assume input order adjacent input q pt pt pt c pt report cpu parameter magnitude large unable collect scale reach iteration instead hundred pre terminate third lead moreover notice time show per iteration require method dimensional moderate computation newton simulate identify input phenotype simulate parent panel additional individual randomly individual like structure correlation pair nonzero size relevant input group subgraph sparsity pattern phenotype correlation show pixel regression lasso regularize fuse output distribute input information illustrative coefficient shown regularize multi lasso reveal clear fuse lasso relationship qp solve structured figure select regularization fuse coefficient fixing fix assume correlate input relevant select relevant every three consecutive group datum substantially moreover notice almost analysis overlap cancer demonstrate employ structure induce dimensional regression solver problem datum breast cancer research devote identify consist particular discover tumor growth group pathway gene provide pathway biological analysis consist pathway approach pathway significance two tumor gene canonical pathway molecular signature lasso overlap gene pathway pathway pathway gene gene analyze pathway logistic overlap tumor type gene expression solve adopt vary regularization cc vary select gene logistic regression different along group low overlap belong different pathway figure number pathway select belong b incorporated select overlap lasso pathway coherent easy interpretation gene pathway independently total compute whole second group perform functional analysis tool pathway breast tumor gene marker belong pathway pathway breast cancer protein pathway involve activity unnecessary protein chemical reaction one protein extensively select lasso marker breast previous pathway find breast relate dna protein relevant growth cell pathway notice pathway relate death occur widely know involve tumor pathway associate pathway case functional signal breast cancer among pathway involve pathway functional gene match biological insight
covariance approach estimation minute inverse twice scale general sparsity fuse lasso group contribution aspect reformulate regularize introduce lead term amenable bregman square root positive major consume quadratic eigenvalue conventional solver first compute eigen decomposition quadratic eigenvalue eigen consume solve eigen decomposition contribution crucial bregman definition problem covariance encourage result bregman method solve fast artificial importantly split bregman term provide efficient popular tool variety construct multivariate mean matrix reformulate aim nonzero entry covariance concentration idea behind explain select simple penalty result note entry learn model fitting model treat use require nonzero attractive account furthermore penalty regression able correctly skew overcome principled find maximize log encourage graph n empirical goal find definite tr trace denote concentration guarantee view approximation penalize formulation guarantee computationally challenge prohibitive coordinate descent coordinate descent lasso implement solve remarkably solve graphical specific limitation prevent penalty fuse paper penalize substantially graphical method broad penalty old gain efficiency recovery completion general optimization reformulate interact thereby enable split first generalize matrix estimation include update consume subsection formula step propose solve result quadratic section illustrate total later equivalent closely optimization multiplier method multiplier increasingly becoming regularize inverse regularization term proceed bregman reformulate general use norm net fuse lasso likelihood make auxiliary coupling amenable split demonstrate multiplier admm presentation bregman use lagrangian lagrangian corresponding constraint lagrangian augment lagrangian term augment lagrangian dual multiplier solve size ascent equation solved augment still function term multiplier minimization run step split bregman definite four eigen decomposition root equation convergence define moreover definite converge decomposition converge write govern initial matrix positive simplicity combine newton square htp newton f covariance penalty correspond special applied matrix satisfy solve htp use newton trial illustrate utility first demonstrate propose definite compare performance bregman inverse estimation compare conduct intel illustrate square root mostly motivate equation implement matlab sr upper triangular decomposition exactly list newton solve square root stop calculate precision demonstrate highly converge step mostly constant substantially decomposition example take newton newton next bregman method penalize graphical trial far solve scale estimation code code link language reasonable time trial despite language penalized terminate relative fall addition primal graphical guarantee matter value affect iteration involve empirically artificial gene expression artificial create generate parametrize inverse diagonal positive random approximately inverse covariance covariance graphical testing artificial less time take second solution second evaluate scale plot cpu
sample try new ibp ordering process exchangeable exchangeable customer consider customer exchangeability customer dimension bayesian finite version element model strongly e integrate sparsity prior motivated improve interpretability element increase reduce suggest uncertainty actually aid interpretation however generative adjust constant roughly source source active hyperparameter metropolis hasting marginal datum parameter diagonal upon current denote ibp matrix determine likelihood integrate th loading exchangeability ibp exclude expression column contribute likelihood hold account factor contain row prior mh integrate either new latent matrix high dimension proposal move note simple prior new slow modify poisson multiply accept take collapse sample likelihood term likelihood scheduling improve scheme describe ibp sample ibp conjugate act harmonic remain include completeness sampling column calculate invert calculate prior g variance allow wish share also put noise additive constrain isotropic b b gamma share ad hierarchical prior couple variance variant fa analysis sparse factor bayesian ibp propose nonparametric simply appropriate ibp toward derive literature binary variance calculate white noise give reconstruction permutation rotation rotation average ten sample infer spurious thresholding omit since consistent ten generate ten number refer use fix share dimension reconstruction different latent ten random average ten sample plain analysis bad factor four spurious noise ard improve long occur spurious feature improve perform well factor ard suggest default actually marginally well find underlie also see place ard control finally perform couple share variance reduce example check infer histogram number latent show significant standard deviation noise sampler infer show skew deviation suggest proposal sensible proposal slow sampler reach equilibrium add greatly increase spurious prior assess performance ground log ten percent remove large mcmc split test mcmc run various gene incorporate although ard sensible number predictive log missing indicate cpu second average across cpu pt algorithms breast cancer datum finite sampler measure used likelihood calculate log number ard prevent overfitte improve slightly model comparable sensitive sparse avoid bad order sampling encounter datum nonzero element sparsity model cpu number straight per increase suggest inversion precision ard negligible double finite cost decrease increase overfitte time decrease cpu somewhat feature birth marginally choose comparable sparse make normal considerable hierarchical dp datum sec either individual expensive gene split ten training test number gene slow large gene appropriate predictive performance note include breast capture variability surprisingly worse latent give log likelihood slow breast improve predictive ibp provide sparsity straightforward factor infer choose breast set ard like prior computationally birth death ibp manual finally feasible prior specific incorporate knowledge improve improve interpretability gene expression set share gene perform suffer slow acknowledgment would like anonymous comment algorithm describe consistent reconstruction microsoft support grant bayesian extension fa datum model superposition infinite process ibp use prior sparsity gene datum matrix datum set analysis pca independent ica superposition independent load usually assume gaussian inference probabilistic model matrix pca whereas diagonal ica latent assume tail integrate fully bayesian whose play role desirable irrelevant
involve whether call evidence alternative study expression effect representative observe evidence associate hypothesis hold interpret apart significance define interpretability frequentist observe favor nan hypothesis amount quantify p indicate evidence bayes limitation pose nuisance frequentist practical implication bayes specification distribution advantage coherence rarely choice improper algorithm apply selection extent divide generating proper expense presence result clear belief potential notably bayes satisfy interpretable criterion hypothesis distinguish pearson minimize provide review lead frequentist yield relevant statement actual example remain repeat cover confidence regard observe issue outline continue development approach inference enable minimax without repeat sampling origin source parametric herein reduce probability countable denote necessarily predictive averaging employ originally theory minimax optimality employ oppose accord length family algorithm efficiently density minimax involve value prove problem optimality nf contradiction assume since ratio n v contradict assumption call observed exceed regret worst optimal frequentist optimality average guarantee individual optimality select among terminology favor hypothesis nan interpretable evidence importantly information optimally quantify predict predictor average kullback leibler unknown logarithm choose interpretation logarithm yield broad enough refined scientific except distinction weak mirror originally cf accordingly fairly constitute quite strong c c negligible moderate heuristic evidence interval discrimination amount nan quantify discrimination impractical application typically nuisance regret relative ideal member hypothesis unless additional available biological normalizing denominator family distribution conceptual difficulty overcome replace datum largely depend nuisance compressed statistic compressed also include relevant weighted variance arbitrary efficient weight focus comparison sample equal share single approximate approximate except difference exact complexity vanish lemma since discrimination w favor generalizing except smoothness apply since I discrimination evidence discrimination observe nx xx I x weight satisfy constraint broadly regularity result extended space open bounded I inequality imply consider hold define together weight comparison equation pseudo statistic might expectation consider fisher weight entail smooth single pseudo assign comparison weight il ii weight case study unweighte since separate analysis comparison comparison population address problem biology illustrate weight reduce consist estimate eight site site nan hypothesis display discrimination discrimination weight assign site hypothesis affect feature measure primary question logarithm abundance treatment disease perturbation whether feature strategy prove direct correspondence proportion abundance ratio interest influence abundance level protein breast likewise breast mostly pr group thus compete th display approximate discrimination favor alternative hypothesis hypothesis protein protein denote right bit abundance level differ disease status versus protein giving abundance level protein visually indistinguishable nonetheless effect weighting size example fig display protein panel depend hypothesis abundance differ disease status use nan versus protein comparison little lose except reduce address infinite section interpretation comparison adjust extent tend researcher report medium large especially adjust evidence favor commonly often weak bayes favor hypothesis strongly favor insufficient nan important discrimination
investigation phenotype development biology wide large continue serve excellent landscape protein network responsible growth systematic mechanism operate protein network gene dynamic gene phenotype response activity response growth response root root velocity neighbor profile image mark back numerous algorithmic substitute often manual make develop technique utilize air structure follow throughout image originally production curvature need automate throughput system image software automate prototype hardware software product modular automate application article provide evidence throughput functional protocol biology quantify subtle trait gene quantify trait example development growth challenge insight image wang department usa database engineering department mathematics mathematics ny usa towards computation algorithm software nsf dms grateful helpful biology early root branching root com grid acknowledge genomic deal screen target genetic desirable biological consume article progress scale image gene protein growth trait implicit environmental condition common one typical scenario trait biology advanced number powerful method gene protein collective protein process trait systematic discovery major bottleneck progress trait method quantification systematic dna signature extract distinguish characteristic specie behavior course outline step identify quantify trait subject force orientation growth quantitative trait informative carry genomic signature vary great complexity simplicity visible start life simplify serve noise ignore root texture also part whose diversity growth dynamic become essential pathway article provide extraction root direct growth response water light temperature study area dark tend root propose quantitative proportional use root region root despite molecular sensing signal response molecular response response broad asymmetric formation include system use protein identify mechanism determine explore action reveal response band induce curvature reveal quantitative gene role describe give application protocol attempt quantify subtle gene protein design specifically focus hardware novel automate image component orient software accommodate essentially analysis ready protocol platform model suitable segmentation suitable application derive throughput simplify obtain resolution allow select delta function movement automate occur accumulation surface cover mean scale quality sharp system algorithms iterative recover fine idea et joint variable fixing overcome follow solve euler fix first switch role class convergence appropriate function turn movement slide sometimes occur automate mostly inside article development develop new object orient code hardware quantify feature continue discover serve trait carry distinguish depend argue generation article robust fast root expansion cell root way root growth direction horizontal region call vertical length calculate length make carry growth segment growth growth region geometric primary algorithm extract growth discussion biological significance mention list feature purpose dynamic growth growth acceleration root root htb follow feature representative feature growth vertical region horizontal region segment root length root growth velocity
mix effective three number require decomposition evaluation single intel ghz cpu hyperparameter update however latent algorithm hyperparameter satisfy datum ergodic eventually explore hyperparameter expensive computation covariance covariance hyperparameter sense every ten update elliptical hyperparameter apply elliptical simplicity surrogate hyperparameter six update representation whitening match site approximation moment match approximation use set expansion infinite base slight advantage notable mining poisson work poorly site lead large bin count mining dataset expand give bin zero count work come approximation site investigation discuss recently cox surrogate advanced applicable process fix practitioner poorly whiten posterior site advanced simple whitening serious performing inference hyperparameter acknowledgement anonymous comment publication reflect school computer science process gp popular specify probabilistic consider make carlo sampling careful slowly require incorporate explain generalize specify split class approximation monte easier develop priori observation update parameter possibility sampling slice use wide variety technical little generative distribute science consider summarize covariance focus popular square cm hyperparameter input covariance joint q new different covariance chain operator initial position operator leave invariant z posterior simulate transition leave conditional invariant fairly markov towards target distribution recent work transition variable covariance hyperparameter need invariant metropolis hasting possibility hamiltonian monte carlo cm use square fixing appeal markov slow joint covariance highly hyperparameter especially dramatically hyperparameter conditional quite constant sampling prior fix strongly couple strategy independent commonly normal draw square low cholesky decomposition would principal square square root behave like power instead link hyperparameter update actually change result trend change acceptance instead respect propose accept propose draw solve pt p h ideal weak restrictive strong likelihood ideal ignore update variable proposal observation plausible noise analytically hyperparameter accept posterior applie create auxiliary introduce surrogate guide proposal variable auxiliary true automatically auxiliary hyperparameter possible draw marginal auxiliary process spherical surrogate update plausible surrogate hyperparameter illustrate leave update hasting term far proposal must crucially careful scale distribution slice adaptive search slice surrogate proposal careful axis align hyperparameter move prop surrogate imply u h sec surrogate randomly h surrogate noise plausible region latent update determine algorithm current latent acceptance reduce uninformative current tend base preliminary likelihood individually fit fit one dimensional site noise site thresholde propagation expensive cm sampler move move narrow posterior jointly value hamiltonian hybrid monte tune hmc jointly hyperparameter target replace often taylor look like laplace likelihood approximate pt current pt fix hyperparameter hope likelihood independent chain mix expand log maximum probit likelihood expansion see give flat gaussian likelihood flat auxiliary covariance surrogate
whether wikipedia action abuse predicting time summary median min max set baseline predict majority logistic content provide information higher predict content c european relatively examine control semantic evolution abuse project wikipedia experiment wikipedia article formalism qualitatively predict segment detection document may reduce operation complete remain pixel successful control analogy likely example wavelet filtering field acknowledgement nsf grant static continuously author representation framework experiment concentrate modeling sequence version control lead sequence document consecutive difference consecutive version keep track control book project com wave control document index locally concentrate continuous generalize weighted sequence version document axis smoothing domain simplex vector mapping capture content smooth probability geometrically control document content mix time sharp fourth address field vocabulary integer word v component represent document vector potentially j control document array row version control information suggest smoothing obtain space vector trace surface normalize normalize vector measure around location thus difficulty length way resolve short version need refer word expand expand result display unit panel display non normalize representation fourth panel display field contour represent panel correspond due version track document term normalize length document make track version document dimensional proceeding describe framework low vocabulary word visualize component component redundant control contiguous segment bernoulli segment get segment length increase rate constant array distinguish display nice total version rate segment third apply four task remain union version document word sharp word three change first word neighboring document position within version shift high technical occur substantial change axis measure instantaneous alternatively provide describe describe total position region repeat substantial substantial integrate derivative curve measure dynamically two book space anchor shift version addition removal integrate axis parameterize anchor position measure anchor point synthetic expect segment magnitude display contour high magnitude segment boundary show maxima portion control wikipedia within boundary panel amount version word substantially relatively third maxima topic segment vertical horizontal line coherent segment streaming message email version control segment find edge consecutive study thing control document segment exist view segment boundary separate segment easily segment segment boundary partially predict edge consider segment task predict section edge segment thus characterize high particular vertical transition horizontal diagonal transition across version magnitude l c c language european amazon paragraph summary max question besides synthetic six article european union european version control document wave amazon control include tag stop number author google wave ground boundary separate display maxima control wikipedia articles maxima segment logistic regression min
amount independent bit capacity channel device problem also quantum physics define principle functional equip call preference order utility line separable measure completion probability dirac comprise affine functional make pre compatible axiom shall formalism functional linearity certain system focus problem preference utility quantum function hermitian hilbert space spectrum total pre eigen non operator early relate phenomena capacity often represent functional shannon leibler axiom axiom logarithm abundance metric variation proper interpretation information generalize resource use shall exposition lot normalization perform structure linear bilinear yx xx use fact space however rich algebraic multiplication hermitian element real every call hermitian hermitian form point convex cone dual yy x measure strictly respect action case act identically element primary algebra continuous locally topological hermitian banach sign non generalization example algebra contain contain unit coincide variational generalization represent outside banach unbounded example generally complex value evaluate hermitian accordingly maximization real normalize e measure set weakly simplex represent extreme point geometry topological different I consider resource hermitian element ht package color conjunction terminal explanation load graphic explanation terminal graphics macro ltb lt lt lt lt lt ltb lt lt lt lt lt lt close weak closed also effective domain shall also hermitian problem generalize value observe increase unbalanced unbounde function empty one unlike special follow eq maximization unconstraine problem therefore feasible special f solution fy show subdifferential transform otherwise convex algebraic interior fy element call subgradient subdifferential functional strictly map single subdifferential example inequality strict strictly monotone concave dual concave analogy concave lagrange achieve shall study monotonic sufficient set belong boundary belong boundary closure half therefore write lagrange multipli lagrange define f fy corresponding sufficient via value equation property clearly existence guarantee restrictive first banach desirable measure often banach complete chebyshev objective utility cost function unbounded polynomial generally unbalanced unbounded consideration motivate functional generalize measure value element call thus functional although address coincide support cx cx mean space element space define gradient minimized function q particular thus unbounded existence close subdifferential empty close empty geometrically f origin one dimensional equation close strictly set inclusion fy prove optimality equivalent property include monotone reasoning strictly fy x generating order dual cone x operator order generate cone non negative assume opposite empty positive measure maximize mutual formulate state operator unit algebra hilbert correspond dual space x restriction super act algebra modular generalization conditional expectation g clearly projection physical meaning always completely positive onto modular invariance refer localization absolute order generating point cone family close strictly contain correspond mutually resp resp f x let localization completely operator act convex case contain mutually absolutely mutually absolutely measure order inclusion resp singleton immediately strict convexity singleton chapter singleton optimal equality cone solution generalize distance resource absolutely belong measure belong case simplex functional cone boundary absolute prove fact image subdifferential absolutely constraint next optimality non deterministic transition convexity contain understand classical preference relation maximize problem may problems information kullback leibler positive equal norm element necessarily classical kl case algebra hermitian trace type convex functional clearly maximize mutually absolutely continuous maximize exist element consider module algebra unique additive multiplicative yy yy terminal option graphic terminal need graphic macro ltb lt lt lt lt lt lt ltb lt lt bp p p dash information dual total ray maximize maximize linear functional element distribution figure dual c mutually absolutely continuous figure family simplex understand utility preference relation include attain represent relation achieve maximization discuss axiom theoretic information strict weak axiom ensure generalize problem ensure directional derivative information appear requirement support utility support operator onto complement element restriction onto complement element measure localization measure treat algebra scalar vector subset every affine equivalence localization measure corollary language composite set transition element decision communication optimal exposition function understand classical however unnecessary shall kernel transition joint probability probability eq event statistically dependency correspond deterministic eq one composite joint define measurable understood channel give notion amount shannon marginal theory refer third distribution transition optimization channel kind kind allow joint kind area decision joint boundary interior transition case fp problem b eq deterministic deterministic transition interior minimize interior fy f fp transform equality f relation separate discuss simplex simplex particular one often choose include element fact mutual exponential example qualitative later transformation deterministic kernel section measurable measurable measurable measurable manifold invertible bp bb kernel mapping b deterministic transition supremum uniquely determine countable omit final show express indeed empty constant empty infinite infinite input measurable sequence mapping constant I without probability hold grow information deterministic transition image notion shannon maximize input maximize proposition entropy potentially loose deterministic kernel sense kernel important transition utility shannon belong condition correspond exponential depend exponential use fact exponential write normalizing integral depend energy db shannon observe utility derivative differential independent distribution product equivalent locally compact utility linear difference expression joint mutually utility constant dirac case constraint inequality qualitative rather quantitative illustration information transmission optimal respect measurable distribution denote conditional maximized hand deviation cauchy assume belong element large unbounded giving infinite image finite argument proposition however amount deterministic utility amount exponential mutual utility translation gaussian expectation derivative amount inverting depend differential entropy construct instance expect kernel finite information utility deterministic deterministic qualitatively markov give generalization function model output machine study system transform output domain consider word call algorithm randomize accord computational terminate reach answer terminate reach computation terminate kind positive size resource computation sequence boolean indicate terminate terminate utility computation boolean utility maximization terminate final deterministic input computation terminate onto final word main deterministic way markov transition sequence deterministic correspond markov map deterministic assign transition kernel deterministic deterministic context variational problem generalization utility theory polynomial error utility communication number query oracle class problem utility duality constraint resource relate study family problem formulae relate free channel recover simply define motivation generalization absolute continuity family establish family separability useful context reason absolute deterministic strictly optimal computational devoted procedure qualitative result suggest broad constraint exist show expect deterministic strict kernel well recently concern information hand optimality subject constraint asymptotic output loose qualitatively
extent understand assumption supervise study form requirement reach regularize scheme study availability condition impose c pf assumption supremum norm eigenfunction integral make explain introduction gradient approximate solution equation subspace closely link square pls kernel pls wide standard property consistency provide pls component assumption latent optimal slightly cg study difference risk derive convergence learn empirical operator empirical reproduce property check adjoint satisfy fact adjoint usual conjugate obtain multiplication advantage noiseless version algorithm wherein replace kernel integral latter hilbert differ inner kernel function inner well integrable follow proposition quantify covariance operator operator integral noiseless denote schmidt bernstein second true sharp establishing abstract context minimal number consider cg algorithm operator datum output discrepancy minor polynomial hold datum cg algorithm consider exactly identify provide satisfied imply ensure precisely simultaneously deviation strong principle rule obtaining result fundamental allow obtain ingredient concentration deviation quasi prove theorem unfortunately necessary present convergence take instead respectively deviation form introduce obtain regularization type inequality value apply outer introduce therefore fundamental idea introduce material derive early stop rule true depend situation adaptive rely importantly idea introduce cg cross model show cg ridge include note selection parametrization threshold rule cast hilbert display cg use norm technical justification focus kernel pls pls future present important effort derivation depend prevent go expectation desirable perhaps stop case choice regularization optimal rate theoretically property hold general strongly hold yield cg regard estimation study whole application proposition property I theorem theorem least square overfitte stop combine rate regularity intrinsic map depend factor true function reproduce fulfil rate rate art operator base least square conjugate goal sample unknown depend satisfie assume belong integrable reproduce hilbert ny response expansion adequate regression closeness target criterion square minimization empirical distribution generating give rise equation invertible perfectly poor error overfitte considerable overview support generalization nf regularize inspire inverse popular case component regression extensively cg learn cg subspace subspace euclidean rescale cg iteration correspond early conjugate gradient appealing construct forward multiplication cg response learn situation relatively depend pp assumption weak noise put measure coincide case kernel intrinsic rate determine rate consider discrepancy sequence threshold actually inner consider discrepancy stop sequence reason stop stop step obtain sc hold probability
backtrack proof store store compactly represent scalable proof set proof algorithm probability discuss storing proof like proof derivation tend share suggest try task try originally generalise recursive structure please refer try automated logic essential property store different path token term token common branch distinguish token storing token respectively internal transition may child sequentially become threshold dynamically index far reduce dynamically expand hash table require require reach leaf node storing list proof proof use fact element present store three proof root six token seven share save common store save common dash false represent set reduce view compact decision tree node label two leave assignment child take assign start obtain subgraph redundant reduction possible node subgraphs root path tool translate generation build execute via utility share file replace child ct ii n c I generation otherwise become extremely memory quickly build logical operation successively detail combine create child child node leave subtree translate result occurrence subtree translation graph result ce ed ac n generate calculate sum child node true false terminal terminal return return assign leaf label finally entire probability receive threshold determine end query stop section width execution quite two number program complete check accelerate process skip cache standard database database sample million proof often whether fact database sample suggest good take whether represent compactly three mean belong call fail directly database sampling fact biological extract around know result graph start three comparison calculate exactly use roughly acyclic long testing database store gb report respectively furthermore index database query explanation start measurement take spend proof spend execute whenever meaningful report exact interval include threshold shrink algorithm interval interval sample path k compare algorithm table result apart therefore obtain approximation proof query increase running path code run widely actually depend interface magnitude converge need remain exact implementation use total reach show fast first good confirm top probabilistic rr c c c c graph around edge inference long show probability node lead also bound reach loose bound formulae encounter process hour cause query backtrack heavily version obtain second require tight c c database cover million probability monte large possible imply success practically branch bind reasonably tight interval path c c give query unlikely consider restrict connect two none exact fast carlo reliably c bottom confirm implementation database original part thereby application language represent logic design scale success query example execute database address disjoint already scale implementation pd tight lead focus connectivity query work query database clean background engine interface promise closely pd semantic even though subtle semantic state art system without exclusive implementation calculate know meta implement sum symbolic build program mutually exclusive collect proof explanation mutually exclusive sum logic programming calculation receive increase logic primitive therefore relationship logic perspective hope general language efficient relational already cp elegant probabilistic representation closely probabilistic type estimation logical parameter probabilistic logical closely inductive limit relational learning inference repeat probability query implementation raise problem deal disjoint interesting transformation could optimize program take situation disjoint currently investigate efficiency incorporate like team logic foundation project ci university fc year relational develop extension mining biological fact indicate fact belong sample kind introduce top entity past database logic develop prominent include logic pd traditionally study focus impose first learnable expressive interesting inference expensive answer long inference inference even probabilistic hard probabilistic logic use context biological network edge label biological protein phenotype extract public database link various treat mutually whether belong randomly sample program success succeed semantic well time database however semantic infinite random variable semantic order inference additionally query represent network mutually exclusive implementation implementation contribute success explanation efficiently diagram adapt approximation low success contribute compute bind proof integration algorithm integration use effectively nod semantic algorithms approximate discuss report conclude upon fact clause mutually different program clause add shall use run encode subgraph coin edge set possible denote fact program since background fix definite interpretation define interpretation generalize restrict fact program ground ask node say randomly subgraph edge tree depict figure fact necessary proof probabilistic fact introduce indicate logic program represent time use goal contain b full program consider rewrite entail vice versa explanation contains outline initially neutral proof yet empty remain succeed clear monotonically never change contribute hence pp subset probabilistic conjunction represent construct know pp pp pp illustration additionally encode e formula full low order implement describe approximate query evaluate introduce sort lead success branch calculate explanation proof cf query represent correspond explanation overlap good proof account add edge cd bc cd propose carlo execute existence query provable compute give use stop carlo similar interval context biological represent monte carlo programs although neither differ mcmc method
well unseen kernel hilbert add sort exactly optimize instance regularization assumption capacity although obtain well sense modify optimization algorithmic tradeoff problematic interested scale well amongst practitioner heuristic speed computation sometimes regularization implicitly learn network iterative bin observe phenomenon application order meaningful domain motivate interested great extent formalize address nontrivial graph laplacian special interest vision vector characterization walk base approximation eigenvector one base computing nontrivial eigenvector graph procedure implicitly regularize optimization exactly interestingly identification relax program enter unit formulation distribution somewhat implication explore intuition matrix eigenvector power method start initial compute weak eigenvector expand eigenfunction iteration approximation eigen eigen draw intuition undirected walk associate usual matrix vector let normalized laplacian start optimization problem remainder subspace reference orthogonality statement impact argument check carry language description three walk matrix naturally heat alternatively write denote heat heat thus describe heat undirecte one walk r multiply connect undirected walk transition hold permit iterate walk except top use place nontrivial achieve initial random heat matrix operator spectral sdp sdp x operation program unit vector density let lx objective semidefinite invertible minimize second statement regard assumption statement kkt condition strong sdp formulation keep exposition decide present easy optimal diffusion generalize f hx establish scale heat tr lx conversely give parameter equation parameter parameter determinant function f f appropriately scale version vary ad conversely x hence appropriately thus establish solution give step depend x vary require ad conversely walk proof large body theoretical broadly inform geometrically learn explicitly regularization improve property act level regularization design ill novel zhang yu boost oppose convergence well describe aspect noise eigenvalue arise estimate walk improved partitioning heat call try community community empirically implicit quality heuristic interpret alternate perspective several small nontrivial adopt perhaps surprising associated statistical imagine one method nontrivial eigenvector adopting perhaps surprising identifying noisy motivated us ask recent characterize cluster large property formalize severe cluster phenomenon commonly observe machine phenomenon intractable eigenvector compute eigenvector rank understand concept extend problem though concept implicit extend obviously vector perhaps regularize intractable
map provide implicit key detailed understanding state notation determinant function quantum system formal treatment reference euler product periodic term power concentrate correction resemble surface case rigorous connection quantum nevertheless indicate beyond axis trace admit expression sum formal eq form formal clear depend hamiltonian maps physics physics devote boundary obstacle obstacle mathematical treatment case convex illustrate set three also near arc parametrize horizontal vertical angle velocity tangent circle green blue correspond region forward bound stable manifold latter unstable reduction quantum map physics study discrete construct transfer operator unitary replace unitary quantum map quantum toy system generalize unitary quantum map quantum physics atom dot map popularity quantum map stems much offer quantum classical level realistic flow quantum organization remainder classical particular introduce refer support flow consider tool need calculus operator scale strategy associate e complicated find formula problem pose serious difficulty suitably projection lead hamiltonian e map summarize acknowledgment thank national foundation dms complete author institute advance national science foundation la thank also fig apply give differential manifold reader interest recall physical operator assume x px pp hx uniformly respect compact compact self adjoint correspond assumption decay long admit extension assumption roughly flow energy encode e intersect abstract see classical symbol assume characteristic surface simple like ensure fix define finitely compact boundary eq disjoint uniquely remain neighbourhood neighbourhood smoothly eq always component component application change dx topological due flow eqs diameter partition realize add component make fig unstable dash analyze discuss quantization identify notation map arrival subset call obviously neighbourhood local neighbourhood neighbourhood totally mutually disjoint notice contain also define trajectory fig sketch definition trajectory boundary arrival contraction imply boundary group call view sometimes map support structural hold flow component quantization family mean thing r n symbol necessity come slightly fouri due exponential operator characterization single operator value quantization integral comment term simplify place confusion context involve function compact set norm possibly neighbourhood symbol make notation finally complex construction affect formulate present material reference proof start symbol quantization calculus class quantization exactly symbol symbol projection main symbol symbol admit power symbol operator principal symbol admits call notion symbol unless norm norm recall operator uniformly characterize symbol say q intersection space concern essential support function wave front replace always satisfie characterize individual family operator h exist specifie say asymptotically q l operator set w st fourier globally fourier global integral operator dependence example family unitary family dependent field real value symbol canonical transformation graph denote usual lagrangian unitary canonical graph locally unitary integral operator since canonical point integral class form operator characterization fouri near origin always possible family neighbourhood construction unitary cutoff operator associate neighbourhood unitary open define neighbourhood say unitary integral neighbourhood equivalent unitary integral construction instance contain self principal flow generate fix energy zero transformation r hamiltonian quantum operator certain space possibly associate neighbourhood fouri integral operator r compose briefly see reference independent operator totally follow coefficient analytically outside operator operator r r simplify r consider existence implicit z analytic region reason property come symbol identification principal symbol outside enough convenient explicitly long near infinity intermediate compactly infinity instance potential direction consider value function calculus functional self adjoint arbitrary write q put simply put equivalent govern property symbol show convenient notion give lead equivalence ball inside past future field need set interior time bound outside neighbourhood rx ik enough neighbourhood rx set notation eq complex neighbourhood drop transformation review result unitary compact direction fouri integral well unitary cutoff neighbourhood define map converse solution flow fouri integral segment relation associate operator smoothed region hermitian inner bring neighbourhood fourier operator symbol inside time terminology ignore symbol write fix writing define similar expression project well neighbourhood cut outside compatible extension solution fourier q property define operator integral l equip want solve solution show obtain integral operator operator correct ensure identity solve system finally solve full achieve notation see concentrated cutoff flow consider problem concentrate neighbourhood neighbourhood aim constraint precise slightly short intersect intersect fig neighbourhood near neighbourhood condition fulfil thank resp dash contour thick arrival connect dash two trajectory inside extend fourier cutoff set operator concentrate take set maximal e satisfie estimate cutoff near arrival due reads parametrize obviously define operator k formula solution k linearity problem let back operator relation identical composition relation associate graph coordinate jk unitary neighbourhood near forward consider property cutoff effectively unity ensure jk intersect concentrate notable state solve inside neighbourhood away procedure notice summarize concentrated neighbourhood datum full homogeneous solution concentrate course reflect neighbourhood unstable describe proposition problem care norm auxiliary difficulty remark end modify norm weight construction scale long q appear satisfy appear discuss still neighbourhood neighbourhood describe scaling weight near local g integral weight depend expansion invariance weight check solution analogue cutoff cutoff symbol calculus jump near return h k respectively argument carry estimate satisfy iteratively symbol change norm hold definition n estimate transform globally require transform acting take place auxiliary subspace compose near invertible construction appropriately build weight construction away modify take allow realization operator obtain modification describe place near let let neighbourhood eq coordinate last independence simplify make trajectory segment outside strict outside particle remain next function define auxiliary element hilbert q globally proposition construction nice space advantage operator suppose support small enough hilbert space satisfy set need adjust hold belong necessarily hand increase see inside layer neighbourhood arrival one appear set inside time outside symbol lead symbol choose know fix neighbourhood negative spectrum together diag g r reference calculus instance globally mod select realization restrict realization operator define put together short space prove suffice inverse rest devoted component operator act cutoff trajectory old project side negligible notice property split component arrival set rewrite data state arrival remark replace operator drastically modify hand space precisely bad case cutoff require ratio enough indeed small inside put crucial rank counterpart nz gx property jk z j trajectory connect jk disjoint support segment symbol multiply decompose near take eq solve calculus spatial cutoff q state adapt small enough cutoff outside norm eq remainder v h occur decompose cutoff q q remainder gm estimate
microarray gene investigate linear publicly microarray set assess randomization expression kf st decode human genome molecular genomic essentially within genome simultaneously provide far small take impossible g logistic regression microarray microarray important interpretation microarray microarray major topic induce classifier e diagnostic learning learned predict diagnostic unseen sample design decrease increase grow refer phenomenon classifier dimensional pool covariance discriminant lda hessian package fail call dimensionality microarray irrelevant gene gene maximal discrimination induce use procedure gene individually advantage low alternative dimensionality dimension extraction aim dimension reduction extraction microarray partial square pls inverse sir microarray investigate area decade paper address fan important future gap propose handle high microarray prediction conceptual modeling begin quite recently recognize variable relate set observable observable variable child intelligence variable assess intelligence ask wider include assessment originally probability correct trait ability intelligence participant response item gene row biological expression gene factor idea approach gene gene e factor account gene similar biological factor protein element factor gene measurement relate gene gene factor meta magnitude prediction carry main microarray gene expression validate parallel pca propose expression step response gene publicly microarray set contain sample gene sample gene tumor publicly filter intensity consist gene expression pi preprocesse datum procedure handle gene cpu consuming require many percentage expression gene two feature selection consist gene experimental set test measure significance change expression define statistic statistic briefly subsection short overview original expression datum trait item person item person outcome denote latent item describe trait produce specific response person score latent score score total person calculate calculation find terminology item gene see imply functional gene vice versa determine factor identify cluster partition gene partition profile mean discover describe need compare datum partitioning gene partition calculate score index let factor sample partition specific microarray dimensionality datum much variation original transform original variable sequentially covariance combination coordinate direction amount original component variability computation vector calculation decomposition eigenvector principal provide linear specify knowledge procedure rule reduce low dimension describe avoid use latent analysis pca factor constructed consider formally consist sample test class sample denote rule implement th sample profile k reduction step short predictor matrix lda assume discriminant assign class widely lda relie assume hypothesis satisfied set demonstrate performance complicate calculation reader factor meta give prediction sensible criterion purpose subset select one sample extraction fit use class subsection successively value run response class otherwise denote select experiment base class randomization evaluation create sample sample learn datum gene gene one method matrix subsection latent reduction loading factor l gene predict subsection repeat step whole predict indicator use measure probability bias skewness overcome receiver roc reader perform false specificity various result use varie class sample two proportion consider acceptable excellent computation carry statistical graphic generic gene function random splitting perform conduct function mass roc interest application microarray preprocesse apply gene subset randomly gene procedure set determine component performance prediction reduction randomization trial value parameter deviation roc see decrease increase size subset predictor include model building pca dimension excellent h legend lda base prediction pca factor present subset accord statistic gene extraction analysis describe roc curve compare method score excellent performance agreement hypothesis classification accuracy low legend mean component area roc data tumor random approach learn class method trial statistic pca prediction random roc pattern performance performance prediction generally lda legend lda error area roc yield figure method regard roc perform excellent legend lda
semidefinite keep considerably vast field optimization program familiar sdp constraint component format regard convert semidefinite constraint augment translate linear every cast finding cast program duality apply semidefinite decade thousand considerably sdp solver toolbox available translate level semidefinite program involve function mention semidefinite whenever union satisfy fisher introduction know space finitely state semidefinite compatible order design characterize optimal problem semidefinite programming support fix optimal assign allow problematic reason support design intractable support semidefinite optimal semidefinite semidefinite finite interval rational rational respect design subset zero polynomial degree whose aside matrix translate matrix semidefinite program example extension proof detailed demonstrate optimal translate investigate high degree rational source solver computer ghz core intercept consider illustrate approach verify semidefinite optimality scalar order hence entry constraint whereas correspondence summary drop subscript anti turned plug computation semidefinite solver explanatory toolbox trace represent high level translate solver leave work variable variable pi minimize pi pi pi toolbox index omit brevity optimal polynomial whose root eight e time polynomial model characterize design follow semidefinite semidefinite scalar first translate compare system equation along real polynomial high designing degree stability scalability high rarely ill polynomial fisher becomes ill condition considerably lead ill condition orthogonal basis look optimal polynomial ordinary basis st tp tn solve program computation second single root polynomial polynomial allow subject rational measuring measurement intensity obey square intensity source total estimate affected source detector measurable measurement detector semidefinite take yield point obtain zero optimal design solve useful zero polynomial might semidefinite simplify essentially calculation set w infinitely correspond design even trivial nonlinear rational model response variable rational function explanatory parameter consider class recently family rational consider satisfy many rational class fundamental non experiment optimization correspond immediate fisher whose estimation design guess useful sequential design concept nonlinear consider readily notation matrix write q finding equivalent linear rational applicable simplification polynomial find easy fisher put parameter nonlinear parameter find design associate derivative fix always one observation optimal chebyshev assumption drop criterion also consider rational rational express finitely constraint computationally work include truly generalize estimate number support root interior chebyshev optimal design linear rational space finite interval finite symbolic form limited interpretation rely effective method whose design applicable problem involve rational treat general generalize rational finitely interval generality far achieve symbolic solution generate program high trivial iterative base semidefinite theoretically guarantee low find globally considerable importance impact study insight demonstrate flexibility robust handle condition analogous generalize linear rather rational fourier polynomial may design nonlinear symbolic question remain extend criterion transformation find equivalent semidefinite representation chebyshev natural importantly idea identify would extremely multivariate location signal source close solution aside involve concrete applicability far study nonnegative know polynomial functional constructive claim apply let exist semidefinite satisfy polynomial semidefinite detail computation reveal characterize semidefinite semidefinite assign semidefinite notation may cone relaxation multiplier suppose admissible lagrangian dual attain finally optimization write eq aside constraint variable rational function variable multiplying denominator equivalent inequality give optimal optimum last way concentrate pt em theorem remark department engineering management sciences rd il linear polynomial rational rational treat zero optimal generalize previously result incorporate various establish previous optimization problem example consider polynomial discuss find locally optimal design rational extension corollary find considerable instance software paper optimal design polynomial rational error model rational rational error normally denominator estimate uncorrelated linearly characterization design translate numerically readily characterization every involve polynomial rational give rational handle run practical optimal design certain model find many component estimate entire identity design mean design show devote finite convex even difficulty arise characterized root support design discover ordinary constant zero degree design polynomial classic determine optimality criterion quadratic discuss design website comprehensive maintain considerable interaction give design odd term multivariate problem cube design incomplete polynomial even general weight consider rational union interval base orthogonal polynomial chebyshev rather scope proof numerically method compute design bottleneck carry finite support canonical carry relatively polynomial concerned univariate np restrict known difficult degree applicable attention turn design comprehensive oriented viewpoint numerical method variant variant classic gauss free maintain support another well current support idea serious drawback care abuse variant know iteration fact iteration coordinate problem iv necessarily converge satisfactory experiment report cf propose model rational optimization involve solve point exist numerical coordinate approach computing coefficient polynomial zero formally derive parameter illustrative case nonlinear involve mostly polynomial stand polynomial denominator function frobenius since decision operator unknown write avoid product adjoint identity cone positive semidefinite denote finitely support notation integral respect order end design call short compatible call small denote arbitrary extended value function finite difficulty design discuss definition summarize affine characterize inequality semidefinite set equivalently allow motivation
theorem eq write bound eq writing slowly slowly vary converge zero rapidly give rapidly generality clear long slowly x nx large eq n x l l writing since eventually decrease hence right asymptotically almost surely theorem write almost since expansion decrease decrease thank david lemma university college primary variation residual survival von limit extreme consistent robust new good see operating explore substantially set introduction tail play important finance pure tail ordering efficiency location value asymptotic maximum medium long tail fr standardize right classified classical categorization law medium may shorter thus distribution classification attempt order work middle ordering see paper propose pure tail pure distinguishing use popular one various behavior tail behavior base residual life hill argue qualitative hill include undesirable behavior hill plot concern restrict tail hill provide apply hill distinguish tail may sometimes hundred thousand large quantile tail actually exceed counterpart tail characteristic simulation work test hypothesis result focus tail consider take hypothesis medium tail distribution long represent statistic sample work section develop discuss relevant discuss alternative wise simulation result type good property comparison conclude residual good type error close gamma gamma medium good interest breaking base argue quantile eq leave inverse tail define short whether indicate write shape let left associated relationship approximate turn restrict monotone density discuss distribution quantile table interest major drawback classify issue estimate exponent use hill early u condition standardized classify come constant estimate analysis possess operate medium tailed sensitive medium separate definition rate refine rp classification q short medium medium medium rp normal short medium medium medium unfortunately rp distribution require value tail behavior es second data quantile interval define define ii iii zero mean connection rate exist pareto infinity failure go infinity converge positive value make rp es property slowly consist rp short medium category rp medium medium rp medium es short fast es medium es slow depend shape distribution medium tailed medium es distribution asymptotic extreme still precision provide sense include medium short long failure make precise es utilize medium tail result residual provide membership definition xt ht function always exist life example consequence corollary hereafter denote mean exp x class behavior es medium testing medium tail medium perhaps medium distribution underlie see long tail therefore pareto es long refine classification asymptotic tail several cccc exponential medium medium medium medium weakly short super cauchy moderately long super pareto long moderately long weakly medium medium medium short medium moderately classification extreme residual scale henceforth tail sense survival function tail behavior arise medium tailed exp unknown need asymptotic distribution medium function slowly nuisance limit estimator fy multiply divide parameter limit statistic es medium must verify medium tail turn fy medium rapidly whether instrumental prove tailed vary fx c cx eventually tail alternative short tail asymptotic simulation previous asymptotic property long short type distribution rejection gamma shape type es tail choose sampling table statistic n x desirable rather reflect es medium tail power statistic es tracking es short distribution serious misclassification serious misclassification classify versa short sample survival great expect l gives state es decrease w w value perfect detect power unfortunately extreme case percentage classification simulation reject classify tail lack table es tailed percentage desirable slightly sample excellent l introduce among es medium tail good distinguishing significantly show capability distinguish address differ find test power short normal approximately block give rise block subsample hypothesis medium sum hypothesis hypothesis es medium tail short tailed favor reject favor percentile size otherwise block large sample without error en stand exp stand table detect tail substantially ccccc exp exp block l l distribution show power short tail sample norm par tail classification classification par increase correct desirable extreme increase percentage similar x exponential selection drive trade power table tailed distribution consideration behavior small level level tail sometimes drawback error gamma medium parameter gamma favor type distribution long tail favor reason since center sequence achieve whether small manner shape quantile shape quantile implement reject close medium medium tail test statistic similarly medium high distribution quantile gamma shape observation gamma cccc cccc cccc devote claim european million adjust discuss exponential right great right tail confirm long behavior plot test tail long tail million shift subtract million claim hypothesis es medium tailed thus tail break strength appear plot tail fall exponential right tail give therefore hypothesis medium tail favor thousand datum fit histogram q suggest tail subtract observation test statistic yield medium f x f nx pz pz pz n pz c pz pf n n x f f ff
noise tend remove difficult actual bernstein establish integrated give mixture satisfy involve average consequence procedure study scheme f integral consume part form absolute taking absolute additional logarithmic relevant proposition ny nx differentiable elementary differentiable get nf jx side na r pf f display useful page let random eq lk lk nc lk display yield integrate get next jensen inequality basic eq give q display obtain obtain successively datum eq argument take j definition lemma http edu support pour difficulty consider achieve good performance minimization computable design establish rate achieve reasonably mild performance result measurable noise know expectation denote define pf p empirical integrate independent choose may treat question rp consider design rp construct index nonzero unknown performance consider penalize minimization pt eq penalization bic pt combinatorial consider computationally feasible example cf chen pursuit vector allow lar establish establish oracle inequality restriction recently propose minimization propose zhang alternative motivate performance either theoretical design van de survey simultaneously taylor framework adapt study condition high weight pac design establish excess risk study exponential first sub estimator initially set note estimator compute fraction consider aggregate scheme datum splitting splitting necessary least dimensional may deterministic pac develop framework inequality excess aspect monte reasonably introduction prior hasting organize procedure oracle design modification oracle excess noise n variable aggregate define suggest prior variable aggregate establish deterministic design display bad notion estimator display combine achieve ep section rate
find subspace generate discuss track motion condition extensive hybrid handwritten digit affine vision motion rise refer subspace tracking segmentation frame object affine camera lie equivalent cluster subspace prove object cone accurately approximate low linear subspace face subspace require partition datum I unknown dimension manifold kf direct affinity agglomerative spectral curvature appear recommend require initialization need collection lie underlie require propose geometric local model component appropriately size global hybrid suggest however neighborhood adaptive sized need recognize quantify study local fit flat art nearly deal obtain art motion particular outlier justify complete particular valid analyze ssc additional restricting modify rigorously quantify main precise local approximately find good neighborhood different extensive datum benchmark face handwritten database datum show particular synthetic extremely previously mention method face indicate fundamental suggest fit heuristic show heuristic quickly determine state find test compare hybrid segmentation video handwritten digits recognition fast discussion describe method capturing optimal neighborhood approximately find neighborhood justify capture fit optimal neighborhood radius neighborhood small around around hybrid consequently local flat neighborhood around underlie fit equivalent though common refer possible guess fix neighbor reasonably adapt look large neighborhood neighborhood dimension neighborhood become approximated sense belong enter neighborhood onto flat numerator flat utilize invariant optimal neighborhood precisely neighborhood near neighbor increase neighbor check neighborhood neighborhood search summarize nk kt kk try justify around r b underlie mixture arbitrary choose adapt probability appendix analog ball case l measure let minimum weak often sort computational preprocessing represent point complexity note limit term replace good candidate describe tuple near flat randomly list pass minimizer pick htbp local scale cluster pt generate contain random random flat power outlier square minimize indeed require evaluate configuration example strong robustness one follow related subspace datum choosing fit arbitrarily thus scale svd decomposition represent since cost complexity come complexity scale requirement organize size candidate storage candidate subspace find neighborhood fit neighborhood finally spectral matrix algorithm replace matrix multiplying corresponding root eigenvalue remark change use euclidean parameter default disjoint approximated pt em normalize accordingly step default input choose similar ssc affinity th entry affinity ideally affinity expect underlying subspace cluster suggest far choice close point segmentation embed small small norm row fit subspace neighborhood moreover force applicability base try enforce practice operate differently though ssc base calculation step limit replace storage exclude algorithm experiment notice without much noise allow allow technique tailor ms r r c kf oracle pt ssc r pt pt kf ms pt pt conduct artificial verify situation accurate mode discuss correct misclassifie exclude change trade accuracy large balance though perfect intel ghz section run intel ghz gb kf agglomerative matlab code kf http edu http www edu db http www edu http edu code motion http www edu http www ssc try modify tailor step refer ms motion segmentation mean ms matrix wise initialize guess membership initialize pick tuple follow kf guess since kf tend kf record run ssc algorithm algorithms oracle oracle noise construction speed database motion http www edu contain frame videos frames video formally frame due camera move distinguish feature background coordinate camera object across frame live affine subspace implement ms min min motion implement subspace linear manifold subspace learn early cluster segmentation copy www edu report ssc ssc ssc ms median misclassification two experiment misclassification record table demonstrate figure misclassification misclassification ssc twice explain ssc ms slightly well ssc besides ms ms run fast ssc ms ssc many energy true ms work good data information spectral combine adapt ms total two advance parameter paragraph ms ssc negligible randomness ms random comparable deviation test ms ms ssc face database http failure extend face database consist image vary condition image repeat randomly face variable lie dimensional nine database roughly slow well also provide example follow http co software table failure two factor sparse per dimensional relative consequently span cluster neighbor closer near distance whereas term consider subspace ssc find discriminate direction irrelevant classification dealing whiten remove exclude reduce greatly become whitening htbp whiten whiten ms voting ssc pt htbp whitening whiten ms ms ms work available http com work contain digit digit reduce dimension rich process record misclassification htbp c pt ms voting pt ssc htbp pt voting pt ms n pt ssc c ms ssc ms ms ssc misclassification misclassification ssc also good ms fast mnist explain use return algorithm point flat among determine add find logarithm e apply ms ms ms part artificial intel core ghz gb artificial user compare ssc artificial http edu sample unit cube corrupt deviation four restrict subspace separation let voting majority neighborhood heuristic true distortion find pca choose try pick obtain low algorithm part high ambient try idea nevertheless well report repeat speed find second htbp c c c minimum ms voting ssc good local flat heuristic distortion help reduce vote artificial option computation reasonable obvious face detect number ambient subspace affect experiment repeat whitening rate correct cluster record ms ssc perform ssc small difficulty indeed b ms vote perfect detection whiten ambient include intrinsic respectively tolerance repeat due error table htbp ms ms voting ssc c subset ms vote ms ssc large real datum except ms ssc demonstrate geometric fix neighborhood neighbor well flat flat continue stop parallel design favor neighborhood data three outlier favor sample rate neighborhood adapt neighborhood average run neighborhood wrong
take induced generalization orthogonal orthogonality follow coefficient function metric k dependence last implicitly prefer dictionary dictionary question arise probabilistic consist value outside dictionary function cover dictionary depend unit hilbert sphere sample dictionary sample length fast draw eq certain generalization representation give straight proposition draw result rate bind let least minimize right dictionary signal high infinite reconstruction treat magnitude feature element inner fulfil representation hilbert concrete word let count well bag imply reconstruct content could decide dictionary bag approximate dictionary suggest minimize eq act norm induce combined cover dictionary reasonable eq product describe remain cover order cardinality cover number generalization prove section restriction orthogonality dictionary rely lastly recall cover number definition wish introduction application generalization cover da sd note banach formalize cover cover lipschitz next clear metric function start define norm maximal bound certain induce qp I geometric representation dictionary induce representation error representation set another unit representation I number symmetric scale point sphere eigenvalue norm theorem eigenvalue particular inverse scale linearly eq norm lipschitz dictionary lemma whose existence conclude next lipschitz constant subset dictionary h least signal assign dictionary basis arbitrarily construct identical covering result need appendix let cover supremum least class function note cover metric minimal arbitrary member ef mf ef mf supremum choose inequality mf ef mf mb mb dc mb probability bound cover substitution metric quantify orthogonality dictionary uniqueness sparse algorithm dictionary proceed discuss elementary proportion dictionary orthogonality term product perhaps simple coherence product represent expressive power illustrate restrict exclude disjoint whose inner first extreme mean orthogonal sign well generic dictionary half proceed prove overcome maxima possibly unit uniformly light variable choose rotation maximum find eq q clear scale linearly triangle tight case estimate dictionary kernel show complexity property simple kernel present feature simple signal unit unit apply nothing representation corresponding representation near proposition euclidean norm dictionary dimension practical issue order address computational generalization bound consider signal represent metric denote show feature constraint pose difficulty beyond common depend dimension simple approach lead weak case space instead use representation dimension consideration map old hold relevance old map enough old old map dirac counter two map geometry sharp kernel older use old condition x root mapping define proposition give cover theorem representation old old cover dictionary sufficient substitution several implication language penalization sparse property dictionary low relaxation mistake g present exclude possibility upper large trivial future exploration view favorable geometry encourage acknowledgment thank partly european community fp agreement appendix justify growth rate instead combinatorial effort constant tight concept need body vector star shape closure sub root non decrease rademacher average integral xx xt te u du tail particular qx qx qx xx xx xx prove core decay complete lemma replace expectation upper entropy observation r applying since f dx department electrical institute electrical institute technology element application denoise separation previously unseen example magnitude assume distribution question generalization learn dictionary expect regularize dictionary level orthogonality assumption strong yield rate oppose localize complexity provide similar set smoothness requirement technique sparse representation combination detail measure often measure vector coefficient determine induced call representation generalization extent
minimization admissible path solution compute relate chain minimizer replace minimize maximization solution maximize bring newly elegant indeed recursion solve unconstrained optimization term risk notational hand simplicity consistency hand involve observe immediately generalize state solve recursion sufficiently small actual probability decode motivate impose positivity easily viterbi j x eq positive prior reduce back clearly start would arrive generally treat previous consider also equivalent newly introduce map viterbi decode combination maximum prior decode every admissible perhaps solution minimize generality assume exist admissible path risk suppose must imply hence imply viterbi decode almost surely take fairly nonetheless x tc tc admissible solution similarly almost surely computed solution find every next let follow score terminal path state completeness break slight abuse notation maximizer let yield every allow interpret recursion hence maximizer form recursion generalize far follow risk problem remark posterior viterbi decode surely decoder viterbi solve refer decoder distinguish generalized accuracy decoder characterize solution function fa ga fx x state completeness write b ga fa assume c c r lemma insight well drawback paragraph elegant ordinary concluding recall adjacent minimize derive loss usual maximization decode path natural think minimizer move towards viterbi indeed minimization viterbi decode experiment minimizer point monotonicity allow I use unlike grow block away replace eventually viterbi certainly code idea well long block result guarantee another risk positive integer generalization first since let q gives hence also divide complete result context path risk risk markov homogeneous priori apply risk decoder viterbi hybrid significance show solution compute fashion theorem risk marginal along inequality monotonically decoder embed risk family q viterbi clearly within case recall end reach sufficiently obvious recalling inequality corollary viterbi since would generally comment definition decoder inequality divide r vx result consider interpretation entire possible least viterbi inference risk approach mainly publication dedicate inference apply transformation clearly lead decode return break power transform nonetheless let function assume j propose return viterbi point motivation necessarily viterbi ignore viterbi symbol viterbi implement viterbi decode symbol sense emission emission sequence nine path viterbi path px symbol aware suboptimal actually viterbi absence symbol symbol uniqueness idea intermediate decoder time attempt viterbi follow map continuous verify function laplace calculus logarithmic identifying recursively backward respectively notation produce viterbi latter transform appear proof lemma notation match lemma leave implicit introduce state part start equation error contradict return essence transformation viterbi clear besides parameter transformation limit special reason really wish idea moreover explain hybrid also subsection become operational subsection operational modify decoder practice transform probability vanish see f moderate length soon exceed dynamic double precision bad computation chain indeed appear function use trick remark intermediate logarithm however transform I mean become transform generation store irrelevant trick require actual value course function operate resolve worth effort complexity issue hybrid jacobian refer via recursive generally albeit expensive commonly k mm irrelevant compute transform forward backward variable respectively claim operational decode trivially state symbolic toolbox machine hybrid decoder base already decoder base magnitude use practice example verify realistic practice symbol decoder vanish joint p ti I ti I py I px scale aforementione emphasize viterbi practice ordinary hmms secondly potentially principle redundant bank secondary structural atomic http ac uk hide represent realization position observation symbol distinguish four enumeration final seven class number five eight long specifically first long similarly last prediction protein divide class go likelihood transition compute situation parameter value repeat leave stationarity evidence appear short display position split piece truth row decoder gain viterbi subsection secondly accuracy transition state five respectively output member posterior viterbi legend monotonicity hybrid illustrate risk viterbi indistinguishable constrain monotonicity viterbi constrain row indistinguishable differ capture row probable neighbor likely illustration idea blind priori pointwise generalize matrix absence ht risk c additionally instead dataset initial emission realization simulate necessarily degenerate risk operate reasonably realization subsection stands give parameter underlie population could risk analytically simulation reality complicate specify cv output wise obviously reality cv rely likely report variability merely observational validation average sensible bootstrappe possibility simulate synthetic hmm distribution freedom margin viterbi round accurate less viterbi e obtain subsample consist realization appear notably viterbi decoder accurate decoder extreme difference short position shorter subsample confirm decoder decoder viterbi examine sensitivity histogram subsample consist subsample extreme histogram suggest replace viterbi decoder examine risk difference largely unchanged minor increase subsample position risk viterbi validation viterbi gain viterbi respective pointwise viterbi come accurate viterbi little accurate viterbi respectively replace risk compute viterbi six next log return avoid switch display return bar bar constrain new viterbi path constrain viterbi decoder return decoder realization monotonicity base notably admissible viterbi simulation compare measure posterior recall block parameterize give viterbi continuous parameterization case display performance member family identical remarkably blind viterbi circle classifier call evaluate sequence write besides interested risk viterbi viterbi path viterbi decode viterbi hmms ad viterbi possess ergodic viterbi see converge random risk imply risk appear actually proof risk help viterbi inference misclassification rate viterbi decode thus viterbi make error asymptotic risk theoretically reality however asymptotic expectation risk even viterbi decoder long run universal prove upon theorem risk risk decode possible viterbi nice prove completely probability prove minimize could prove risk combination three constant intermediate case together long vx predict outline say constrain decoder constrain viterbi decide like risk x tt active word computable latter measure x line risk member multiply prior risk vector transpose provide state undesirable transition recently loss consider provide generalize follow perturb version assess stage exploration want use output produce loss instead one risk ordinary risk combine family risk transformation minimization immediately member multiple readily corollary chain represent markov proof path extend annotation already mention namely assign problem solve already labeling priori admissible positivity class extend loss generalize incorporate since successful generalization offer possibility potentially arithmetic ts ts ts certainly risk dependent tuned cross generalization present extension base e could author grant nr sf visit ii author grateful thorough review work suggestion dr manuscript point subtle mistake mat simplify emission alphabet symbol idea relevant situation emission alphabet big emission continuous emission distribution unconstraine return decoder positivity return path despite prior probability zero posterior decoder constrain path estimate hmm dataset describe l decoder probability l decoder decoder pt hmms paper hide primarily posteriori viterbi path pd around dedicated application decade careful several problem issue propose practical furthermore simple hide show compute present interesting pd algorithmic interpretable bioinformatics task generalization admissible path hybrid interpolation accuracy symbol posterior decode viterbi besides application process communication reference become computational bioinformatics security hide field model influential spatial image segmentation hide idea year success distribution system possess albeit whereas marginal markovian posterior independence observe lead hide naturally special hmms particularly fact indexing realization relatively straightforward computational approximate technique configuration hide field posteriori efficiently dynamic name simulate anneal useful semi factorial applicable extension exposition ordinary adopt respectively convention commonly let include brevity cause ambiguity write matrix hide markov regime non emission write shall assume emission borel measure count integer often stand observe unobserved hmm notation addition largely function require argument shall expression stand stand unconditional call datum segment path view segmentation region label far solution digital literature problematic path viterbi viterbi viterbi commonly think viterbi posteriori viterbi path may reason sub expect viterbi path typical indeed might probability add representative general context viterbi path particularly parameter transition probability probability coin hope typical realization maximize individual posteriori well maximize expect characterization estimation mode biology know decode pd successful posterior viterbi wide biological application dimensional consensus absence centroid communication hmms largely influence term e decode viterbi decode natural question improve maintain computational viterbi inference negligible conclude remark make though decode optimal rate viterbi conclusion binary leave room inference differ gain clear viterbi fact viterbi writing application viterbi use return systematic viterbi apart aforementione soon version article matter example sense maximize time possibly viterbi easily subsection besides infinite subsection constrain decoder path specifically call admissible explain prior note form aggregation force path appear possible course understand chain refer decoder admissible constrain also early ignore distinction posteriori admissible protein rise latter distinction decoder distinguish posterior mode distinction decoder path guarantee decoder obtain pairwise use suggest description detailed context path pointwise nucleotide level answer respect map address move thus example aggregate individual small effect map state mapping annotation viterbi inferior map annotation practically hmms np chain unlike viterbi symbol annotation path extend admissible path also know satisfactory although viterbi decoder still popular biology demonstrate various example recently demonstrate surprisingly viterbi predict protein restrict decoder admissible strong map seminal sensible vanish albeit leave unclear viterbi could viterbi inference logic true possible produce prior path path aware moreover related concerned map decision theoretic extension partially asymmetric loss incorporate prominent secondary possibility viterbi inference interesting need intermediate mode publish propose interpolation algorithmic interpret general importantly claim explain apart trivial situation despite raise discuss consider new unlike analytic measure family datum family optimize performance measure advantageous aware emphasis automatic decode hard valuable state sense identify cluster posterior recently thorough geometric general appear beneficial family smoothly optimization criterion understanding gain path similarity score context relatively become inference mapping eq decode refer classification decoder cause naturally formulate whereby risk family viterbi furthermore
example evolution along branch distance date plain sufficient specify set marginal specifie covariance class turn offer equally evolutionary inference value trait separable exist function covariance space separable separable component specifie follow corollary wiener evolutionary evaluate follow isotropic meaning isotropic gaussian degenerate process mathematical find supplement part proposition help autocorrelation establish convenient range covariance inference reduction spatially spatially supplement detail illustrate way make functional trait modelling spatially simple space trait trait correspond markovian markovian gaussian kt evolutionary proposition obtain note isotropic end belief variation trait spatial stationary trait hyperparameter characteristic combine summation homogeneous covariance property lose covariance control delta functional sample regression datum regular lattice illustrative develop supplement construct univariate specify priori alternatively eq give representation functional function inference provide question straightforward trait principle evolutionary trait trait proposition theorem section section biological object feature correlate relationship flexible combine bayesian placing rate evolution brownian extend function relationship example functional unobserved functional root infer sense temperature therefore ambient temperature growth rate heart series process assumption effectively specify character evolution ii role evolutionary avoid confusion indexing model flexibility mathematical substantial use gaussian prior nonparametric view account functional relate spatial involve multidimensional index topology conditional independence particularly use evolutionary versus regression considerable recent cross unless specify mean common gaussian specifying encode depend equal argument covariance co might make random co co evaluate pair mean combination independent long model variable dimension evolution trait indexing point observation surface
assume number candidate expressive exhibit content cluster bit communication sensitive mutual approximation concept physics set ensemble mechanic logarithmic correction calculate factor know factor joint mutual relate boltzmann identification ensemble access rich physics analogy theory mechanic particle role free energy reflect code logarithm partition markov monte employ analytical technique deterministic function reliable capacity sensitive high data tradeoff rank describe paper explore set string symbol alphabet significant communication comparable respective reliably bit sensitive information fluctuation identify individual suggest solution set replace method question naturally choose capacity selection combinatorial depend noisy noise level estimate average ergodic exploit mechanic enables study properly valuable david partially support fp international theory mm validation suitable control order depend information thereby uncertainty clustering model high fluctuation superior selection generalize comprise cluster optimize directly employ centroid method normalize cut linkage inspire principle linkage base various clustering ask principle method source viewpoint conceptual root assumption ultimately complexity endow ability validate set select channel good robustness clustering employ code minimizer approximate clustering model yield highly although selection still measurement relation describe structure g three application use distinguish measurement object relation assign group parameter uniquely identify object measurement cluster require characterize product assignment exploratory selection assess criterion e mean measure prototype theoretic validation assume simplify notation explicitly hypothesis minimize cluster approximation reduce two statistic deviation subsequent respective object I furthermore uniquely sufficient set training noise erm unstable cluster cost training correspond measurement clustering training hypothesis reader might indice nearest cx cx cx enable basis training datum determine overlap mean empty intersection datum become measurement tradeoff control constraint describe communication receiver generator generator serve channel receiver take optimize cost calculate code channel characterize clustering determine function index receiver agree employ generate receiver problem calculate permutation define nr behind cover clustering respective nr communication succeed stable stochastic fluctuation criterion reliable ability receiver specific permutation codebook shannon setup receiver set available determine membership clustering receiver communication depict generator generate cluster worth conceptually restrict problem optimal optimization necessary sufficient reliably identify set reliable define protocol coding analyse probability capacity capacity determine precision scheme select receiver event introduce calculated receiver
minimizer visualize priori minimizer secondly minimizer either formula powerful application subgradient know belong strictly question arise naturally constrained devise study online task generalization tackle processing manuscript extension mapping let map dynamic priori study constrain convex continuous differentiable nu n cluster lie nu tackle task stand manuscript organize include special wide application online find section complexity increasingly system recovery task sequel denote n associate symbol stand real equip denote classical dot lt stand bx v v b v cx cx subdifferential subdifferential value continuous differentiable singleton nothing differential well subdifferential function close cx x dx c tv verify close ty call quasi call equivalent mapping mapping particular iff mapping share relaxed projection nonempty associate metric point quasi mapping follow symbol weak strong subgradient convex exist inner limit b symbol stand closure set fashion subset subsequence repeatedly sequel assumption nx mapping n subsequence contradiction lead follow hence dx establish choose arbitrarily subsequence clearly nx arbitrarily mapping later sequel another satisfie relate assume mapping existence sequence dx dx necessarily exist n sequence projection satisfie assumption assume point nonempty let hold true weakly sequential sequence nonempty nu nu assumption hold nu nu nu assumption hold u assumption nu nu nu nu nu combine easily verify recursion follow stand relaxed subgradient projection mapping verify nothing fix arbitrarily hence convergent establishes assume nu claim theorem quasi easily subgradient n merge cauchy definition side assume moreover side claim n easily obtain time map one word theorem side establish theorem u u nu nu verify inequality n become guarantee nn order tt tu verify using easily verify consequence assumption mapping stand time priori meet signal processing application whose impulse also sequence rich quasi priori mapping context incorporate devise ax monotone monotonicity ax induce definite monotone give stand nothing map minimizer proximity x inconsistent knowledge piece surely ideally quite case piece priori inconsistent tackle follow convex proximity everywhere fr differentiable differential rest mapping hx xt tx hx length convex identify convex learn instant introduce weight du il collection e q fix definition lemma true arbitrarily assume previous establishe arbitrarily calculus since form sequence nonempty set elementary algebra recursion ambiguity case define also u nd hold n hold nu assumption follow apply task assume equip hold n inequality claim deal observe establish impose however clearly suggest boundedness true previously already du nu n mind theorem become direct utilize regard assumption indeed notice suggest u nu v previously notice establishe guarantee consequence previously adaptive low basis retain exploitation recently interest previously importantly recovery realize efficient minimization recall integer stand define cs generate real stand stand norm vector norm k stress batch recent cs appropriate operation predefine cs recover update improve measurement feasible development importance engineering time vary storage resource deal available take study framework cs scenario major limited letting design additional task capability track variation place require real time batch develop cs e time become exhibit convergence counterpart operation devise albeit requirement demonstrate incorporate set algorithm close associate quantify closed l metric hyperplane scale linearly aware mapping need sort operation multiplication due relaxed subgradient reason introduce unweighted enhance different ball help incorporate radius suggest algorithm n w necessary condition radius ball follow invariant save couple extensive spirit reader methodology compare couple belong minimized accounting build classical employ weight norm term scoring solve lasso variant word respective sub account instant infeasible implementation benchmark nr total case invariant whose position nonzero zero equal one vector zero equal tag curve refer case lasso inherent tuned way produce low error iteration although parameter expense demonstrate propose projection lead performance propose drop exact mapping accounting sort operation necessary computation instant nr experiment similarly refer nonzero value coefficient change filter tracking performance practice use realize equal coefficient equal odd coefficient set change instead set tracking factor forget concentrate variation reduce factor expense choose value factor propose system would discussion dr information matlab manuscript theorem theorem problem wide non incorporate knowledge manuscript project subgradient priori mapping benefit class way cast priori introduce mapping nature information property special case potential propose scheme demonstrate system recovery task multidimensional method infer time vary mapping dimensional system contaminate noise show distinct difference counterpart follow batch scenario time instant newly incorporate process sequential prescribed efficiency online become tool case dynamic scenario change nature environment variation signal past put receive become clear tool need like aware
pm multiscale air fusion bias coverage output magnitude coefficient appear predictive include proxy covariate covariate focus attention proxy useful fusion proxy scale sufficiently distinguish proxy diagnostic discrepancy fusion united issue highlight implication spatial lack circumstance prediction proxy scientific goal spatio prediction fine pm effort smoothing use recent consider optical deterministic pm rather complexity temporal individual separately spatio little accounting focus average air relatively usefulness change correlation differ short period gold quality first variation month mid optical depth moderate proxy snapshot approximately value location vertical average base light available km analysis proxy spatial variation month km proxy regular km grid vertical level low pm ground average short term air quality pm day average second pm output report third day source process analysis review health effect fig show pm raw markov monte appendix basic spatial grid represent sub single simplicity pick element observation proxy account spatial proxy grid particularly relevant sensing grid relate grid sub heterogeneity term normally p remain simple smooth nonlinearity I thin spline cubic radial function basis control able small variation thin spline mid knot scale mrf mrf thin spline mrf weight generalize discrepancy represent smooth variation scale provide I analyse integrate spatially integrate instead focus understand particular car often weight realization asymptotically resolution de show approximate mat ern range go infinity differentiable path surprising realization car heterogeneous regardless value component penalty induce thin henceforth mrf mrf extend detail exact include fit posterior standard whose smooth car mrf fine albeit mrf give variation suggest alternative small scale spline reduce krige capture large discrepancy understand proxy interest implication proxy implication discrepancy application involve combination white proxy fine scale pixel noise scale discrepancy proxy reflect small scale spatially model without white gold variation proxy spatial model treat spatial ignore proxy without gold small scale proxy improve achievable variability proxy estimation tradeoff gap clear proxy help variation improve gold scale discrepancy proxy accurately reflect discrepancy correct proxy constrain large scale difficulty occur scale cause discrepancy tradeoff proxy prior remove scale proxy occur spline set uninformative weakly scale flexible discrepancy represent proxy largely proxy proxy sample scenario spatial scale discrepancy spatial diagnostic assess performance relative observation minus output observation diagnostic output sum ratio average analog quantity diagnostic use l scale proxy proxy quantify diagnostic appealing indicate variability proxy explain discrepancy ideal situation interest contain process trade addition dependence hyperparameter marginalization joint convergence high marginalization base detail mrfs effort analysis hour day respectively mrf informative simplify present scale discrepancy scenario construct mid gold proxy observe mat ern vary value simulate surface independently discrepancy operating scale cell differ spatial representation population road road class average square scale fitting road road first misspecification generation road analysis include capture generate near major road size fit replication surface component normally proxy white noise pixel variability covariate empirical true large correlation spatially proxy varied replication correlation exclude correlation replicate correlation occur residual spatially occur chance vary appendix mean square base replicate table variability difference error scenario replicate reasonably proxy include replication minus proxy spatial discrepancy full lack identifiability play role difficulty estimate quite sensitive inclusion proxy substantial discrepancy residual covariate posterior value one signal proxy term subtle make proxy resolve induced simulation covariate natural indicate discrepancy pick already need operate small proxy discrepancy scenario proxy scenario assume large fixing scenario model run add improve discrepancy vary scale fixing fix scenario surprisingly exclude unfortunately identifiability reasonable cross assess discrepancy greatly relative difference scenario remove entirely include proxy proxy small discrepancy scale discrepancy proxy covariate reduce model discrepancy scenario spatial proxy proxy covariate always small model reduce without pose take proxy ability range median difficulty exploit proxy correlation proxy high analysis proxy proxy recall error proxy still benefit proxy prediction complicate fashion signal proxy informative km grid large surface km mrf rather thin spline computationally mid represent small km posterior scale calculate covariate km cell fall value g km fall km grid covariate km weighted cell four km grid seven negative drive strong apparent ad hoc manner hoc formal orthogonality practice specification near would hope somewhat variability scale pm scale suggest resolve variability discrepancy surface fig scale real north proxy simulation sufficiently interest strength gold able exploit proxy pm extreme refinement might improve proxy term find validation without suggest term restrict observation pm prediction poor simulation adjust discrepancy dense proxy signal raise identifiability proxy fitting identify observation proxy penalization latent process tradeoff sensitive note trade occur analysis relative proxy data identifiability spatial difficulty exploit analysis include signal proxy small scale way discrepancy interpret relate proxy shrinkage proxy complicated proxy improve relative proxy information despite concern identifiability signal noise fundamental task know ideally literature lack identifiability scientific flexibility process model present assess proxy proxy spatial process result highlight difficulty actually improve prediction information concern proxy able information proxy variability well conclude variability gold note daily proxy value day observation local variation pm identify yet poor job capture variation proxy variable add useful absence part covariate local contribute proxy mask relationship proxy mrf spatial scale lack mrf recently mrf mat ern approach explicitly gap might limitation proxy scale think handle identifiability discrepancy impose identifiability signal discrepancy term seem unlikely substantially well identifiability treat proxy greatly complexity simulation proxy discrepancy discrepancy decrease proxy improved prediction scenario covariate purely understand proxy regression may good drawback sense cover sort one proxy subtle involve scale proxy adjust leverage proxy prediction phenomenon simulation concern proxy plausible large small spatial proxy interpret analyst proxy additive spatial proxy alternatively proxy measurement error coefficient limit gain scale signal carry proxy proxy discrepancy principle albeit practice explicitly decompose separate allow successively predictor suggest acknowledgement liu processing overall lead thank environmental power institute grant health effects institute spatial mid additive matrix representation explanatory proxy pm cloud site correlation site prior dimensional mrf fixing coefficient spatial implicitly give limit five knot knot knot covariate equally space spread covariate penalize exact informative component exchangeable amongst term follow I integrate normal q prior carry marginal likelihood result exponent calculation inversion combination form contribute infinite avoid determinant improper prior column collect sum block tune proposal determinant calculate operation simple proxy dense correspond basis computationally total must latter bb consider diagonal I equation recall quickly spam calculation remain normal matrix calculation describe draw indicate variance ii kn variance daily subsampling average daily month I simplify daily dl give kn month I advance locate enhance identifiability co locate integrate value add primary mcmc avoid calculation take vi involve diagonal analogous daily pm average pm work pm alternative modify log transformation scale single accomplish correlate residual tail approximately albeit skew outlier worth somewhat base pm several km class area national density value centroid fall grid population density road density extremely truncate location contribute mixed spline represent effect location emission strength source represent effect strength distance strength weight distance fine pm primary km source five proxy grid point grid box average integral emission hyperparameter informative variance low upper prevent part location distribution away toward smooth simple suffice even improve run burn th iteration storage cost reasonable size multiple month validation pair note justify follow computational represent grid parameter covariate km base pre advance average km pixel overlap knot include represent joint ac pp follow calculation integrate determine remain normal remain note grid cell km entirely water treat km grid include km cell intersect computational require year within km use rather run iteration subsequently cost reasonable environmental space gap observation proxy call discrepancy proxy complicated spatio dependency extend popular discrepancy employ little field thin spline capture scale discrepancy computationally conditional auto flexibility lack inherent context matter output estimate small discrepancy predictive improvement modeling observation proxy identifiability prevent prediction result
cdf pdf rr sd pdf pdf representations implementation regularize distinct comparison square rmse respective sampler use e round discard burn difference cdf pdf representation binomial term rmse cpu many rmse gain pdf draw proposal ratio even show slice never reject expense inner slice slow overall sampler hand maximum take time long base mh absence slice assess visually due autocorrelation nearly slice mh fully full map modern regularize logistic synthetic like vector zero unit metric approximate respectively prior precede burn mcmc round estimator find initialize value mean except mle package efficient omit cv penalty parameter cv give final reliably intensive pick use mle l avg mle irrelevant wherein apply dash red summarize low across repetition fully hand table employ penalty average increase hand region table due many case qualitatively estimator though former fix fully behavior choose offer stability setting bayesian preferable map hard increase uci contain classification treat comprise cv estimator bayes estimator burn round attribute expand map calculation avg section absence presence wherein estimator dash line summarize thing section good predictor relative regularization expand automated simulation logistic inferential goal development everything prior coefficient conjugacy gamma art context describe attractive implement generate pdf mh rate ess nine coefficient ess spam lead fast well movement lead around extension example handling convention simply likelihood independently trial logistic extend ordinal response probit logit cdf pdf applicable ordinal break regularization implement add sampler goal far approach regression via similar spike grant handling code associate valuable comment expression namely term quadratic collect pdf inverse kind inverse simulation school business usa paper logistic exploit normal carefully mixture implement mcmc say posteriori include flexibility computational applicability uncertainty assess optimal methodology modern word augmentation logistic numerous posteriori bayesian regularize bayesian collect include handle proceed employ advantage previously encode predictor I penalization amount regularization relative solve discuss lar subroutine scale inference covariate follow shrinkage finite intercept practice indexing ignore simplicity offer fully coincide help map simulation efficient logistic posterior write augmentation hereafter combine augmentation suggest finally recognize observe otherwise framework traditionally hierarchical include marginal expectation give context logistic outline scheme illustrate empirical section direction future package available mle logistic inspire g absolute tool calculate likelihood bayesian concentrate whereas mode motivate equivalently estimator solving recognize use computational response follow regard fix discussion defer likelihood yield bayes subsection augmentation primarily concentrate double possible briefly logistic represent latent product logistic write integral mixed queue primary interest include generalize rely pdf generative likelihood drop cumulative zero yx yx establish model summarize conditional z identical generative extract suggest alternative show represent inspection eliminate integral eq avoid analogy problematic fortunately mix long extra pose practice specification implement p regression heavy around mode origin towards provide discussion notable laplace typically proceed cv vary assess estimator pose difficulty power offer tractable uncertainty lead efficient option ig prior shape base second option ig identical identity tail purpose efficient draw unconditional yield overall simpler proposed reject mh cdf replace adaptation may auxiliary random although section scheme fast behave similarly cdf multivariate prior combine op represent significant formula pp instead pdf penalty likelihood observe enter representation adaptation result conditional reciprocal inverse ig conditionally outline multiplied ba b b r kb r p sr variation slice former replace draw approximate uncertainty include via anneal sa sa posterior increase schedule possible start systematically monte expectation determine threshold iteration initialize iteration chain chain procedure style importantly sa know optima certain optima although quick sa however burden schedule regularize short safe often perhaps em option beyond infer option classical annealing option experience former work dominate yield far less provide find predictor thousand require extension repeatedly contingency table typical un regularize way proceed describe section scheme require turn feature observe likelihood subject equivalently term
regular algebraic construction review handle fractional factorial design effect valuable characterize design effect carlo effect observation use gr basis order division fractional factorial code factor code complex unity factorial algebraic statistic set equation polynomial vanish factor factorial level factorial polynomial design ideal polynomial unique factorial two ideal basis theorem basis coincide fractional factorial calculate intersection consist write intersection gr gr respect elimination gr basis design resolution define relation q follow element contain reduce gr gr reverse reduce gr hereafter write level code indicate relation factorial fact design obtain eq ideal see design ideal fractional factorial design define addition non polynomial polynomial regular non design obvious regular gr fact hand gr argument gr gr basis next gr basis regular regular design express effect membership calculation gr basis design ideal give polynomial lt belong gr gr respect set standard lt theorem gr basis basis identifiable gr write x ideal identify effect effect high hierarchical resolution relation ideal gr whether define gr include gr follow level level th unity code satisfy representation correspondence algebraic fractional factorial design classify function concept design resolution orthogonality regular naturally see addition incorporate algebraic naturally function indicator equation characteristic indicator factorial function factorial design factorial example indicator exceed b e f generalize regular factorial design assume factorial fix hold work author design regular fractional factorial factorial full factorial design factorial conversely regular factorial factorial design fractional factorial satisfying property factorial simultaneous identifiability factorial main simultaneously full factorial full model fractional factor dimensional factorial mod determinant realistic situation unknown rely robustness interest require number active factor consider present evaluate fractional design minimize diagonal appropriate effect compare criterion dimensionality gr basis design experiment procedure construct give publish topic basis another branch investigate basis apply basis design section work integer regular fractional design simplicity run natural run variable mutually independently distribute expression treat nan write way nan statistic specify valuable fitting exact infeasible markov key conditional sufficient statistic eq basis calculate reversible conditional ideal illustrate fractional factorial fractional factorial relation several model effect seven may column interpret main effect also interaction effect two interaction among effect column note correspond additional two want relation effect membership factor complex specification sample space fractional design level code parameterize example constraint simple treat column covariate total code part conditional polynomial algebraic statistic work algebraic statistic close establish branch argument branch except design theory row often ideal therefore design factor code complex indicate end effect factor without factor argument reflect ideal algebraic design fractional factorial purpose design gr
text corpus scientific paper specialize motivate desirable datum able live node tree infinitely exchangeable breaking play prominent constructive dirichlet draw dp base eq stick length break call stick place break piece break right piece random view sequence include paper accord partition natural number idea distribution infinite partition possess topology sequence index partition pt construction string index node stick breaking size beta branching interval onto construct string eq decision branch determine child mass sum must satisfy mass depth throughout remainder induce partition sometimes chinese illustrate set order branch tree bias draw node child stay child later stay n previous stop child chinese restaurant customer stay child accord datum part popular without structure prior tree live arbitrary node infinitely exchangeable exchangeable chinese unbounded depth path distribution nest partition binary tree live marginal cluster exchangeability however topology hazard live internal seed gamma seed seed gamma pdf decay seed decay seed pdf gamma seed decay pdf stick break labeling mixture th continue direct requirement edge must tree give global diffusion transition coupling complete infinitely examine generalize describe transition fx nu weight challenge assignment frequently integrate infinity slice dirichlet path draw variate break imagine top need conditioning previous enforce tree draw slice th currently initialize determine currently must current determine correspond lexical prevent represent slice appropriately perform discard hull assignment use stick crucial include perform biased permutation set repeatedly draw imply weight keep invariant hasting proposal slice aforementioned hull slice far markov n inference cifar image dimensional factor bernoulli real component parent place diagonal monte hmc slice robustness stick break use modern mcmc slice approximately represent provide capture high low branch texture shape material bag word topic model thus multiple share node multinomial symmetric dirichlet kind chinese restaurant node topic path visualization datum document subtree histogram show year document node expand version supplementary material secondly assess predictive create ten partition perform inference predictive take pseudo improve lda number believe improvement constraint topic topic seem may term topic performance tree good lda fold figure fold procedure train follow first run tree word association association vary additional full posterior somewhat multinomial parameter
maximize improvement contour branch bind algorithm simulator run overview process branch improvement expect contour estimation present compare branch genetic test conclusion outline computer simulator I without input hypercube output gaussian spatial hyper estimate maximize close hyper maximize good predictor square power exponential mat ern correlation see choice correlation omit use near singular unstable overcome ill condition well section include gp expensive model simulator feature global maxima contour sequential budget simulator setting data process model new trial improvement estimate run simulator trial exceed limit develop new gain computer ei minimum simulator improvement interest criterion minimum maximizer denote cdf attractive ei exhibit second local performing attain reach simulator run become ei multimodal peak trial branch optimize ei review often global optimization present find real dimensional hypercube component branch pruning branch rectangle randomly bound gx edge guarantee pre specify tolerance nontrivial appropriate bound ei criteria specific feature pruning remove upper rectangle algorithm initialize counter list pick l ii min ix step branch bound pruning ei simulator function monotonic w take lb lb lb lb needed replace bounding ei bound easier estimate deterministic computer simulator next ei ei contour development maximize ei maintain versus simultaneous global specific scenario weather low maximal corresponding expect improvement ei ei criterion individually estimate maximum global subsequent neighbourhood dominate global otherwise explore overall ei global optimize upper accomplish partial equal straightforward show f monotonic monotonic w every bind monotonicity close bound equal ei contour simulator motivate involve bad performance server one server queue simulator estimate contour yx x sx neighbourhood around contour r normal pdf cdf respectively ei additional trial trade search gain computer motivated modify ei facilitate still uncertainty neighbourhood contour without criterion ei obtain drop change modification tradeoff section piecewise monotone bound let partial sign mean derivative cdf plot worth note height peak figure increase nature remain always may display maximum using contour maximize improvement outperform save propose genetic ga ei various approach ga design ga ei method long demonstrate compare true case use output ga implement maximize initial mutation augmentation x x augmentation cross ei retain ei produce ga magnitude mutation perturbed dimension information optimization ga trade search contour consist search balance search table denote proportion choose specify contour proportion new trial computer simulator output dimensional present average realization random hypercube choose ei ga contours interest respectively whether difference c cc cc contour add consider function add new notably contour implication integral contribute global desirable characteristic improvement criterion simulator local global reweighted reweighted ei enable branch ga ei start strategy fit surface fit ga ei estimate ei two ga estimate directly comparable though approximation estimating gp fit outline comparison present ei criterion ga budget budget entry table ei ga feature contour maximum function dimensional hypercube result gp fit complex see ga exclude ei row table ei ga ht cc ga ga contour ga ga table function maximize ga close initial maximum method indistinguishable ei ga ht cc contour ga ga analogous dimensional contour determination ga ei achieve ga ei criterion note design maximum fixing ei evaluation consequently iteration ga tolerance true design surface compare interest add design compare static approach hypercube serve ideally ga outperform sequential next eq global attain ga maximize simultaneous estimation global minimum hypercube design size trial sequentially maximize ei criterion realization bar denote clear ei criterion lead simulator red ga optimizer baseline hypercube estimate minimum seem though attain close ga turn static somewhat design sequential ei ga contour height measure estimate contour denote discretize static result average random design display contour ga similarly static contour quickly contour contour corner design hypercube design likely place contour static dimensional global ga ei criterion simultaneous size winner well much simulator ga static design size average hypercube design minimum blue lead contour ga static black design minimal improvement contour simulator careful ei save simulator result design hypercube sequentially highlight optimization evaluation ei figure display ga design simultaneously
person repeat game payoff average advance existence prove see notice existence strategy external use existence strategy also construction strong property call internal player regret external stage regret introduce shorter calibrate strategy complete survey sequential converse convex calibrate strategy statement prove construction strategy consistent construction game monitor I action receive idea follow asymptotically external well see strategy external prove explicitly strategy strategy signal depend play opponent restrict finite regularity assumption main give idea construction partial monitoring repeat game choose probability furthermore belong prediction call depend past h nh n stage calibrate eq word strategy calibrate close close go strategy expect general define calibration player stage choose partition go formally player every diameter player calibrate strategy calibrate definition calibrate diameter calibrate calibrate calibrate strategy respect grid grid yet algorithm polynomial calibration result condition replace payoff player player choose payoff behavioral h player payoff almost surely uniformly respect notice purely condition state define subset separate informally point matter outcome expect payoff moreover player player complete convex player particular player prove consists call proof person choose vector payoff internal average internal player strategy every increase payoff game state existence prove note auxiliary payoff strong converge rely result existence chain non therefore sum every player payoff stage respect player player useful strategy originally q norm get grid exist strategy player respect resp choose generate payoff auxiliary every grid close soon finite irrelevant proof calibrate game conversely existence calibrate give new mainly proof advance play payoff prediction since multilinear introduce auxiliary game action outcome player forecast let action game depict choice player close calibrate soon small probability therefore least inequality bound imply q sum cd exist play accordingly necessary convex proof work von allow projection onto gx z minimizer implicit construct sum program big elimination repeat polynomial fix compute obviously reduce solve solve polynomially elimination payoff one arbitrarily aim algorithm construct strategy approach stage without explain maximum external match dark coin choose coin otherwise coin payoff cc cc c payoff nature good payoff receive signal whose law evaluate payoff internal action stage partial response want framework family player assume player trivial guarantee converge behavioral stage player intrinsic player uniquely sequence I sl regularity bl ib regular ball good ball enough strategy part firstly main observe generate idea calibration transform implicit constructive corollary game predict grid choose sequence close play asymptotically action belong different calibrate slight time armed calibration regret chapter survey every define uniform every chen white calibrate know close soon big close therefore ii equation hoeffding every construct strategy external describe know compute regret accumulated block beginning decide next play follow fine external regret try payoff abstract abstract game monitor action almost calibration therefore internal quickly slowly zero depend drastically converge unobserved mapping star behavioral receive signal play nx equivalent nx receive stage signal mixture behavioral behavioral conclusion play strategy strategy player use I compute know prove continuous compact grid since define imply assumption application gx x I iw rr g require face opponent regular consistent strategy mention action denote mapping define choice receive law belong assume player player range polytope convex hull finite range sense p continuous bb good player playing need repeat game example player odd stage player must fact polytope definition continuous true strategy payoff strategy actual give easily show see every diameter every random history resp actual payoff rely g strategy player accordingly block accordingly h partial history long regret accumulate block stage trick prevent rate regret two restrict strategy actual payoff definition calibrate strategy monitor linearity payoff strategy property monitor regret payoff consistent respect actual payoff consistent
picture gibbs conditional distribution exactly hasting distribution simulate algorithm proposal surface see log center eq approximated h generate candidate acceptance mode full th give accept reversible mcmc autoregressive design proposal autoregressive good autoregressive refer reader work extend autoregressive process order autoregressive restriction nevertheless conditional autoregressive constraint deal stationarity suggest move innovation suggest parameter jump make rate move employ suitable order close order proposal newton approximation posterior let innovation lag innovation term root stationary innovation associate thank parameter prior reciprocal satisfied prior jump allow need order uniform process assume proposal truncate autoregressive autoregressive jacobian acceptance simplify identical identifiability parameter acceptance log one normalize log evaluate derivative u x convexity bar move control parameter vector propose jump propose vector assume unitary jacobian parametric choose approximately equal rate note gradient split nan define notice opposite mode dim variance simplex system hessian respect ta apply modal value approximation approximated recursion gradient respectively provide analytical choice period posterior mode reach acceptance rate move zero effect parameter jump gibbs simulate thus burden within mix well acceptance size convergence diagnostic first proposal hasting prior setting experiment random iterate typical present root inference choose regard correspond highlight precision frequent next higher consider exhibit fig precision dataset parameter setting efficiency mcmc truncate normal truncate typical raw chart gibbs mcmc parameter average observation simulate bar sample mcmc rmse average acc effective order rmse iteration discard number still work combine graphical inspection average kolmogorov statistic convergence rmse ess mcmc parameter rmse axis modify color circle display display precision rmse ess prior since fig show improvement rmse ess low circle depict order output necessary different chain estimate probability true order result associate ess markov mix aim dataset estimate modify gaussian observation chart subspace chain chart chart probabilitie ccccc probability bar order posterior mean size base specific last show secondly standard deviation decrease increase concentrated noticed bias base integer beta autoregressive high dispersion shape obtain section intercept certainly economic model challenging recent advance nonlinear characterize brief period rapid contraction long focus let say transformation usually recently beta consider mainly economic cycle observation source aggregate study dataset challenging amount represent issue european bank european statistic modify autoregressive autoregressive discard estimate model us acc acc rates capacity acc dimensional capacity rate production capacity production factor I e force stock capital economic capacity economic time series forecasting indicator issue role practice economic demand capacity level economic activity adjust inspection chart trend deterministic trend could naturally beta conditional imposing constraint intercept impose specification focus autoregressive parameter residual regression capacity beta beta fig rate autoregressive focused mean process allow easy order parameter deal parameter specification informative studied jointly estimate parameter different estimate true mixing reduce near boundary within informative truncate beta autoregressive allow evidence one united area paper autoregressive beta particular inclusion order beta conditional extension explanatory reciprocal jump autoregressive coefficient identification recursion constant belong simplex condition positivity satisfy appendix k pc correspond iii de de beta autoregressive restrict attention estimate due suitable specification moreover particular mcmc metropolis within gibbs solve reversible jump mcmc beta autoregressive reversible jump define interval proportion challenging issue year issue modelling transform real line standard example box cox early contribution follow contribution along series seminal beta consider instead employ mixture extend beta propose beta apply context beta recently markov switch autoregressive mean autoregressive parametrization autoregressive without attention appeal challenging nonlinear modification procedure consider autoregressive become autoregressive moreover reversible jump markov carlo beta autoregressive autoregressive jump iterate chain view kn x likelihood indicator distribution markov sampler state form accordingly markov probabilistic reversible jump see fan jump beta autoregressive process follow reverse jump space dimension jump acceptance move contribution propose combine find outline beta parametrization bar show beta autoregressive algebra denote beta refer parameter past transition density autoregressive process distribution fairly flexible unimodal family review identification issue part beta determine parametrization context variance quadratic location precision x k convexity path volatility chart chart conditional mean process anti unimodal switching type behavior fig anti unimodal switching bar
sake brevity complete result instance result quite conservative performance either method standardize compare show observe standard standardize especially high noise also slightly run comparison empirically yield odd ratio unclear whether rely simple avoid difficulty coefficient situation un square error odd sample estimate especially combine odd ratio need explain correlation closely consist regression return coefficient coefficient quite odd ratio model modification rely correlation regression l l association cause death possibly relevant death dataset cause death record death france year coverage death start cause death death death record analysis cause code international classification disease total category code analysis apply entity appendix therefore death record death five cause frequency report appendix cause death decrease heart failure heart disease disease diabetes cancer death clearly upon age gender vary age gender decide whole population gender sub consider age sparsity yield association good model different many association occur little association especially ratio ratio unbalanced compute take respectively analysis window machine computational computation un estimate derivation step reduce second final retain intersection derive apart obvious association association diabetes disease cancer disease disease negative association worth association cause death biological plausibility old save take second method estimate second detect agreement confirm agreement return ratio paper empirically compare approximate method association performance term f slight moreover method case could similarity term computational disadvantage especially low parameter method pseudo coefficient adaptive derive theoretical pseudo alternatively cost aforementioned lack cross start combine good time fast particularly truly might slow mention implement package think package glasso rely coordinate interestingly observe different especially ratio example decide retain intersection model lead conservative competitive example approach optimally approximate regression attain performance reach exact computational result gaussian ise absence justification recommend theoretical study enable retain cm cancer cancer cancer cancer cancer breast secondary c disease diabetes disease use f disease g disease g disease heart disease heart disease disease disease j status disease disease due disease system disease disease k disease disease disease finding elsewhere fall fall x association statistic due amount biology domain application recently gain popularity purpose literature exact inference slow approximate recently extensive study method rely approximation achieve association death death biology researcher involve study relationship example system biology underlie variable way study relationship total number cell contingency free greedy selection testing even contingency table plus hypothesis testing issue originally penalize enumeration profile plausible large alternative speaking instance tree become dense approximate inference notably recently rely two distinct penalize regression graphical model three derive maximum solution show extensive study solution either solution derive fast ever conduct thereby conduct outline principle slight modification fast application association death contain vertex independence binary focus q strict therefore independence consequently complexity need principle pt valid interaction also higher handle dramatically extensively hereafter terminology use estimate separately rigorous dimensional setting grow consistently graph sense solve separately asymmetric combine way possibility belong alternatively estimate belong comparison study recently maximization solve problem enforce symmetry differ neighborhood still represent set spin elsewhere resp th column implement maximize however pseudo exact less pseudo method refer interestingly point easy association minus twice compute obviously pseudo coincide interaction may sequel completeness shall maximize account spatial example genomic study describe ising suggest result put descent specific sample upper partition obtain likelihood solution solution handling compare original add glasso develop question work moreover related use put decide model pt computational precisely spaced scale small run window intel ghz gb intel ghz core gb ram mac windows cell draw multinomial distribution different normal distribution subsequently odd retain compute lead association potential greater retain lead pt graphical probability px panel graphical representation panel edge coefficient logit gibbs px evaluate rely poisson value tuning lead gaussian approximate likelihood log precisely quantity consider stand sparsity induce lastly define q k approximate likelihood base furthermore obtain covariance add term close exact undesirable effect obviously try conclusion consistently confirm finding corner exact black solid pseudo dash circle pseudo likelihood blue likelihood dot green circle solid solid performance oracle score slightly fourth simulation design focus save procedure although one insufficient c method acc score cm l cm l pre acc cm l cm suggest section half likelihood table grows achieve model un slow approach
shrinkage conditional mode motivate bayesian also average new penalty mixture iid interest make center omit important propose penalize residual sum solve minus mu mu mu plus mu minus mu control regression lar provide implementation lasso consistent choose yu show vanishing regardless choose due wang et consistent model inference select may bring undesirable introduce sub lasso interpret posterior use lin study lasso explore prior previously consider generalize way coefficient focus em algorithm somewhat broad mcmc prior coefficient choice explore model purpose generalization linear lasso originally property hoc thresholding rule difficulty bring recommend credible interval variable selection fail explore uncertainty space spike mixture lin increase plus mu minus mu minus mu mu mu plus mu minus different penalty naturally covariate wang preliminary treatment argument distribution draw al subsequently obtain array estimate gibbs permit unify flexible penalty least wang cox model outline novel structured penalty group lin variable selection yu rest organize tuning vector explanation discuss average analysis dataset present unified deal selection model software deal many penalty represent normal exponential motivate mu plus mu mu plus mu mu mu minus mu mu plus minus mu mu minus minus mu minus mu mu mu minus improper introduction use shrinkage motivate structure plus minus mu minus mu mu plus mu minus mu mu mu plus mu mu minus mu major difference allow intuitively penalty mode wang gibbs bayesian full multivariate conditionally conditionally inverse gibbs fast choose framework bayes prior shrinkage close deal approximated sampler rule mu mu mu mu mu plus mu minus mu minus mu th expensive obtain hyper parameter make hyper plus mu minus mu minus j mu mu plus minus mu minus mu gibbs size choose scope practice trial join minus mu minus mu mu plus mu advantage use algorithm conditional parameter rate parameter specification join although number increase use specify alternatively estimate rule mu minus mu minus mu e minus mu plus mu minus deep hierarchy allow demonstrate gibbs sample plot versus expect would put likely zero phenomenon demonstrate plot sample discard distribution central c trace plot likelihood versus decrease increase demonstrate signal may shrinkage allow minus mu plus mu mu plus minus plus mu become mu minus mu minus plus minus mu mu full conditional lead amount shrinkage coefficient choice choose demand lasso exploration model fail method hybrid coefficient hereafter suggest median posterior explore frequency choose consist less median mp surprising original lasso uncertainty make inference may set help improve bayesian framework average use therein refer formal treatment uncertainty use ensemble sparse mode observation predictive give plus mu mu mu plus minus mu mu mu minus mu mu performance scoring use predictive measure smoothing parameter mu mu plus dd mu nonnegative kullback leibler sense offer average sample example case prediction estimate gibb give conditional replace integral integrate accordingly fact predict draw select draw gain predictive condition fix compare lar example preliminary use marginally compare original measure fit replication summarize lasso mean example consistent seem go increase perform well even large area coefficient median lasso generally superior summarize example perform outperform variable selection ability experimentally conditioning suppose dataset square minus mu plus plus mu plus minus mu minus mu mean original similar average various size set experiment well well lasso summarize surprisingly due use outperform percentage important health require special percent percent body summarize carefully et al omit nd diagnostic percent age weight cm cm intercept lasso x correlation additionally remove help small bic ht proceed let model w shrinkage model mu mu mu mu pm dd mu plus usual formal mode uncertainty smoothing parameter gibbs straightforward high uncertainty high mostly account approach use predictive median predictive clinical measure weight age seminal score percentage response consider standardized intercept exclude smoothing coefficient smooth put predictor estimate ht obtain convergence mean give estimate correspond zero shrinkage shrinkage propose strategy include variable lasso choose present mostly select select presence account considerably different second examine form training rest prediction median well far regression complex generalize cox group lin composite yu unify broad context likelihood wang l mle approximately need keep specification discuss novel flexible frequentist mu plus minus mu mu plus minus mu plus mu mu plus mu minus mu minus mu mu mu plus mu mu mu mu mu conditional specify plus mu p j j mu minus mu plus group lin minimize plus mu minus plus plus minus plus mu th mu plus mu plus minus mu mu minus mu r minus mu mu mu size group identity conditional block size j j j mu mu minus mu consider natural ordering group extend yu vector plus
cutoff order spectral observe fig markov explore burn follow always initialization stationary run consider initialization length precisely concentrated small interval burn independent exploration burn period sample accordance width instantaneous burn chain burn period accordance rate hasting algorithm quite rotation hasting law simplicity counterpart rejection parameter accurate realistic image corruption white close visible give capability realistic coherent build hyperparameter mean minute assessment law addition unsupervise image frequency invariant encounter medical optical property methodology valid mean optimization penalization penalization penalization overcome difficulty development unsupervised deconvolution prior unknown parameter difficulty efficient unsupervised plan extend law penalization limitation function impossible resort idea overcome finally cutoff within hasting sampler laboratory france constructive suggestion author grateful laboratory vector covariance diagonal gamma write case hold mean precision pdf know student law conjugacy pdf give hasting law multiplicative law proposition follow law name metropolis law acceptance get simple ne le universit de universit paris sup place du france france pt national sup france phone r r pt de de france mail cm ref remark rgb paris de la france paper deconvolution framework infer law effectiveness parameter high spatial coherent deconvolution active field decade medical imaging generally problem induce observation precise importance critical development quality moreover poor limitation pose acquisition deconvolution class accordance information dedicate image numerous ill different bayesian information design build representative law maximizer optimization latter lead integration two addition firstly depend response devoted deconvolution contrary devote unknown know main extra practical model description usually relatively field addition deconvolution strategy parametric physical parametric practically parametric need interest refer difficulty blind ambiguity even resolve resolve moreover design nominal addition algorithmic respect contrary blind algorithmic linear unknown element despite difficulty format blind format moreover blind extensively case secondly law name hyperparameter variance law tune deconvolution hyperparameter approach devoted parameter recent address experiment laplacian address deconvolution hyperparameter image interest variable rule law noise uniform parameter regard attention parametrization facilitate conditioning degeneracy estimate choose simulation algorithm enable distribution despite firstly detailed law establish sec finally sec devoted consider square nx stand width convolution transpose convolution fourier image strict description spatial fourier domain coherent efficiency notational convenience x fouri component n coefficient account penalization differential operator law consider prior parametrization next parametrization facilitate law integration hyperparameter moreover result law parametrization sec parametrize read design diagonal domain sometimes refer law focus positive pixel penalty differential difference pixel scale build diagonal elementary parametrization equal corresponding absence frequency consequence vanish still despite degeneracy law format rule incomplete embed degeneracy format remain yield rely energy extra proper classical frequency approach degeneracy rely frequency determinant expression factorize control converge meet precision parameter degeneracy law chain provide compute law marginal representation analyse information uncertainty mark ambiguity sampler conditional law law burn complete joint next step posterior law eq q transpose regularize square solution wiener invertible describe sec finally diagonal matrix even law law update law c jeffreys k need law parameter direct metropolis algorithm sample propose law divide proposition sampling iteratively number mean approximate image spatial recursively single end describe obtain completely need entirely simulate study respective give protocol control entirely also different low signal broader different realistic show case image generate white laplacian image numerical profile fluctuation hyperparameter set non informative jeffreys law fourier normalize illustrated fig nominal rotation convolution add white smooth fluctuation corrupt level image nan coefficient h nuisance dirac sec fix fig line finally situation step sec line ignore obtain sufficient exploration difference successive sample compute approximately matlab law successive empirical give case respectively deconvolution notable profile order reconstruct profile match precisely fig pixel visible visualize circular average power spectrum hereafter radial frequency spectrum fig respectively spectrum image retrieve radial frequency dominant information lose produce correct properly wiener hyperparameter concern comparison difference fig visually indistinguishable especially euclidean image true report deconvolution
variance work former latter analytic sequential stationary variation design result design modularity sim easily package implement agnostic sim analytically ei generalization support far extend gp sim detail ei package ei classification model prior soft max logistic categorical imply sim sign easy say alternatively calculate matrix differ negative diagonal panel panel show sign b cluster relationship synthetic row panel sample imply color accord average posterior matrix rest identical c involve sign matrix set sign full sample index science sg sim parsimonious nonlinear univariate simple canonical use setting focus drastically simplify generalize sim favorable illustrate computer two package facilitate pursuit index sim parsimonious multivariate dimensional predictor variable response random predictor formulate apply section projection mx provide flexibility ad hoc step fit may author prefer sim interpretation sim computer run input configuration choice gaussian g prediction integrate base incorporate uncertainty fit calibration make one reason considerable insight nuisance quantity comprise projection index sim nice review use spline link sim illustrate empirically offer widely univariate limited suggest gp lead motivation model clear spline already naturally pose computer show gp sim error isotropic correlation sim complicated spline canonical unnecessary simple sim sim spline primary experiment formulation framework see multivariate computer package herein one suggest extension remainder organize follow hierarchical reformulate monte illustrative synthetic comparison section computer computational work sim experiment gp sim presentation notation version ig conditioning presence relevant finite shorthand fx nn correlation prior gaussian correlation prior interact identifiable choose flexible von sphere sign lead posterior setup carlo proceed chain metropolis gibbs iterate conditional actual expression conditional gibbs whereas metropolis mh proposal rw fashion von modal add lack identifiability sign switch offer post suggestion cluster mode posterior proceed krige equation discuss detail reformulate follow lack identifiability without unit ball gp constant quantity identifiable sign sign matter mcmc explore possibility direct variable selection sign pose primary identifiability heuristic allow explanatory effectively remove treated function none proposal interpret correlation gaussian inverse matrix sim odd equivalent combine iid q x correlation preferable equivalent I choose von recommend available sensible sensible design scale lie unit make sensible much easy benefit yet readily one fewer eliminate latent setup implement sim code trivial turn implementation consistency reformulate continuous uniform essentially prefer software interpretation simple inferential advantage augment analytically never predictive posterior stand imply jeffreys preferred may result require decompose newly avoid unnecessary eqs save slightly significantly carlo scheme comment start mh good rw uniform slide window reject accord implement metropolis draw similarly correlate update rw center choice know pilot crucial good hard von sign sample estimate pilot run sign compound product experience change mh acceptance ratio pilot run try always propose require probability easy value student degree freedom minor classic krige extremely sample predictive quantile summary basis path extension extremely original leveraging gp sim try forget root index collect location useful assess explanatory aspect identifiable sim real computer primarily gp sim isotropic separable separable scale isotropic see superior multivariate explicitly state software sign initialize allow compound involve sign diagnostic show predictor mahalanobis output predictive location estimate covariance location sim predictor carlo design conditionally form training row true make mahalanobi ht sim min rd time three mahalanobi summarize figure ranking mahalanobis isotropic observe sim distance cm cm display realization performance also fit index see b index prediction credible interval ci provide look advantage ideally well variability axis visualize mean horizontal dot prefer index though comment cause lack sign index correct sign mcmc heuristic index build true hard get nice plot index response adjust figure arise obtain ht negligible correlation obtain run new proposal seven possible von ht result normalize lie density sample length square histogram panel eight lie rectangular domain experimental obtain without highlight sim compare far outside sim class approximate eight comparison sim even see testing sim pilot determine mh proposal also indicate discard predictor ht b sep sim min st max competitive well worse separable project measure separate direction ht distribution describe contribution aspect sim show component sign rarely sim separable good gp sim arbitrarily sim experiment benefit estimation especially play predict try sim computer experiment computational back detail simulation relevant response speed angle attack shall roll detailed later portion like pose challenge experiment sim canonical experiment fold fold inside fold predictive fold response hold test code available aspect mahalanobi play major simply covariance explanation sep min median rd min max explanation improve visualization time fold generate mahalanobis sim summarize left part top version mahalanobis sim small indicate reliably project input correlation well axis align spatial ht aspect figure way towards sim mahalanobis model non estimate relationship suggest index explain deal exception relationship index cope accommodate section figure posterior index suggest fit mc mass origin suggest input mean lift response exhibit broadly summary mahalanobi nearly roll response relationship sample five response figure roll omit brief lift response probably roll roll seem input quite roll
input describe beneficial evaluation agreement ideal moderate draw arise evaluation random act representation produce version mutation efficiently mechanism mechanism mutation landscape ideal mutation jointly evolutionary perform class mechanism representation perform essentially pac provide target notation evolution randomize hypothesis mutation mutation class randomize representation neighborhood non negative mapping desire value predict context boolean consider function boolean disagreement loss early hypothesis range quadratic relative domain admissible explicit z use mutation beneficial neutral use beneficial candidate neutral relative neutral beneficial mutation tolerance candidate selection algorithm evolution output take concept mutation converge evolution probability evolve monotonically fr single evolution emphasize distribution require evolution refer learnable basic polynomial evolution require initialization performance process start reduction somewhat derive independent monotone base usual independent refer th point point point performance modification equal decrease combination modification converge loss modification improve type formalize nf definition case exist ni xx q bind low combine choice case eq compute convert evolution exactly done exist tn pn question whether characterization understanding learning address first low way distribution use specifically iteration turn boost agnostic particular distribution new agnostic believe insight learnable concept elaborate concept monotonically include addition independently boolean suggest need consider simple loss result subsequent work monotonicity monotonicity imply robustness evolution change function give along evolution monotone evolve les valuable comment I also grateful anonymous useful correction theorem fact definition bound characterization preserve characterization give characterization agnostic boost design demonstrate existence evolution loss phenomenon reason learn query restriction estimate unknown query oracle oracle query satisfy tolerance demonstrate convert robust small theoretic barrier query powerful algorithm fact introduction albeit know also extend noise scenario find preserve system crucial theoretic learning learn al prove give relatively simple statistical nearly uncorrelated gave characterize weak statistical query require bound query work notable bound dim upper easy exist strong uniform polynomial function require number query query dimension complexity specific invoke equivalence weak explicit respect recently derive result complexity agnostic informally characterization state learnable value close semi function x boolean idea say large characterization lead f correlation product function refer orthogonality efficiency learn efficient achieving error orthogonality easy analyze prove agnostic replace concept least model hard confirm dimension agnostic even generalization achieve task learn dimension simplified enyi elegant characterization maximum enyi proof complexity preserve efficiency computation proof accuracy preserve comparable direction distinguish give weakly approximate query execution imply desire property inner learn process closely area close closely core construction boost new type boost weak weak function namely explore learnable computable imply learnable learnable pool adjacent type acquisition particular converge rely decrease evolve monotone monotonicity allow adjust existence ability environmental monotonicity basic requirement recent show depend performance hypothesis namely uniform positive algorithm design specific use hypothesis equivalently agreement measure quadratic hypothesis function boolean loss vice versa fairly translate demonstrate concept monotonically quadratic robust mutation mutation variable correspond hypothesis exploit evolve list range range work scale quadratic hold define rely heavily particular fourier transform list uniform distribution formal model one preserve characterize proper emphasize characterization agnostic recent boost positive integer denote problem function product easy version dot semi g ff domain refer together represent concept representation representation complexity describe number drop length introduce distribution produce approximate oracle upon example independently output satisfy fx convenience hypothesis needs value think value hence fx expect polynomial pac pac learn respect weak learn disagreement target less precisely produce fix agnostic introduce situation agnostic class concept give hx agnostic access hypothesis pac run generally agnostic produce fix advance statistical query access oracle target concept respect distribution instead pair value fx convenience range extension say learn tolerance learn place oracle make tolerance evaluate minimum learn say extend agnostic analogously example denote agnostic learn statistical query alone relate al weakly use statistical exactly polynomial dim almost orthogonal use product gave characterize weak query weakly concept say set concept weakly number possible relate dim directly maximal almost approximate strong version let generalization orthogonality characterization query generalize characterization class simply use boolean define value exist either small give simple characterize query tolerance learn decompose value tolerance simulate every x x g provide simulator return therefore combine giving claim inequality standard confidence transformation unchanged establish direction every build step find point g every every tolerance g p desire query iteration establish every f xx fx x f cl iteration use query property give efficient convert algorithm statistical query vice note circuit efficiently size unlabele access learnable produce circuit run circuit give simulating would simulation easy replace easily access yield learn place estimate increase circuit generate learnable characterization orthogonality convert approximate every uncorrelated extend definition dim real say exist function set state boolean boolean easy generalization follow f f modification relation set minor need approximate every denote ensure f base obtain characterization concept learnable pn pn class require weak function function weakly agnostic imply learn formalize cb dc b word agnostic
huber asymptotically alternatively huber multiply case interest novel inconsistent meaningful globally convergent sensor sensor sparse outli km km km explicitly understand solver minimizer denote vector find everywhere subdifferential define differentiable subdifferential yield positively consider admit block minimizer huber evident rather surprisingly generalization huber capable l contribute specify vector sensor likely presence regard cutoff tend contain outlier suggest residual criterion sensor classify reliable heuristic rule practically select scalar mention sensor roughly know prior alternative solve prescribed note group lar descent warm grid value efficiency problem assume specification across case sensor across huber case contrary remain colored modify early solve interior solver exploit structure advantage successfully solver block variable block descent separately involve minimize keep fix step denote minimizer quadratic second minimizer sensor li jointly instead update residual thresholde offline demand product operation presence exploit save computation block coordinate complexity iteration readily small predefine termination vector affect robust huber appropriately argue non yield improved estimator appropriately initialize interest explore surrogate recall seek end convex follow solve initialize unity demonstrate solver lasso propose property validate weak weak refer occurrence single keep long large validate independently simply sm notice due normalization variance select identify uniquely solution empirically satisfy correspond intensity indicate east east fig integer small choice probable recovery black solid accord circle success probability validate moderate rs develop setup network collect observation previous sensor ls solver solver obtain ga ls ga implement serve simulation insensitive sensor successfully classify sensor residual reliable novel advantage even iteration evaluate solver reliable sensor mse empirically average comparison include ga ls conventional huber solver vi solver turn critical cutoff huber estimator worth coordinate algorithm snr db snr time time plot consistent sensor outperform task reasonable performance show serves good solver combine absence model sm solver modify remark mse curve snr db correlate setup superiority solver prominent classify sensor reliable classification solely sensors reliable ga attain correct sensor classification setup differ reliable db laplacian set solver iv residual huber scalar measurement sensor consider list reliable sensor yield sensor huber exploit scope exploit sensor outlier give rise compressive sense reveal sensor sensor show hold gaussian reformulate subsequently cost estimator derive solve efficient simulate design theorem recall imply minimizer reliable sensor minimizer vector indeed attain inequality necessity proving must cost attain attain conclude continuity hold inequality trivially appear last lipschitz continuous suffice continuous bound express supremum infinitely subgradient norm norm subgradient expect next introduce random entry functional triplet expectation exploit property readily proceed need I j c index set lemma verify deduce side establish exploit separability optimization yield hold n conclude ht line perfect circle fail ga ga ls statement edu sensor distinct sensor recover specify sense formulate finding feasible proved link establish signal sparse rise strength first relaxation measurement recover obtain concave fourth scheme tailor noisy cast combinatorial subsequently fall framework block capability verify sensor network robust convex compressive recent advance technology sophisticated task environmental surveillance medical imaging typical heterogeneous sensor signal term sensing entail operational measurement comprise observation snapshot view impulse response underlie multivariate costly delay stationarity constraint reduction cope curse large sensor observation sensor failure sense device sensor communication interference reliability sensor fusion aggregate instead sensor vector reliable sensor henceforth refer rs context establish sensor area even outli sensor network rs rs task one sub optimum rs section cone relaxation hard problem relaxation compressive cs cs vector satisfy equation pursuit bp block comprise variable non sparsity herein residual recover block signal develop solver generalize equivalence alternative concave constitute rs solver surrogate concave sequence weight third vector select sufficient establish whenever sensor per sensor world application contaminate quantization dynamic identifiability scheme signal ratio sparse cs noise lasso vector block sparse cs placing sense noise robust fourth initially interestingly cost huber function attractive block alternative non stand multivariate notation n equal collect vector possibly subset sensor compactly state I scene interest view possibly image system domain sensor environmental monitoring could chemical compound field green capture measure sensor sense due propagation effect failure even irrelevant rank either infeasible full th linear ignore admit check whether satisfie retain proceed check challenging admit infinitely easily handle focus note bandwidth delay stationarity row match henceforth proceed whereas subset aggregate matrix rigorously pose otherwise strictly positive minimize linear solve rs concave linearization utilize local let minimize cost minimize minimization drive equivalently weight error residual weighted indicate high reliable let r consistent total sensor total unique minimizer first thing consistent uniquely recover may minimizer assume uniqueness characterize vector satisfy every minimizer ss empty intersection belong minimizer avoid cardinality recover reliable second require stated next subsection partition hold exist exhaustive exclusive subset arrive contradict critical whether former hardness random assumption decay start equivalence condition provide follow equivalence theorem turn generalize minimizer worth range impossible useful establish characterization equivalence subsection another remark range valid constraint non lie ball though reduce set probabilistic bind subsection valid extra impose early practically infeasible sense prove condition decay exponentially entry summarize pf lt deviation result sufficiently multivariate generalize refine expression proof partly analyze unknown relate isometry sufficient relaxation independently minimizer low success suffice condition fail subset
empty network maximize maximize expand instance simplify diffusion inference interpret propagation tree note cascade contribution propagation tree take span tree network cascade remain stop cascade thus maximize maximize edge tree maximum weight direct edge observe forward node already infect cascade edge graph dag use edge acyclic direct span dag dag weight incoming maximum score sum node parent handle dag maximization create proposition span propagation income efficiently compute direct tree dag put together find likely cascade cascade cascade aim log observe monotonic add complete maximize monotonicity observe convert tree maximum span network usually edge infection cascade propagation maximization graph search would optimal np hard max collection element cover set index pick maximize produce cascade cg contain incoming max whenever solution max correspond efficient could decide whether value efficiently find add score add enable let cascade edge cascade cg cg submodular argue direct span dag obtain assign incoming incoming weight maximum proving maximize marginal stop return marginal gain edge heuristic think graph value adding change iteratively hardness however fundamental et monotonic return fraction moreover tight dependent decrease number edge gain great np hard algorithm network thousand node speed improvement cascade cascade infect cascade network may cascade cascade cascade cascade network localize update drastically behind evaluation sequence gain add graph whenever marginal gain greedy element gain insight exploit maintain queue respective iteration greedy high weight since decrease remain high algorithm decrease queue later run several implement formulation addition parallelization likelihood cascade edge big make structure scope empirical runtime cascade likely ci g g ci proceed experimental diffusion real show surprisingly outperform heuristic typical network experiment synthetic exponential performance simplification namely exactly likely network edge general proceed follow diffusion simulate cascade aim recover cascade node recover poorly error edge cascade parameter cascade direct generate namely forest produce kronecker core hierarchical structure generate network power law simulate cascade need choose parameter need fix cascade control control cascade cascade make fail get cascade cascade edge important cascade never trace infer choose amount principle need fraction pick start cascade cascade cascade cover edge look cascade forest node edge cascade edge take cascade cascade follow law median cascade edge forest flat cascade node exponential case set infer diffusion edge cascade propagate edge pick edge score baseline performance two optimize optimize exactly online fraction correctly log plot bind np red curve narrow return diffusion online optimal actually value generate present fraction edge appear recall cascade scenario node per cascade infect per cascade cascade mean generate cascade record infection time fraction cascade miss cascade cascade infection cascade percentage missing node naturally amount choice basically propagation likely thus give cascade chance propagate external source various percentage node infect external source influence external note appropriately diffusion due external strong infect million news article million web create link refer transmission cascade start recursively post trace chain reverse identify cascade cascade post short million distinct cumulative million phrase phrase cascade cascade phrase cluster website mention phrase cascade general methodology handle concept cascade cascade cascade truth create create post link construct take create receive connect post site link cascade right notice perform improvement medium perform prefer create site information ignore site satisfied life break point good result hard site mention particular infer underlie baseline fair synthetic realistic synthetic function bind tight synthetic least shape trace remainder medium site number document dataset cascade dataset diffusion edge track proportional strong strength cascade add influence medium rarely network bias notice main side medium devote distinguish media deal post play central role side site part dominate news establish rest figure connect network phrase track medium infer diffusion analyze information network first show node diffusion direction node shorter infect slightly sum node set reach node propagate influence diffusion network site split site medium link news medium say medium notice many medium later media medium link tend pick early seem later pick lastly capture medium mass show type site spread source distinguish medium medium site quick another even infect slow infect whether information come medium infer quantitative qualitative insight surprising use insight several diffusion receive considerable attention shape cascade infer diffusion formulate rich predict occurrence individual predict independently pick hyperplane predict vi link probabilistic infer cascade programming solve include transmission rate transmission rate computationally edge think degree probabilistic graphical fundamental difference graphical learning direct dag network learn direct first cycle edge direct reciprocal allow infer network direct acyclic dag undirected typically network bayesian heuristic network network framework however hill search offer performance work novel formulation relate static method unsupervised method relevance influence graphical relate visible although function maximization find work network diffusion formalize scalable network cascade good likelihood exploit objective infer exploiting localize able scale cascade accurately recover relatively number drastically outperform maximum evaluate real news exhibit medium site topic technology site capital site future work utilize etc accurately propagation network dynamic interesting assumption system protein protein infer connection result promise process base acknowledgment thank resource microsoft yahoo grant nsf nsf nsf fa foundation la infer infect di vi influence underlie network spread unobserved tackle path infer propagate node become infected explain observe infection time np million news article year flow media diffusion news media site tend media rest site circle influence medium acting mining behavior idea innovation disease setting spread news collective spread disease challenge address need actually moreover identify successfully task automatically large scale identify phrase place edge epidemic however place unobserved commonly infect observe infected propagation discover infect infected propagation people get without know infect people adopt behavior explicitly cause especially study network path diffusion information influence wrong inference complete identify propagate relatively web supervision global structure medium site interact site play diffusion influential question underlie influence propagate static node get infected piece product observe infect interested path take reconstruct propagate propagate infection recover b propose make perfectly recover edge disease work propagation network node infection unobserve figure create trace cascade cascade infect subsequent layer depict cascade create true connectivity infection cascade infer formulate spread node infect could network large surprisingly computation super exponential tree cubic tractable function exploit speed exploit function evaluation broad greedy hill contrast infer optimal synthetic reliably propagation influence network overall synthetic dataset outperform heuristic correctly apply information news article online news network news medium medium tend news keep discuss news medium infer understanding network gain various play diffusion process assess devoted statement optimization section optimization propose solution shown conclude discussion section propagate static direct create cascade infect unknown network adopt particular infer carry semantic influence give hide direct observe cascade triple cascade initially observe time reach node reach infected neighbor infect observe triple infect infection different recover network use cascade practice simply nod network cascade relatively node cascade observe cascade vector node infect infected infer describe cascade tree simply specify infected cascade occur show near maximum cascade occur probabilistic build cascade infect neighbor choose implicitly cascade infected neighbor neighbor infect cascade fully describe direct spread infect illustration cascade create implicitly cascade notion new gets infected currently neighbor cascade necessarily get neighbor yet infect cascade influence node observe pair node direct since propagate forward node property intuition decrease infection infect arbitrarily disease propagation scenario attribute could information status would strength type allow cascade infection depend property symbol description spread node infect cascade cascade node cascade cascade cascade propagation node e e purely intuitive transmission time distribution time argue particular multimodal interpret sense cascade cascade explicitly stop cascade stop never reach thus pass want cascade simply infected element spread cascade stop simplify simplification empirically work moreover rewrite edge fail observation come examine edge possibly probable product optimize specify cascade cascade gets influence parent likelihood aim cascade infection tree infect tree figure three cascade propagation tree combine propagation cascade consider propagation tree cascade spread cascade trees subgraph cascade even range span case inconsistent assume basically skeleton cascade propagate know tree cascade really propagation sum direct span cascade occur cascade occur independence cascade formulate transmission solve node infection cascade eq q edge equation constraint number seek sparse graph section intractable eq cascade
specify poisson follow theory rate many investigate expression count correct cluster signal switch induce ensemble equilibrium response neural furthermore periodic input profile derive periodic steady impact transmission interact population neuron effect short coarse dynamic abstract exhibit prototype phenomenon generally define rescale hazard drawback operational time inter event example constant period maintain dependent hazard function underlie approach regard choice hazard enable dynamic analytically investigate phenomenon pde dynamic analytical rate generalize compute hazard time neural activity ordinary equation like change rate transmission periodic input fix qualitatively dynamical class turn cover range generate hazard event instead history dependence hazard define hazard function component process density equation boundary rate fraction process age imply boundary interact population dependent solution uniquely lose trajectory determine evolution represent fig feasible hazard first obeys scale auto delay denote auto domain find constant due hold apply change use fig numerical display exhibit frequency ensemble gray mid dark gray periodic fourier steady active periodic ik obtain function mutually infinite ik spectrum different couple inverting give spectrum cosine obtain recurrence relation unique manner relation within require tolerance spectrum output display amplitude three low function emission rate maximize slightly hazard enhance emission interestingly second harmonic amplitude harmonic activity twice ensemble perform period maxima dominate maxima harmonic intensity process anti rate drive multiple distortion light mid dark amplitude phase dark gray mid light device neuron time introduction generation independent probability density duration remain event follow inactive see generalization normalization analogously obviously recover localize hazard generalize periodic random class density obvious hazard theory dependent hazard time expectation hazard function function hazard function fig special describe b time transform system equation enable replace differential equation constant tt temporal yield switch give simulation distribute function upon fig via qualitatively change spectra time easily localize pdf original hold coefficient emphasize validity output arbitrarily investigate pdf pdf laplace process call hazard process differentiable expression suitable sense translate hazard fulfil process inter pdf parameter choice illustrate know concatenation interval gamma frequently time cell phenomena hazard identification gamma entail generalize rate define inter event unit random number pdf parameter accord x study l line hazard solid upon simulation process hazard average bar denote deviation trial output point generation action potential release particle device ensemble produce delay delay describe property rate relation express dependent hazard result display stochastic might reliably rapid change demonstrate steady input adjacent couple rigorously solve term recurrence explain activity slow reliably fundamental stimulus frequency subject frequency contain stimulus stimulus
index step step subspace develop belong scope constraint form create associate mathematically efficiency converge direction em subspace former superior parent obtain mle acceleration minor situation illustrate example dynamically formulate generic solver linear system explore context em however later suffer hence inefficient common direction context solve analysis accelerate demonstrate dramatically popular perhaps effort gradient implement evaluation code monitor extra implement search em variant new efficiency cpu remain arrange follow motivating propose several conclude remark neighbourhood em eigenvalue determine show play illustrate acceleration determine magnitude dominate em growth assign group treatment weight mixed one fix effect effect stop example dramatically show figure take converge second converge em direction em entirely subspace fix since em induce implement step along large become eigenvalue observable nine loading call loading row em significant mixed convergence em explain fall entirely subspace add dm table eigenvalue remain unchanged eliminate dm eliminate fix subspace eigenvector eigenvalue span gain em choice converge depend potential acceleration dynamically information formulate generic iteration algorithm compute original b call large appendix obviously insufficient small neighbourhood fast direction lie iteration mle proof reveal illustrate relaxation factor generate nine example simulate em small neighbourhood mle mix one algorithm movement towards small line connect show two show implementation proceed repeat dash include iteration line search line connect numerical another way red dash first conduct along connect become clear generic cm search calculate demonstrate efficiency neighbourhood mle exact conjugate direction efficient near mle note much easier implement em algorithm typically code line search required effort note expensive optimisation one evaluation stand dimensionality descent view method system certain discuss bivariate adapt figure relatively fast second em second use second slightly em mixture popular machine pattern recognition extension dramatically fast class efficiency em ten norm cpu use em implementation dramatically iteration cpu overlap among first somewhat rare phenomenon may conduct suggest acceleration framework develop acceleration method em motivate development implementation nest equivalence conjugate justification also numerical simplicity attractive literature analyse optimum line however exact performance efficiency note acceleration work rate specific make work broadly leave future investigation chen david van zhang zhang zhang suggestion da establish mainly adapt neighbourhood may prove determined mentioned fraction name represent order matrix mle definite com com com com com com com pi com com ti com com eq equivalently determine eigenvalue simplicity eq make com immediately eq q eq two conclusion com dm obviously dm com I dm q hence prove statement conclusion dm statement conclusion induction proof generalised direction certainly true search em direction true hessian prove along com expand conjugate conduct achieve high routine may various line routine desirable trade propose transform constrain unconstrained implement search constrain along commonly software optimize narrow control optimize advantage force accurate mle magnitude constraint feasible find interval univariate several determine induced intersection follow vector represent direction cm degree boundary type e solution inequality handle way degree freedom linear mixed distribution current dimensional dimensional size covariance enforce spatial linear eigenvalue mix dm cc
tell singular simple approximation identification row orthonormal contain singular get eq row orthonormal orthonormal construction orthonormal mind let address restrict problem ga k da kf simple c c definite later simply choose comment condition showing actually formulate consider otherwise vanishe everywhere mutually finally distinguish like actually curse achieve sort decay consider discuss role contrary learn approximation require consequently sampling evaluation desire query necessary sake address simple g ridge mathematic ridge name projection pursuit recovery pursuit problem ridge reference therein notation assume column stand ingredient taylor giving access taylor expansion bound consider contain derivative matrix structure g g informally term approximation due compressed tell recover stable minimization precise get approximation set maximal derive z z final control satisfied shall hoeffding g formula reliable tractable approximation eq shall hold practice approximation application uniformly smooth additionally obtain evaluation compress sense matrix entry isometry least natural number part e reference part theorem immediately apply equation error recall copy lead assumption lemma relate normalize stability fix maximal ingredient bind j hoeffding recall reader convenience hoeffding assume surely exist scalar hoeffde inequality let hence c g violate hence ready ensure sign use ga c ax estimate g collect comment differ domain make heavy completely find address follow opposite arbitrary sampling hold success parameter choose say prescribe least posteriori sample properly calibrate otherwise beginning purely view arbitrarily derivative small extent appear ratio return detail stable reconstruction choice consider practical basically choice sphere haar rotation I borel rotation rotation ok identity natural become concentrated around sense fix measure dy r k informally matrix exploit ga ga long scale copy row via coincide span moreover eq right vector accord z singular contain define final kind accuracy compress span well singular equivalently separate related approximation use define r depend show theorem lead satisfie decompose may lead similarly put j z possible might norm singular tailor equivalent relate value necessary matrix singular decomposition understand size perturbation bind follow useful stability subspace eq say speak separation equivalent say away apply final ingredient value provide generalization hoeffde generalize semidefinite matrix improve technique chernoff dimension value sum matrix proof application thus get study random surely x g show recall lemma first singular vector define therefore priori follow posteriori ii far straightforwardly dimensional simple kb ga ga x calculation vanish due symmetry dy similarly expand taylor approximation computation deduce radial nearly radial respect able actually suppose neighborhood exactly discuss limitation firstly numerical secondly deal body sketch still rigorously interesting discussion evaluation precision difference measure lead sense substitute therefore estimate observe take effect noise unfortunately sketch natural random analogue identically variable mean variance regularization whose observe scale sparsity therefore noise error scope numerical minimization coordinate choose axis success white black successful mention free last picture increase decrease behavior careful inspection method modification replace point therefore prefer able even suppose actually x modification true circumstance easily enough consider convex need yy thus execute define xx yy yx hold drawback clearly understand subject I functional unit ball replace dual diameter always bound norm clearly investigation sketch special would get result depend acknowledgment like kind warm part acknowledge financial start award anonymous valuable mm mm theorem theorem conjecture notation remark continuous define ball budget point approximate algorithm point suitable approach compress sense chernoff semidefinite classical bound invariant decomposition subject key compressed semidefinite stability decomposition life capture approximate relatively smoothness regular accurately numerically complexity sampling measurement map notion performance grow exponentially result learning phenomenon curse dimensional problem exhibit eventually behave respect problem appear tractable approximation dimension behavior cf polynomially notion sort sum physics involve describe dynamic function tractable refer class consider q negative model various unconstraine address
volume optimal choice strategy function statistically asymptotic analysis ever portfolio market especially strategy increasingly research research adopt standardized variety influential financial financial mathematic economic market something statistically market et et et name develop add piece formulation treat market impose establish work research appear market within medium country economic medium portfolio package business economic modern stock access resource identify great deal situation competition choice worth portfolio market character share market comparative choose share package portfolio return request portfolio pass ahead familiar share share package characteristic valuable possible choice stop package portfolio choice base characteristic act fast successful go evaluate potential affected financial constraint essentially request request subject portfolio share consider return portfolio share stop share bank thereby bank compete bank medium relationship fairly typical modern valuable share employ shall minimax optimization rule major mathematical capable accurately market employ fairly mathematical prediction formulate strategy stock portfolio bank problem package parameterize assume customer financial sufficient several variant within portfolio analysis optimal ensure stability financial strategy behavior compete portfolio model assume package bank algebra markov suppose assume two stop interpret agent monitor concept p n borel moment lie mean monitor monitoring process refer value stop monitor convenience negative value include observer n definition material sequel hold q expression convergent mathematical expectation ij rewrite calculate let next small vx fx inequality infer calculate limit find excess function supremum q choice use act take every solution operation mean virtue problem validity vx fx n vx nonzero price analogously function price stop price zero observation solve stop markov observation fx fx formulate valid case find power probability ergodic asymptotic equality om ordinary scalar monitoring opposite operator exist strategy observation conclude proposition markov process state essential vector boundary quantity form least next consider portfolio share price constructive together constructive elementary event subset define space quantity form virtual possible process family important definition tt find fx subset map sx sx x usefulness choice database package bank portfolio supremum take markov stop stop choice valuable share bank portfolio stock market compete stock market bank package use stock valuable share portfolio make package package permutation number stock condition important thus round mathematical share package p desirable implicitly complicate package return portfolio prescribe induce event j taking involve valuable mathematical expectation price share bank share coefficient package stop moment detail choose valuable use choice valuable threshold stop markov first valuable share calculate probability associate conditional price x stop package choice candidate share figure main sequence chain additional add valuable package consequence follow relationship decomposition space subspace price set probability lk kolmogorov substitute get equation easy calculate valuable share stop moment find solve inequality verify induce decomposition inequality package obvious symmetry choice valuable simplify bank portfolio contain number share package stop moment satisfie take stop heavily bank assume low bank potential behavior bank portfolio share share package whose compare portfolio compete market stock appear bank share package notice somewhat simplified behavior priori information possess financial capital share bank portfolio financial constraint subject portfolio share package price prescribe strategy essentially bank portfolio process develop compete share package ranking usefulness characteristic package independently within portfolio permutation order number possible choice package n moment mathematical large price q bank transaction share basic
neuron create question parameter logical aspect formally database management importantly cognitive phenomenon simulated observer discuss updating involve formulation question realization consider period event clean observer record event algorithmic physical reaction observation possesse occur result add increase precision sensor language observer share question formal prescribe alphabet tag subset set endowed denote structure subset shall associate let happen toward observer underlie p correspond value gradually negligible limit face ignore treat irrelevant might boundary become transform nest complementary degenerate correspond face convenient corner corner full induce proper precede discussion pp mind distinct proper interval see probability probability diagram corner zero leading say denote q graph every relation every contain face main process introduction move removal redundant question question face infinite simplex regard observer leave abstract observer observer structure evolution observer study observer every face mean observer update either imagine alternate move state observer maintain whenever small grow become provide possible human issue study phenomenon bar change circumstance figure amount occur graph achieve update implementation achieve reconstruct complete picture reality price pay low benefit price turn update logical context elementary updating process vs natural aspect memory abstract observer observer consider negligible alphabet question group skeleton union irrelevant skeleton language role language formation formulation parent child question available begin environment lot parent serve child result parent memory sensor mean population common pattern memory agree meaning sound blind exist already decide population synchronization among set population describe child parent population subsection capability employ observer create question old main observer need capability exponential maintain efficiently prevent observer subtle gets incorporate discussion space impose algebraic boolean alphabet formulae alphabet together require admissible pa aa bc ac bc construction compare option call database arbitrary observe environment follow observer evolve observer interact observer assign observer goal capable dealing mean treat accord significance observer rather importance tune contribute accurate record part observer trait separation database physical observer vertex weight borel algebra represent observer picture mind life observer moment evolution actual physical physical observer relevant current observation division stage totally dependent physical stage algorithmic mention potential speak approach software hardware enable study question hardware support note observer choice boolean algebra sensor principle considerably fuzzy logic still one define throughout explore observer observe fuzzy phenomena record deterministic quantum temporal replace logic expect self structure could aware allow necessary role interaction database realization logic another logic question regard motivate propose enforce deterministic introduction strongly frequently observe thought include serve test title capable demonstrate aspect proper language develop summarize front put speak accordance update face database corner observer less implication unless force corner statement every act conclusion update lose manner least implementation imply propagation observation search reach either incoming unless mechanism value high ignore possibly context realistic implementation contain reason formally speak feed conclusion understand implication occur easy cascade propagation reach observer reaction wave reach like motivation network coherent family simultaneous observer relate introduce sensor correspond sensor automatically deal human difficulty able think write create loop reading arm update notion reach song stop material read work heavy explain precede paragraph stress biology physic totally stress peak become input periodic enough ignore stress capacity parallel input evaluate resource resource evaluation may search turn creating factor observer appeal serve mathematical precede computational propose biological force interact separately notion language level back stage evolutionary process language relate realization g underlie structure language language english understand natural language language acquisition individual restrict language capable produce thm proposition thm thm remark cm cm observer evolution state real process biological serve evaluation mechanism relevance observer observer assign recognize observer serve geometric produce serve transform model flexibility efficiency possibility neuron observable human thought reasoning loss acquisition dedicate memory complex extremely structure element comprise mention exclude possibility direct piece piece capability input desirable outcome specie exhibit remain evolution central ability roughly brain algorithmic underlie answer question goal intelligence whose na construct machine capable develop tool describe principle turn improve formation acquisition structure logical variety become desirable formulate answer main principle principle define answer invariance abstract mind realization us search exist discard tie physical incomplete put use inconsistent prediction database store maintain store memory retain handle necessity principle human shape management distribute among reasonable produce restrict unclear maintain principle responsible sentence contrast general capability human strongly say possess rather inherent example formula tune head failure phenomenon well difficulty else normally memory merge powerful tend management result evolutionary make desirable replicate undesirable effect daily occur principle resource indeed since constitute set admissible admissible situation regard storage ability explain functional occur frequently goal formal evolution environment environment state correspond observer additional information content updating involve content add receive evaluate mean procedure replace mathematically speak category belong database various possibility transform algebraic topological category correspond principle category consist every pair object composition define order triple bb c g say distinguished element triple compatible meaning axiom regard object compare improvement see difference two common may observer objective reality sense category model motivation source shannon introduce theoretic database observation space endow corresponding algebra observer sensor imagine explain notion entropy sensor absolutely preference know sensor know input correspond complementary producing become family power partition induce join sensor observer partition reason visible view possible computing observer approximate minimum expectation binary question ask order higher despite ability database put set edge construction space necessarily bipartite perhaps observe person observe space number precise north west imagine observer able ask point south west east us answer open number characterize quality well observer occur reader verify graph graph one well picture idea central geometric combinatorial content mark vertex visible state observer categorical follow sensor list list refinement hence onto induce every visible contain category modeling bipartite develop searching turn computationally observer come possess grow situation bad assume job interpretation business observer complementary subset probably never sensor go home fact observer maintain content without content vertex state make repeat observation probability vertex final track objective observer numerous time evolve add vertex think sample question result answer update quickly minor small cause practically big large intractable skeleton trait retain encode implication update structural e vertex visible interest detail close objective success paper underlie graph well study reader detail duality median treatment intersection precisely denote rectangular grid grid positive integer sub inclusion relevant consistent state respect realize inclusion vertex state one integer make straight segment figure demonstrate median three diagram interested set graph wu wu finite set coincide arise realization relate principle introduction restrict structure shall proceed idea median graph structure aspect duality translate median versa median graph relation impose prove restrictive purpose define property set embed proper element theoretic length moment henceforth coherent apply select precisely vertex visible denote family consistent lie illustrate example induce embedding belong box observe inconsistent vertex become reduce corner inconsistent look picture precision answer question correspond contribute circle obtain precise answer proof separate logical structure realization result distinction constitute introduction case representation realization realization seem figure ready observer seem memory environment classify individual environment may observe stress result context regard universe ideally ideally coherent note human hold coherent rarely hull observe family inclusion fact easy via intersection family relationship observer observer vertex observer event observer observer theoretic every observer state small observer aware source stress provide observer change stress try equilibrium regard plausible observer theoretic interpretation convenient describe phenomenon observer attract state observer choose result observer precise date correct probability state universe consider possible situation observer allow state universe ask observer view universe inconsistent state observer observer exist state universe impossible event probable situation motivate observer precede concerned observer representation would structural component rule bad common share logic inherent notion consider observer implication relation equivalence hold together automate reasoning environment action understand state matter logical effort understand operate combine inherent structure directly set cube example assign unit assign leave want disadvantage assume coincide inconsistent objective well way ask standard situation like observer needs identify answer come position would reasonable maintain relation increase keep ever encode record balancing exist implementation come possess observer observation search observer may update
become quite part suggest mcmc ps high true bad differ ps reliable relatively nested ps ps sc ps ps thing tend go wrong valid bayes factor validation explore change mcmc focus seem since probability one purpose bayes good discussion finding hope communication density feasible induce constant base first heavily good good moderately multimodal introduce bf sample implemented expand note often choose big correctly densitie density markov chain converge start state number chain converge similarly design converge iteration converge follow path different spread big contrast try directly marginal marginal thus fisher estimator bic ignore poor suggest simulate mcmc independent p variance convention apply need evaluate term care know approximation also usual compute information computation use neighborhood use see mcmc run sample laplace modification give behavior role selection ps big maximum likelihood regularity condition kullback projection easily expansion pn substantially unlike asymptotic mle still work pt pt pt pt ask manner procedure justify ensure role model joint prior usual density default choice jeffreys variance acceptable thought early regularity ingredient namely wrong bayes sampling factor true ps sc mcmc support conclusion computation bayes advance mcmc compute posterior major bayes complex phenomenon numerous paper bayesian book quick calculate posterior interval calculate select ratio marginal model probability ratio measure bayes factor modern recognize recently principle calculation bf calculation reversible jump lead different state chain connect space differ prevent mix another popular bf major explore nest dimensional last reference example geometric mean arithmetic path arithmetic path bridge modification usually cauchy recommend examine less p like popular recent popularity relative ease multivariate apply finance see ps implement integrate score discuss ps bridge summarize introduce path essentially except path need ps hold likelihood factor generalize prior behave mcmc dramatically properly big heuristic correct inaccurate wrong grid mcmc time start point ps work avoid propose ps well implementation point discuss validate heuristic projection likelihood gold sc draw burn burn suffice ps sc go show modify namely ps sc ps topics grid sc simulate ps sc choose conservative life ps relatively well section reduce sc say ps marginals preliminary screening comment briefly importance ps appendix one familiar asymptotic maximum course point asymptotic would point like ps sc validate partly partly subsection ps path selection model ps mostly ps review ps toy relate introduce introduce showing validity ingredient mcmc argument later section subsection among sampling p simple bs bs know difficulty suitable sampling try difficulty introduce act numerator extend connect bridge get numerator denominator generalized bridge ps idea estimate calculate construct bs path give unnormalized normalize constant taking derivative identity order integration eqn log normalize bf sample converge point mcmc bayes factor commonly scheme ps modification ps providing compare unfortunately example orthogonal bind one path convenient generally ps amp resp density factor later one common density parametric arithmetic path rao mean importantly mcmc amp ps regard degeneracy ps ps study discuss ps toy selection ps fail modification ps sc difficulties calculation bf later begin consider true wish independent wishart help bf conjugacy take wishart appendix arithmetic definite combination matrix path scheme describe early bf ps report define bf value bf bf mcmc true factor factor loading model imply outcome uncorrelated latent without rotation post triangular restrict parameter reciprocal entry may boundary equation easy calculate definite mle neighborhood example factor global give specification element normal prior diagonal low triangular conjugacy simplification idea loading scheme mix expansion induce class sign loading suitable prior whole range degree go use sampler stick convention factor define complex define factor depend say threshold path explore path tune grid sample arithmetic path path along construct gm prior estimate turn bf song along expansion respectively connect path score along path simulate sample distributional model point fy fy ps model normalize log mcmc factor parametric arithmetic assume r eqn path integral namely eqn notational write respectively show dt fy fy quadratic quadratic function apply moment dominate dct integral statement extend path remark like ps related theorem ps study estimate bf simulate report salient feature bf tend rather subsection happen simple prior scheme operate path path nest model small change contiguous scheme ps sc precision subsection ps sc ps sc demonstrate underlying issue load diagonal also mind remark tune namely grid size bf discussion remark path sampling converge finite ps cauchy cauchy moment integrable enough degree freedom f sensitivity bf change f table priors continuously show rd rd bf change prior report mcmc run mcmc standard deviation nd size mcmc size l pt major size fine grid deviation bf estimate differ order magnitude special bf correct focus prior grid size bayes estimate much value mcmc mcmc remain early ps explain bf low dim reduction model prior moreover cause relatively two ratio space properly divergence take likely make seem plausible stability lack bf show ps modifications idea ps remark identify cause poor subsection subsection ps try solve deviation ps sc ps pt factor factor give prior try compete keeping estimate step ps implement path reduce variance dimensional vector triangular correspondingly perform path computation I step score ph true close suggest fluctuation bf stability bf dimensional notice proportional computational grid size regard produce curve grid bf value set model standard estimate ps deviation ps argue ps sc bf keep loading diagonal table lie range respectively ps sc generated factor bayes ps sc parameter factor factor factor pt bf generally bf expect pattern estimate bf ps estimate bayes bayes usually choose true often ps ps bayes true ps sc well ps choose equally ps sc detail sc ps factor along remark study figure proxy fluctuation loading latter infer denote vector view sort mixture log sample nonzero come cluster present range appear representation vary moreover see fluctuation value represent near model ps tt use ps sc ps sc stand ard deviation avoid standard concentrate ps sc ps mode poor ps ps toward mode study nice mixing sc simulate use sc poor plot different lag sake ps except big ps sc figure ps slightly explanation mixing miss probably explain discrepancy notice bf ps ps sc look factor loading load top row show autocorrelation believe call autocorrelation loading dominate tend big simple cover model simple
infinitely element fmri environment improve resolution development genomic array increase distinct unit appear repetition copy briefly region genome span never contain genomic type genomic absence hence collapse per genomic discretized array number consist probe status hundred thousand region unnecessary appropriate mixed method point step resolution use reduction latter probe collective genomic neighborhood lose note high differ keep either number collapse mathematical correspond belong group similar secondly region use since permutation test independence valid permutation invariant wise error motivation spatial illustrate pair discretize correlation diagonal part notably wise structure span oppose spatial modeling describe apply purpose adapt take physical region practical concern leave region plot color correlation consideration spatial location th region first cluster besides class distinguish unconditional test usually margin margin permutation label permutation summary distribution testing association cluster permutation pearson statistic determine region motivate hierarchical cluster region procedure control firstly hundred fdr reflect highly share estimate fdr subset secondly cluster fdr usually subtle conclusion relevant plus permutation control feasible testing clinical permutation invariant testing invariant rejection combination improvement hierarchical context apply value threshold neighboring reject likewise connection cluster emphasize guarantee give rejection cluster cluster iii hierarchical reject j region region reject step cluster region less focus behavior standardize effect wise scenario cluster region scenario testing analyze region row er er negative row sample column involve concentrate per constrain sufficiently large decrease finally lie nominal correlation cluster validation precisely consecutive argument outline probability consecutive next need low use cluster region range confident create genomic dependency association label result cluster unbalanced hence peak differential proportion proportion small less region summarize identify cluster identify identifies region identify identify region hierarchical significant hierarchical identify cluster contain hierarchical comparable clear identify type type proportion differential scale axis dark grey significant mark bottom proportion gain group proportion c raw conceptual two separately array rigorous region versus unconditional available hypothese methodology apply unconditional cluster algorithm separate conditional unconditional likely conditional hypothesis cluster approach value testing acknowledge accordance hypothesis might prefer unconditional emphasize tumor importantly clinical good tumor large sample heterogeneous genomic location occur detect unconditional attractive external applie extent unconditional differ split part use test unconditional part significant reject confirm unconditional average region member reject reject conditional approach confirm unconditional cluster latter conclusion strength help region significance reflect wise adjusted wise extent distinguish genomic response neighbor far molecular level decide genomic dna really relate prefer discretize clinical information cluster useful would characteristic correspond copy rather gain consuming unlikely one digit digit loss normal would keep digit gain two gain double dna probe design strongly verify usually costly straightforward genomic cluster sample clinical outcome genomic individual small simple selection dedicate combination permutation multiple resolution clinical introduction resolution datum e massive parallel sequencing acknowledgement associate testing van partly software hierarchical testing package contain actual implementation software package site http www ii permutation test short prove distribution permutation permutation may conditionally sufficient allow principle improvement assume imply intersection relationship improvement dependency fully example x often wise random covariate dependency appendix multiple control consider case correction provide proof use also aim pre step hold condition region complementary sum appear hand c logical relation implie fact dna array tumor contiguous dna contiguous principle clustering pattern moreover really capture
full feasible speed carry point label classifier subset great magnitude practice fold variability minimize rate bx estimate reduce rate correct estimate x calculate validate addition original set feasible time note ten would estimate shall demonstrate application cancer bioinformatic al contain expression normal patient et al gene min max min minimum expression across sample gene level cell contain label breast cancer label sample separate give label et al gene small round blue cell b double firstly unit deviation row cross validate versus five machine multinomial plot list validate bias correct form guide level list centroid near unbiased estimate consider al question inspection heat illustrate heat sort visual separation negative top similarly separate pathway et present extremely dimensional need insight differentiable approximated ab reference lee parts object factorization nature van c university broad gene tumor arrays national sciences usa lee non advances system mit zhang localize base representation international conference pattern volume decomposition advance mit improve molecular class discovery factorization bioinformatics inferential microarray bioinformatic transaction intelligence component canonical matrix classification international conference bioinformatics chen california pp improve collaborative collaborative temporal conference discovery paris c g bias gene expression precision bioinformatic form plan r p j molecular discovery science discrimination method journal american yu early breast breast r diagnostic prediction gene expression nature r diagnosis cancer centroid national usa available representation molecular specie state sciences usa national zhang tumor use nonnegative transaction ccc svm square loss global iteration blue dot sort negative sort top iteration text latent factor ever demand differentiable bioinformatics factorization gene xx observe observation variable microarray gene might thousand let transpose overall zero reduction primary perform give form minimize frobenius square factorization replace b restrict nonnegative factorization nmf shall call approach factorization classic svd van exactly column matrix diagonal rank specify minimize al large essentially svd factorization applicable function square limit global iteration element factor fix loop minimize individually rather total second perform rank contrast factorization task minute demonstrate reduction supervise five know ab spirit variable interpretability consideration nonnegative b advantageous view interpretability lee lee improve interpretability whole nmf procedure lee parts subspace local localize et al et recently variation nmf datum mix combination al penalize decomposition describe function derivative member note limit loss since range algorithm follow input factor rate correction initial b generate randomly cycle step cycle step internal update ab loading derivative see total parameter loop fix role iterate difficulty convergence unconstraine iterate iterate switch partial process factor update recommender technique predict netflix note recommend maintain value correction rate illustrate figure
grid feature visual center rectangular partition whole partitioning cell cell compute measure local addition pyramid decay computed kernel level kernel kernel pyramid kernel band linearly spaced histogram combination sift result average active kernel mkl difference decrease explain result image classification generate normal increase band parameter weight sparse spectrum remarkably mkl true spectrum hand mkl row well often find consistent elastic band width weight dataset similar column medium mkl net regularization mkl change parameter mkl mkl two sample kernel mkl seem favorable dense kernel kernel neighbor intermediate favorable preferred result prefer ac trade uniformly learn mkl elastic depend weight spectrum also mkl often outperform weighted mkl however mkl helpful feature lot computation accuracy elastic regularization interpolation extend regularize mkl problem sparse mkl poor mkl small mkl among candidate reproduce hilbert functions mn nx iy minimization member rkh logistic regularization mixture determine trade rkhs non zero regularization square norm mkl correspond weighted mkl take th matrix nz weight define concave linearly concave substituting unchanged rewrite equivalent uniformly call elastic mkl mix hope mkl approach weight penalization al
event vector investigate jj second increase resolve step proceed iteratively loop step store orthogonal shall specify maximize separation wrong similar determine statistic compare k previously update remain consideration k kk substantial growth term send correct send hard weight false alarm indeed unlikely distributional property discuss define separation send eq specify matter grow near number normally induction high send send covariance joint constant density arise constant send exceed room stay case positivity separate gap choose consider quantity capacity capacity mean likewise play nearby compare growth case bound g lx analogous suitably alarm accumulation order source drop lemma non minimize evaluation favorable function make decrease skewed choice overall positive alarm satisfied produces indicate reliability successive normal bind deviation full manuscript pick satisfied generally incur error probability eq kullback leibler divergence bernoulli least investigate invoke preferred producing shown produce helpful simulation channel superposition code successive decode subset message index superposition code decode exponentially capacity presentation give decode vector partition power algebraic message string encode realize concatenation number bit specify power receive distribute map decoder produce estimate overall fraction section mistake reliability mistake high string supremum channel achieve arbitrary decode moderately decode moderately mistake adopt channel noise constraint code appropriate involve internet wireless phone space communication scheme practical scheme capacity equal sum simple allocate variable slight variant allocation power across agree setting find summarize finding capacity achievable albeit order gap capacity reduce factor mistake rate conference refine obtain exponentially equivalently journal measure gap capacity benchmark scheme include related superposition code theoretically optimal code gap capacity order initially compute receive inner comparison perform inner incorrect allow proportional constant variable mistake exceed result sign superposition improve subset without presentation prevent superposition code distance rs code overall interpret take either code fraction concatenation code mistake error exponentially small code good low iterative decoding mathematically capacity limit channel superposition convex arise statistical iterative preliminary iteration high inner residual communication reliable capacity express recovery combined signal convex accurate provide design establish complement typical recovery non lead minimum capacity superposition channel channel feasibility identifying
exist x measurable sigma x particular remove underlie loss generality ergodic eq f one point countable borel define elementary notion segment join family disjoint third tree establish segment gap subtree good binary rational integer define definition segment adjacent interval topological regularity family f countable countable family regular element ergodic join family disjoint join empty intersection lemma establish useful join segment suppose sub integer join cardinality remark intersection contain indexing without selection ensure tree binary tree locate refer distinct child exception two child internal leaf leave adjacent proceed depth short necessarily path depth node length show collection leave correspondingly nearby depth suppose leave exist child sum child section present arbitrarily tree segment segment root leaf call intersection principle ensure adjacent join final remove regularity satisfied internal child equal adjacent intersection empty interior pair non adjacent integer join adjacent cardinality every appear together establish order assumption regularity countable subset element join measurable borel lebesgue ii inverse borel measurable iv symmetric countable section multi identify join adjacent element differ least sample relative union expectation limit via stage important splitting identify stage split stage follow proposition treat value difference precede identical difference adjacent segment particular use hierarchical appear require involve binary keep difference substantially result detailed completeness countable borel measurable ergodic f dyadic consist countable lebesgue remove use splitting construct stage proceed f let join dyadic segment function pointwise sample join function fashion relation many average differ proof simplify apply follow state measure b tight subsequence weakly see absolutely stage appearing th produce define suppose splitting stage eq join fashion define lemma continuous set away integer intersection interior closure set proof condition internal set adjacent segment path empty interior intersection empty proposition let depth assign level begin child segment non empty construction lr subsequence set identical include exist assumption finite argue intersection adjacent segment measure exclude possibility segment adjacent segment intersect interior one segment segment ax ax k third yield suppose argument like g contradiction segment assignment set child depth property node appear select interval r subsequence inequality display sufficiently lemma l j proposition child follow fix proposition suppose interior node label segment adjacent pair non integer iii particular inequality node show integer argument integer subsequence adjacent embed subtree label child easy segment node appear root construction ensure member join empty tend infinity theorem statement f countable contain identity satisfie shrink diameter let map statement lemma process f define let f segment define sequence rational number include let fact interval family finite mod fix sense moreover rational family approximation argument proposition exist join obtain segment examine segment end h j kf argument zero join everywhere mod zero form join eq cell cardinality lemma completeness proof let ensure q follow display collection join include dyadic join diameter lebesgue bound two inequality sum join contain proof support nsf grant dms proposition family measurable combinatorial extent predefine convergence average ergodic sampling resolution ergodic average eventually within f gap finite average bound countable place place exist run title dimension number process complete separable f countable measurable sense every ergodic limiting difference expectation f discrepancy f theory inference consider identically substantial discrepancy mix focus ergodicity summarize ergodic call asymptotic discrepancy omit mention confusion say provide combinatorial quantity know scale definition family f gap arbitrarily mean et
h r together linear regularization model learn kernel mapping way regularize mkl weighted target one omit case optimize show equivalent strictly replace block criterion include net thus investigation criterion recover elastic easy section generalize begin expand value slack incorporate lagrangian multiplier incorporate equality lagrangian problem partial lagrangian kkt n rearrange lagrangian conjugate thereby remove dependency function follow inf convolution moreover conjugate dual generalize arbitrary regularizer loss solely correspond dual regularizer pair offer conceptual practical contain implicitly output optimizer induce form kernel need actual parameterization primal kkt optimality feature model focus case consider elastic net multiplicative elastic net regularizer optimality translate hand numerically optimal coefficient solely mean use identity efficient quasi newton hinge hinge optimization substitution dual inf convolution problem hence express k mx nm ms supremum call approximate give kernel formulation rademacher complexity let rademacher rademacher classifier literature far constant induced kernel q classifier normalize upper bound improve compare main rademacher regularizer term norm regularize weight influence regularizer recover approach term capacity elastic net regularizer decrease w b show first apply old dot c apply twice mean account factor analytically underlie mkl mkl elastic net mkl mkl unweighted balanced sparse intuitively non mkl robust mkl achieving less perform worst sparse mkl perform counterpart section prevent perform well block mkl error aim detect site rna bind genomic dna start regard key detector thereby rely task employ kernel represent shift st spectrum energy draw roc repetition experiment block norm elastic elastic elastic net net mkl norm block mkl norm block norm mkl vary model net approximate mkl norm classical outperform unweighted sum accordance highest surprisingly considerably recent norm mkl remarkable significance domain unweighte recently confirm state program detection experiment describe consist http traffic institute first http request randomly traffic traffic generate exploit buffer attack use virtual environment http gram avoid dependency http request length test example roc repetition auc elastic net elastic elastic net block norm mkl mkl mkl block mkl show net relatively typical detection net relatively reach mkl well sparse mkl mkl version bioinformatics mkl predictor present framework line mkl variant plug mkl variant term give concentration match previous bound mkl mkl depend scenario compare mkl bioinformatics network surprisingly mkl sparse practical future translate function hinge area roc curve ex e cs berkeley multiple lead kernel regularize minimization approach objective present show formulation special case analytically empirically select kernel task difficult view choose selection research allow predefine typically come criterion kernel leave variable base classifier approach take second optimize base norm trade contribution sensible encourage weight way extend
finding basis contribution improvement prevent local mode period jump secondly involve full two write application configuration point centroid configuration protein identify site atom atom sum match requirement match approach use shape et configuration green inference match treatment match configuration match regard multiplication rotation translation size translation chapter riemannian size rotation translation r dr td l ml translation q linear denote point parameter tx sx pm isotropic gaussian model constraint protein box obtain multiply direction protein likelihood variability lx v tv al assume use assume distribute possible gamma match metropolis position particular point match match probability select becomes match accept probability al metropolis hasting match require whole calculation alternative hasting match accept carry match ensure remove brevity shall refer model matching use computationally fast approximate get mode reach position satisfy certain propose much big move translation flip four reversible big jump effectively help subsequent match near match three translation flip step rotation near nx mx x nx distribution define phase jump phase interaction behind explore region big immediately else allow home big jump n p p match accept new match exactly configuration point turn green configuration body transformation co remove rotation matching translation perturbation uniformly volume concentrate rotation translation xx z axis euler angles euler angle haar perturbation underlie perturb mutually take haar density condition rotation angle hasting draw perturbation angle haar rotation group update lx match otherwise exactly mcmc update configuration similar appear note green matrix rotation update rotation angle green matching possibility configuration match situation al density rotation translation nuisance inference nuisance integrate marginal obtain optimize nuisance different density laplace method theory perspective rotation form explore performance bind site protein jump protein p criterion efficacy determine distinct configuration select random measure correct match figure configuration converge correct match reliably surprising configuration allow run continue million iteration monitoring match match reach allow continue big use initial histogram iteration success big configuration encourage big jump include success increase result within million run start point look run posterior go three thing look result compare result firstly effectively convergence method angle gibb make matrix convergence form proposal close formulate poisson much big start take lot converge jump jump let algorithm run period introduce jump converge big jump could increase despite take converge jump big jump always always since proposal accept half translation compare configuration scheme run start algorithm protein green accept match match principle one tend run run five proposal move status match value correspond fix effect protein independently run matrix match appear occur divide match match everywhere else match configuration change prior effect match model green similar configuration match readily relationship long suggest variability burn consider know algorithm without perform fix deviation point cube corner subject distance point uniformly cube corner start matrix match start record proportion matrix match record proportion matrix repeat match experiment choose four deviation iteration configuration estimate match reliably model match configuration give look increase perform model match reference illustrate study pose relationship simulation appear value two swap conclusion method jump however comparison converge reliably protein match optimisation
smoothness smoothness base since smoothness slight variation finite difference instance z x fx fx fx denote continuously imply induction generalize nonnegative z x continuous dimensional let countable product eq supremum partition il b I sf jj sum integral eq q finite coincide let integer ds k defer appendix lemma result let dd b ds lemma ds l dd ds b sd p sd sd p sd sd sd sd sd sd extend standard estimator numerical toy exhibit rate apart obtain ask use obtain question since variation describe alternative use permutation therefore another implement reduce still line simplify apply rmse mc integral approximated analytically digit mapping digit expansion form j b j get expansion digit expansion digit hence x dy countable exclude sa nj j b c j result b u u u c u u b u digits component b b b use schwarz b b last follow cauchy schwarz inequality dependent relate divide I since operator suffice b j jj kt l q ft u ft element case divide triangular supremum j inequality supremum take admissible choice b k sf j sd sf sd rd df technique integral statistic sampling instance root randomize monte achieve rmse strong achieve rmse derivative smoothness general additional smoothness achieve square integrable partial mixed rmse smoothness numerical rmse approximately accordance upper integral statistic instance estimation involve smooth standardized approximate unit cube transformation different carry bad power arbitrarily even mean satisfie smoothness yield improve instance algorithm achieve convergence smoothness know mc integral monte monte quasi uniformly criterion kolmogorov decay randomization achieve rmse bounded randomization method use apply digital net improvement digital net local therein rmse domain latter method mixed order coordinate smoothness use algorithm assume digital finite digital net base n dm dm concept classical digital net net suitable construction use expansion give function proof remain item dp digit also extend ds rmse introduce follow describe shall permutation digit permutation index mutually independent permutation permutation point drop subscript analyze generalize applie see
employ focus parametrization variability quantify bandwidth dimension approximation indicator note normalization function expectation respective derivative involve free leibler z z frequently negative pz constant ignore offer clear energy provide objective readily calculate well depend derivative equation equation proportional minimum bias information free visit neighborhood connect landscape unify criterion pose kernel equation representation discuss add detail carry expand nevertheless capture atomic end propose perform step equilibrium monte carlo scheme create advantage effort estimate readily change optimization aforementione algorithmic module contain propose scheme equation follow carlo available propose gradient estimate current employ early iteration gets discard gradually place recent p relate free energy cardinality expansion least way firstly salient ensemble secondly give vocabulary basis identify kernel surface divergence computational combination hierarchical add greedy procedure maximum generality isotropic kernel parametrize location propose selecting maximize expect value target current reason maximization readily carry overcomplete basis employ e wavelet propose efficiency free landscape naturally small successive fine detail capture e offer approximation successive equation kullback leibler assess general analytically whereas normalizing sampling augment efficient computation appear depend expectation analytically scheme degree force simulate suffer well know every change routine reason rely sequential sampler smc range path sampling work allow difficulty retain interact molecular dynamic dynamic hand change give approximate update mechanism proportional weight update particle dirac center particle expectation particular algorithm iteratively step descent successive z mp algorithm compute expectation carlo estimate mp building bridge base recover auxiliary intermediate affect overall smc automatically determine intermediate process ess intermediate priori reciprocal priori size I extreme arise particle whereas rest weight provide extreme informative weight ie dramatically accordingly p sense energy base produce high gradually towards energy temperature guess energy density employ well add convergence building distribution p purpose schedule employ adjust aforementioned difference eq demonstrate efficacy begin temperature remove straightforwardly general reaction coordinate reaction coordinate evident q integrate reaction see define straight forward pdf coincide derivation point adaptive smc mcmc mala sampler equation note technique finally parametrize free energy surface reaction artificial clearly recover value surface would complexity exploit parametrize rather construct gradually move large guess smc ensure level play inverse temperature wish free free upon simple used potential depict fix inverse region two apparent particle smc depict landscape capture less particle series pick three depict value select great majority free energy rest correction estimate kernel show number quantify equation offer great gain kl divergence periodic side interact pair give atom potential barrier separate two equilibria atom energy q ensemble volume probability atomic position temperature system assumption atom reaction norm box densitie high atom employ various box equilibrium move probable barrier slightly decrease equilibria move leave probable probable dimensional pairwise interaction play respectively potential system finally particle follow global truncate energy number minima parameter initially eq coordinate structure two separate dimensional reaction q domain temperature learn adaptive scheme automatically determine intermediate whole range aforementioned step mala adaptively adjust high approximately step nature smc scheme largely rest intermediate figure similarity energy surface neighboring optimization intermediate overall cost free energy surface four report study truncate correspond latter free landscape calculate previous assess result profile reaction perform numerical integration reaction coordinate depict agreement different e method leibler rigorous minimal adjustment representation free offer optimization conjugate employ wavelet sequential believe free system challenge subroutine uncertainty quantification propose free motivated molecular dynamic estimate energy markovian employ make use scheme sparse multidimensional case employ adaptive monte couple molecular dynamic sequential parametrize capability energy potential central concept physics rigorous consist probe experimentally global application force sample impractical infeasible overcome remain compute surface integration integration require long trajectory order user
friend walk union section walk different l friend event friend friend event group avg friend avg avg last fm identify fm store discover possible fact ground fm increase date assign exploration exception service sequentially rarely non conjecture non close account assign latter sample last fm user repeatedly million discard repeat uniformly space irrespective user refer examine population vs inactive vs solve fm valuable asset must efficient fm bit infeasible summary fm table week later capture fm estimate number fm internet use process substantial process fm evolve duration maximum fm period average therefore population day reveal distribution study fm network period ignore unlike graph collect fm stage discover graph user collect list friend neighbor list user list graph friend friend neighbor graph accordance stage event quite user group event thousand hand friend equivalent enumeration approach enumeration user uniformly event carefully implement action event study member group page return marked return cluster machine execute simultaneously walk walk randomly web site music country rely special purpose outcome seed seed let per online describe eventually collect sample user l type friend event group graph friend friend friend group friend summarize collect type observe repetition walk range event neighbor fraction event group neighbor relation besides event dominate combine group dominate friend occur many event hundred participant walk neighbor percentage user graph g event user play recorded expectation importantly table well type relation friend leads estimate exception percentage friend group thus weight emphasis compare table approximate percentage consider friend user fm plot number type ground friend event portion group average portion base however relation helps considerably utilize friend group close truth approximate fig close truth shape nevertheless probability distribution gap cause ii compare fm validate walk absence remain track popularity track type fm report automatically mention track tag track chart people plug service provide site fm stream validate track week week start track track popularity percentage top track available fm rank percentage track rank curve friend track quite actual additionally lie top last fm graph friend neighbor quite actual elsewhere get close truth early towards degree recent bias bias walk network graph walk fast mix case possible explore unknown graph sampling explore dependent improve goal instead achieve multiple relation sampling idea benefit technique couple improve across thompson condition cluster sampling broadly within design note interest confirm fm week apart consider study fm include track tag user user link explore fm crucial single part recommendation work etc sample multiple relation walk relation sampling connectivity relation fm approximations sampling majority user fail synthetic improve believe grow paper demonstrate utility sampling algorithm implement sampler performance expect correlate prove effective question helpful designing proposition corollary problem edu technique social rely connect time exist relation user induce perform perform select benefit fm internet music fast fail highly cluster service fm walks graph social decade popular present hundred continue study interaction behavior despite limitation service g limit treatment impossible obtain account estimation precise relatively ability draw know lack frame user directly principle especially focus limitation current scheme definition social graph need frame node initial explore produce poor cover body systematic walk social user posteriori know sampling ultimately dependent walk yield representative social speed walk target three group event union definition group union select pick relation connect user link social tie group either exploit graph compare typically graph graph naive collect relation exploit acceleration approach combine relation frequently connect graph graph require enumeration relation propose third stage walk select relation practice equilibrium distribution internet fm highly relation give structure methodology evaluate synthetic graph methodology fm practical recommendation discuss finally conclude different membership undirecte special relation several graph union union contain edge merging also implement sampling relation however union graph helpful conceptual tool seek sampling multiple naive way run random walk collect sample graph restrict never bias seed f graph allow quick walk practical walk union quite expensive require enumeration edge adjacent costly depend relation cost relation union graph employ vertex another edge enumeration node instead two f enumeration within neighboring edge save bandwidth neighboring relation high relation helpful offline application survey enumeration select select neighbor equilibrium process connect walk irreducible recurrent presence triangle issue need completeness briefly repeat parallel way walk traversal show walk sample weight weight hasting paper employ classic inherently degree per show apply throughput combination post hoc walk recommend obtain representative formal assess quality approximate hence stop use multiple walk critical scalar diagnostic walk third diagnostic across ensure walk demonstrate key benefit improve connectivity underlie graph cluster random determine quantify look component belong walk related node run walk axis characterize connectivity fraction belong er heavily connectivity relatively graph say approximately fast asymptotically union trivially exceed threshold characterize well relate associate top significantly drop grow new edge rare connection speed conduct experiment
chapter py augment define simulated weight region choice generally recover marginal integrate abc act practice discard realization dataset sampler simplify important way weight representation see simplification matrix replace statistic significantly sufficient statistic utilize kernel norm abc variate approximate computation use conjugate proposal reduce kernel sampler framework obtain variate proposal sample stable theorem non framework rejection proposal abc present day datum consideration comprise conjugate stable proposal propose variate proposal day assumption allow sample metropolis abc tolerance anneal tolerance utilize sample inversion proposal parameter propose sample propose unconstrained intra day vector perform involve numerical variate generate two heavy tail stable day versus approximation approximate ignore posterior assess bias inter significantly demonstrate day min line estimate posterior shift statistical variate utilize first impact bayesian framework appropriately shift next noise comprise stable innovation noise level shift illustrate deterministic time trade inter day market methodology extend modify apply conjugate bayesian transformation symmetric heavy tailed skewed non approximate mcmc utilize incorporate case asymmetric verify stable mmse matrix variate performance sampler gaussian financial demonstrating mark hence applicability trade framework perspective justify vary depend regime detail suitable setting fundamentally distinct describe property series deterministic series close market account fundamental appropriately underlie price model consideration mixture student innovation intra allow capture certain market still maintain conjugacy comment suggestion addition perform aspect cm proposition remark mathematics edu capital st electrical statistical auto extend incorporate price series shift accurately model develop variate comprise error intra day mixture normal stable inter day innovation inter boundary allow skewed current special price series shift either shift estimation shift market bayesian shift series inter fit variate inter day jump variate stable journal trading trade statistical pair trading consistently assess trading representation co integrate see overview address bayesian bayesian trading real price pair statistical fit portfolio shift jump series evident rank also shift economic social factor attempt economic occur close market trading time secondly appropriately shift co integration framework develop demonstrate robust statistical practically important observe series occur open market asset pair price shift solely market asset period overlap market open particularly accurately shift pair ignore shift consequence trade result affect carry effect design trading begin inter shift multiple contract segment span year demonstrate statistically significant level shift price flexible interest flexible term gaussian member fit stable price shift market trading demonstrate period imply fitting demonstrate distribution appropriate capturing shift typical statistical multi variate gaussian innovation fit basic utilize incorporate innovation thereby parameter period asset market include stochastic time point change switch trading tail simplification trading addition statistical generalize innovation symmetry realistic model novel selection generalize variate demonstrate data shift price reflect large model mean trading system arise day break frequency estimation several min simply discard shift occur significant trading trading close shift trade day discard period perspective shift integration issue address regular change portfolio transaction related trade volume paper price series make modeling price shift intra market paper variate distributional co normal family variate estimation newly aspect variate abc aspect involve standard variate model abc methodology aspect capture abc stable intra price shift contract segment price day model asset stable intra shift stable fit perform set frequentist due price series shift develop process simplify multivariate skewed conclude actual pair row denote random column successively kronecker product model furthermore integrate vector linear relationship assumed multivariate lag express format row dimension dimension represent trend autoregressive long multiplier long multiplier important root full univariate co occur co integration vector stationary univariate contain specify equilibrium variate gaussian capture capture stack variate model stable area data skewness heavy scale distribution consider shift include special sub purely incorporate composite process intra inter asset dependent denote respective market day univariate stable typically specify rate tail sign skewness scale term exponent tail analytically tractable member admit close evaluate gaussian member characteristic discussion intractable efficient observation abc develop day shift quantile stable http american stable corresponding paper market ignore roll effect historical behavior allow day shift stable fit sequentially window length fit historical assess evidence day shift substantial becomes contain reduce analyze inter shift extract open asset inter shift daily market jointly level cd note mini tu us year asset vary consecutive period segment end contract day market include shift contract day shift fit asset comprise inter asset table report parameterization propose fit stable model stable stable day shift series assessment stability constant period cccc result suggest shift distinct gaussian confidence even mini tail several demonstrate inter boundary day data pair series realization length error independently series asset estimate shift see series stable innovation raw versus stable price equivalent th sample stable fit asset h compare know procedure utilize posterior perform mmse specify assume knowledge nine stable hence assess impact noise display histogram bayesian mmse estimate group dash represent mmse set gaussian innovation inter day noise mmse top model mle mmse presence ignore vector presence day noise avoid appropriately inter day shift price us model introduce model assess admit ability apply long point wise would generalize simplification instead formulate bayesian specifically thus distribution intractable range context review review identical bayesian singular long run multiplier indistinguishable unique restriction identity sub skew bayesian prior distribution variate composite presence day conjugacy however conjugacy case via scale normal representation n variate mean positive definite matrix variate definite conjugate derive noise process framework specifically observation obtain matrix obtain covariance lemma variate likelihood factorization prove transform uniquely give transform transformed derive model meaningful allow variate important posterior instead significant equation strictly ie involve auxiliary framework mixed day intra innovation denote observation intra day observation vector include row subtract location parameter stable intra time exploit conjugacy beneficial decompose form specify theorem trace identity determinant identity x complete group inter day boundary grouping observation intra inter day variate likelihood price represent variate present combine variate statistic variate independent diagonal variate group preserve conjugacy result day prior reduction observation conjugacy remark gaussian model independence
mean equal slightly large detail almost estimator straightforwardly family nevertheless redundancy show conjecture propose modification ml put estimator contrast ml text bit outcome shorthand expectation shorthand finally discrete value write read mass however admit let countable x model family relative statistic countable sufficient remainder omit poisson multinomial gaussian variance suggest natural parameterization mean onto infinitely often differentiable define universal recursively note plug universal family common phrase plug universal constant eq plug plug plug model constant understand sequence plug introduce outcome outcome code ensure plug ml outcome practice hold redundancy define redundancy universal minimize follow definition kl exist unique redundancy major type code part specifically mild universal code depend log case major show model ml behave redundancy significantly examine expand taylor get exponential coincide another parameterization dm last step redundancy plug code family satisfy parameter exponential sufficient space widely satisfied exponential define bernoulli put easy x large variance take see family bernoulli long relevant statistic parameter estimator exist p set lebesgue immediate plug code plug behave family unless bernoulli fact exclude small modification put lead claim consider family equal estimator albeit lead conjecture something exponential modification ml plug estimator properly family I nm almost ml hard ml bayesian universal ml achieve redundancy satisfy statistic parameter denote q unbounded md dm validity whenever fourth rough express relative redundancy plug sum redundancy ordinary difference n ix v get expansion mm pt concern ml parameter establish plan analyze line conjecture analyse plug code sample redundancy behave inferior bayes slight almost resolve universal code code forecast code sequentially code previous outcome take equal ml maximum call code plug paper redundancy expect ml variety model family example paper plug redundancy behaviour way see plug calculate three code appear argument yet parameter model redundancy mp cite yet universal code typically behave differently substantially inferior plug selection
auc jensen mining difficult proportion actual link possible relationship observe although author conference another improve near comment manuscript laboratory research program laboratory company united energy national nuclear security contract ac social base link structure exploit consider link link structure predict link time evolve consider link year know moreover scalable truncate decomposition usefulness exploit natural temporal numerical tensor despite difficulty particularly periodic pattern numerical algebra author national nm national different web instance link exchange phone call g correspond phone call link exploit link handle task introduce predict link link relationship problem context movie interest user aspect predict picture link extend state temporal link step relationship periodic arise mail interaction pattern forecast make prediction far organize third array simple case tensor weight predict time analyze link publication link dot denote base slice approximation produce truncate singular consider bipartite proven method scalable approximate method cp higher prove successful network interpretable factorization temporal step periodic link web page web visit history place month computer network traffic computer science conference publication publication produce author conference pair prediction score impractical due requirement store score answer question conference year year practice problem predict link periodic example daily recognize use make heavily versus service summarize outperform summation bipartite use devise forecast cp expense periodic datum forecasting forward scalar letter vector letter e capital column matrix denote tensor euler letter slice entry element periodic conclusion different present unweighted weighted year score extend show approximation straightforward entry tensor motivated link backward call slice weight great link see demonstrate ct paper matrix specifically size column sum htbp score predict link calculate score low prove effective semantic indexing rank arguably one outperform undirected node link node node path term adjacency graph consideration adjacency first scalable inversion rank method applicable square represent situation rectangular represent q become generality diagonal matrix orthogonal incorporate rank rank replace diagonal base submatrix adjacency technique technique adjacency equivalent computation discuss specifically dense dense far consider bipartite represent adjacency eigenvalue sort magnitude matrix square matrix diagonal bipartite graph replace approximation result submatrix interesting except change singular rank via method require adjacency bipartite advance factorization factor amount storage rather operation explicitly model dimension one model cp also review symbol j k individual cp cp tensor svd tensor svd rank decomposition orthogonal svd orthogonality despite orthogonality cp permutation condition cp uniqueness cp tucker factor forecast easily depend rotation whether applicable topic make extract cp score outer quantify vector may different trend profile heuristic average choice component follow temporal heuristic alternative discuss temporal cp method periodic pattern require period method tool predict future period score compute vector step study correspond trend additive blue step red cp forecasting method analyze temporal work applicability leave cp predict cp link predictor matlab processors gb ram alternate tensor toolbox organize third let decrease large number slide seven contain correspond test year keep author least year author available train c new link exploratory temporal interpretability example component cp capture signature pattern publication author cp capture link factor mode third bottom list author conference combination early mid author conference discuss author trend mid relate primarily nice cp link predictor whether conference test year treat regardless indicate author predictor ct systematically use need determine observe entry magnitude slice form heuristic scoring first predict address well predict operate auc view presence imbalance link show predictor term predict link bar bar ct weight improve figure receiver operating clarity omit ct predict link method initially unchanged understand link predictor show cp achieve close less remove link test although accuracy seem order magnitude expect chance imbalance note ct bad predict link also observe among look even start give year ct ct link fast ct ct ct sec sec full evident reveal pattern section datum periodic pattern exploit set periodic play periodic predicting period length period ghz processors ram connection set entity year simulate day correspond roughly seven day might entity service service yahoo google service group people business service email service represent motivation temporal often service schedule may cache well business motivation traffic computer contact computer link could load balance predict link crucial matrix resp least note decide component entity choose strength pick length assume period train period use periodic pattern pattern correspond activity correspond different entity experiment show generate repeat adjust increase decrease neutral show column row train pattern tensor train make significantly train randomly swap select add percent entry instance cp large noise factorization predict generality e link zero positive trend approach nonlinear tolerance normalize gradient maximum iteration maximum technique obviously clear predict possibly week parsimonious prohibitive extremely difficult across comparison value train predict extremely randomly gaussian fit average recovery follow k assume vector ambiguity must condition permutation permutation well mention average yet cp ten temporal original blue pattern generally show line cp
show gain segment nearby lasso attractive ordering coefficient naturally order regularizer fuse solver difficulty optimization problem introduce auxiliary constraint medium derive bregman solve fuse fuse fuse fuse vector classifier preliminary experimental occur real iterative easy variety fuse minor modification aspect require equation elementary c involve nonnegative separately also theorem omit system fortunately solve efficiently ty x ty ty ty ty see ty matrix still solver energy minimizer subdifferential calculus differentiable subdifferential p contraction actually fact p get remark corollary fused exploit difference regularizer fuse solver deal medium fuse matrix split large fuse fuse lagrangian genomic array many solver efficient encourage increasingly classification procedure minimize usual procedure toward fast efficient solve million attractive large introduce fuse take placing coefficient assume standardize different lasso find encourage coefficient toward smoothness naturally real feature variation pattern order molecular order ratio dynamic inference gene distant exploit surprisingly application area wang fuse detect tumor array comparative genomic fuse tumor area fuse denoise social network quantitative trait strictly optimal solution computationally optimization one computationally demand solve lasso approach solve regularize problem include group elastic net lasso logistic coordinate fuse lasso loss regularization guarantee fuse name fuse notice et al pair observation develop fuse generalize fusion work general fuse lasso fuse paper propose bregman iteration solve fuse gain show compressed sense completion fuse reformulate split bregman apply organize lasso augment lagrangian fuse fuse lasso algorithm effectiveness section describe additional implement addition relax order along allow order arbitrarily g fuse error encourage sparsity variable specify toward fuse zero everywhere reformulate bregman convenient split bregman generalize lagrangian note lagrangian inner lagrangian finding saddle solution saddle iterative algorithm primal update base current lagrangian update relatively ascent step iterative problem solve objective induce involve term quadratic completely soft thresholde optimal efficiency iterative entirely minimization fuse analytically theory alternate minimization primal htp k convex split bregman hold furthermore regularization larger long choose fuse constitute special generalized fuse differentiable thus step quickly largely store minimal equation solve efficiently cholesky solve equation x z mention compute equation use step implementation introduce predictor identify primal stay precede matrix require solver split loss problem find optimal differentiable involve difficult hinge diagonal unconstrained reformulate function constraint derivation saddle iteratively update eq supplementary solve modification proposition prove q update yx get result proposition htp solve linear ty ty kb w k k yx property algorithm follow supplementary assume property furthermore whenever illustrate fuse lasso trial artificial application matlab windows platform core ghz fuse procedure frequently fuse design quadratic constraint package fuse additional objective solver implement implementation warm whenever evaluate compare path terminate stop convergence guarantee used choice choose parameter rate identify convergence select procedure certainly improve empirically work gaussian predictor outcome coefficient motivated coefficient change table cpu consistently test ten fold due second p plot cpu take solve figure cpu average run different parameter cpu success thresholding iteration figure solution thresholde htp htp equal use fusion solve art gaussian nonzero test p hour work work piece fuse fuse mass ms great identification protein application motivate illustrate fused problem raw mass site reconstruction equally spaced mass correction remove systematic
non dominate sense small parsimonious prior calibration elastic net selector associate predictor classical impose fundamental relevance keep within discuss application variable regressor microarray genetic deal poorly ill pose recently frequentist among paradigm demonstrate recently lasso literature induce question actor solution prior negligible effect jeffreys illustrate pay prior avoid problem consider variable purpose frequentist view regularization slightly great full comparison consider outcome dominate counterpart therein hierarchical frequentist real dataset conclude exclude vector correspond conditional variable symbol prior classical average observe regressor traditionally front fisher avoid would allow specification observational unit virtual pseudo fundamental feature prior improper probability uniquely improper framework allow improper simplicity reduce prior input often parameter center flat stress p rely namely model q p however impossible infinity eliminate influence jeffrey end information dependent amount observation schwarz criterion criterion g p connection criterion regression resort technique base particular nonetheless involve since author cauchy correspond augment form constraint must connection processing intercept measure specify used choice mostly jeffreys justify possible jeffreys prior indeed eq subspace span jeffreys leading jeffreys prior integrate detail model representation proceed close form p straightforward matrix explanatory predictor n predictor derivation estimate predictor exploit consistency factor prior depend despite argument location formulae jeffreys ensure invariance order would necessary center completely create location quite specific negligible situation location alternative consist exclude prior part center ensure present numerical popular method point selector elastic selection iy identify influential positive influential influential six dataset benchmark table approach variable naturally model averaging predictor l aic criterion prior g global base bayes take intrinsic hyper hyper exclude jeffreys selector net numerical parsimonious procedure overfitte fp solution method f scenario behave slightly slight tendency select somewhat seems tendency viewpoint model perform except note bic reject bad aic lead performance select achieve close systematically aic systematically notice mse compute well otherwise recommend one view ccc fp aic bic mse fp error htbp relative select aic bic fp htbp aic bic example ccc fp aic g lasso number htbp aic bic frequency bic fp relative frequency select htbp ccc fp aic lasso number aic lasso frequency oracle mse fp htbp aic relative comparison location impact last argument go relative distinguishing model example criterion summarize variable intercept otherwise criterion instead htbp ccc mean mse number htbp frequency replace moderate observation correspond aim percentage body regressor weight extend ten validate prediction parsimonious standard show mse stress compute bayesian highly remain open take daily eight variable concentration million response variable pressure height wind international percent air temperature inversion height inversion temperature pressure gradient mm original contain observation study observation bayesian opt five lack difference bayesian variable poorly selection parsimonious relevant perform point view
markov factorization respect factorize pa suggest across constraint respective subgraph pa gx implication independently individual valid give section parent follow clique sx vx gx sx pa gx corresponding function type graph parameterization complete parameter obtain I pa gx low introduce artificial direct operation add edge make child factor drop relation previous exploit probit gaussian generate singleton artificial figure illustrate describe section tie joint describe copula among dependence perspective put copula marginal joint distribution simply transform cdf incorporate encode return markov marginal dependence binary ordinal conditional function parent regressor fix basis among adopt copulas clique copula clique show copula plug become q cdf copula marginal maximize plug maximize via although likelihood consistent estimator substitute something feasibility hasting mh mh message pass scheme formalism compare fold validate probability dag ground truth non set uci repository ordinal parameterize frank copulas convenience average difference per r e driving alarm dag leave breast datum list introduce structure dag capture broad dependency broad parent small dag th predictive tell dag case log predictive significantly compare dag ordinal breast uci repository dag figure repeat procedure dag table perform use uci repository use fashion incorporate validation dag follow bi direct residual direct fix technique copula dag result validation fold bad computational efficiency alternate direct bi bi direct try fit fitting residual compare suggest dominate acyclic translate advance gaussian introduce family independence extend expect learn hypothesis prior knowledge structure play multivariate introduce inspire advanced structural procedure develop proposition factorize gx state except along parent let total say set x vx main result theorem induction probability accord must child acyclic except hence induction hypothesis marginal minor hold pa gx ix x order ordering accord respective subgraph fx pa gx valid non since vertex adjacent vertex pa gx ix side factor process repeat remain give cdf sx pa vx pa gx sx gx pa gx correspond cdf transform direct set marginally relation complete complete parameterization describe purely bi bi joint binary vector indicate summation contain equation interpret cdf transformation rewrite parameterization parameterization summation appear conditioning respective parent subset come necessity enforce independence path cdf clique accounting constraint hence construct different factor element enforce unnecessary understand coincide first figure specification marginal px parameterization come factorization fourth markov although level parameter coincide complicate extra complete reflect c cdf evaluate give generalization department statistical university college ac uk computational college ac uk computational college popular express direct mixed graph generalization much set conditional implicitly paper cumulative network copula propose construction encourage powerful framework encoding multivariate family acyclic dag undirected property dag monotonic sense know network allow flexible alternative direct acyclic mixed marginalization reading separation property practical bi direct model bi might much compare include obeys constraint exploit paper flexible approach copula literature review formalism copula summary graph clique px marginally relationship complete parameterization parameterization value cdf inclusion px x parameterization enforcing
dynamical give evolution relaxation regime angular momentum angular momentum different vary average change angular change slowly long lose intermediate regime enhance angular momentum evolution recognize relaxation angular radial take major uncertainty relaxation boundary especially intermediate coherent related relationship magnitude question dynamical momentum intermediate evolution angular suggest draw conclusion regard work angular evolution intermediate behaviour near angular momentum give brief review suitable angular evolution memory limitation object exhibit process define hausdorff hereafter exceed calculate dimension sketch suitable purpose self part self copy trivial correct segment copy rectangular cut identical give technique well consist measure detail length example measure give shall copy small copy general modify satisfied readily generalize self convenient curve check decrease total go broken going step term modify become determine simulate centre star semi major axis function final test star star star pick exact matter much somewhat resemble star centre uncertainty exist would around explain detail elsewhere field star potential include gr correction lead star individual star angular momentum consistent decrease significantly employ body star field adopt cluster gr parameter star axis gr behaviour nystr om discover part increase avoid scheme advance element correctly except truncation accumulate error star case treatment gr share last star curve record energy angular momentum star figure star plot curve fractional brownian angular momentum test measure star start feature long value gaussian change momentum unlike walk long decrease slope point determine accuracy star slope never evolution angular never slope reasonable vary star look evolution star expectation slowly star coherent short much star reach randomization dominate short randomization motion resolve intermediate angular enhance angular momentum regime rapid growth evolution brownian motion fractional brownian describe motion give generalization produce curve mathematically elegant simple easy physical brownian particle motion consist step duration choose increase decrease decide action sample step brownian motion walk small component brownian increment use repository average determine correlation strength length characteristic angular momentum long intermediate occur sampling match slope intermediate regime coherent slope walk walk start unit whenever take roughly walk show angular angular momentum near system test star cause back reaction cluster star mass spectrum apply possibly coherence potential angular evolve decrease randomization cluster relaxation gr comparison centre star mainly randomization would angular momentum body none mechanism angular momentum evolve evolution calculate dimension angular reliably regime key evolution angular result arise
share similarity occurrence multiple expansion functional whereas spline intensity link predictor internal covariate process develop package linear intensity expansion usage elsewhere focus theoretical likelihood dimensional setup treatment maximum show basis appear say penalize problem spline framework generalize banach general result similar result penalize maximum space solution infinite algorithm interpret reproduce term functional express integral functional consider trivially present involved property space reproducing filter addition homogeneous local vi take derivative martingale abstract always able decide likelihood martingale play count intensity theorem vi banach banach algebra linear functional observe belong measurable map limit hold continuity limit process process point intensity leave intensity require limit ensure wise interpret parametrize definition include continuous equipped uniform filter poisson present reproduce space give w filter analogy ordinary transform process terminology call inverse terminology evident negative strictly speak encode representation penalize play negative concave turn concrete give result suitably time log non second integral integral formula follow bilinear neither one deal derive derivative hilbert prove derivative log iterative formula put parameter stay formula derivative possible make continuously result formula play discretization eventually compute root smooth around root likelihood twice assume differentiable moreover root path second homogeneous move sample alone differentiable derivative differentiable martingale part equality side realization functional show counting case norm turn inner reproduce kernel moreover fix orthogonal onto hence give rise project penalize negative likelihood denote count minimizer span function practical estimation problem form q work trick outcome theorem q happen intensity however take explicit dimensional subspace log continuously differentiable explicit derivation consequence penalize solve minimizer minimizer must belong dimensional inspire gradient propose generic approximation minimize gradient differentiable need determine known literature fulfil choose h return continuously differentiable convergent strict convexity minimizer conclusion towards unique eq give brief treatment process intensity additive intensity function corollary linear functional equip v hilbert orthogonal wise minimizer negative likelihood span also generalize choose lead dimensional ordinary lasso implement model mark intensity likelihood parameter likelihood form case reduce separate example additive filter point example turn reproduce kernel hilbert q furthermore inner space reproducing reproduce onto order find integral description jump spline knot due see knot cubic spline mostly integrate enter r r function function polynomial motivated function specification structural algorithmic estimator result establish generality integral respect basis counting also expect integral negative observe reliable approximation penalize space integral establish functional jump trivially elementary reproduce requirement reproducing bound term term integration interval root derivative enough eq integrable useful straight norm though product kernel argument convenient continuous straight forward norm reproduce hilbert characterize reproduce evaluation cauchy schwarz since already integration linear continuous let functional precisely schwarz u h z z functional operator g g root continuous eq first consider z continuity follow embed proof continuous left limit strongly linear operator recall define outside mention gs gs g u gs denote translation act strongly unitary consider limit left limit let side prove sg sg sg I respectively hand side prove argument continuous continuous turn right limit requirement define moreover gs u gs z gs td functional represent hence combine conclude eq linear functional pg td pg td pg equality minimizer theorem result semi martingale give elementary gs regular weakly q verify check get right u g u u u gs u u continuous term conclusion lipschitz continuous since twice continuously sg u g
explicitly causal present finally amongst detect influence term contain measure interaction system prefer interaction causality equation causal little knowledge consideration fellowship dr foundation lead regression equation run invertible square invertible simplify extension trivial expand expansion repeat sum regression coefficient equivalent show partial may calculate matrix formula rearrange rhs simplify show prove rearrange multiply rearrange get follow immediately expand establish tr ga cd uk bn uk causality direct interaction causality univariate condition interaction take group ensemble establish causality interaction seminal causality support comprehensive theoretically causality individually specific illustrate multivariate motivate causal reveal type functional system mn keyword challenge many dynamical among acquire simultaneously increasingly aim inference system complement connectivity reveal dynamical analysis however interaction possibility reason usually incomplete operate typical fmri identify region fmri voxel voxel comprise change assess connectivity derive extract principal alternatively perform voxel approach voxel comprise range area economic biology principle assess causality causality causality particular g causality time autoregressive basic series causal incorporate inter manner predict future predict said predict importantly causality orthogonal causality among causality description generalize propose totally indeed recently appear numerically implication analysis section measure accord covariance matrix residual autoregressive total explore advantageous trace formulation transformation extend determinant important causality extend previously causal interaction show causality enhance g causality carry notion causal independence discussion summary use mathematical denote vector quantity matrix vector consider vector vector symbol transpose determinant square jointly multivariate covariance term random variable q identity useful g causality process focus use notation analysis concern thus contain comprise residual model uniquely specify zero regressor via obtain find stationary process z want autoregressive model process firstly lag lag lag useful version simply omit causality predictor variance regression notation multivariate write last equality formula variance know transformation asymptotically distribution central predictor long causality valid causality standard causality possibility multivariate square residual call multivariate causality note causality choice fit measure univariate residual uncorrelated minimize square nonetheless quantify reduce univariate autoregressive variance list causality generalize maximum likelihood z distribute large justify choice g causality henceforth appeal z unconditional q together cause cause identity subsection property causality simple factor transfer entropy quantify stem entropy proportional logarithm determinant conditional involve gaussian proportional logarithm determinant crucial equivalence justification measure causality entropy property causality development regression entropy extension causality covariance residual covariance linear account regressor formally ar general process behave conditional multivariate x pf partial covariance transform simple linear transformation matrix determinant find group causality measure causality unify rather arbitrarily define add component add difference think invariance transformation one angle preserve consequence broad transformation insensitive discussion one variable actually invariance mean compute observe principled component change value practical implication affect difference mechanism differently differentially context sensitivity connectivity content respective magnitude worth transfer symmetry group non transformation entropy transformation causality version transfer transformation predictor expansion depend decomposable product decomposition logarithm obvious expand combination expansion entirely sum present past entire variable past iv appendix univariate total multivariate help univariate predict help multivariate univariate support total univariate helps predict degree predict current plus help predict current whole predict implication high suffer stability univariate suggest expansion indicate extent present effect predictor enter equation stem determinant residual causality root lie outside lag exact exact lag lag since practice lag regression rely crucially trace density ii analogue analogue satisfactory trace causality remark g domain trace analogue form residual matrix simulate unstable process reject lag lag empirically achieve integral compute quadrature display confirm accuracy finite lag relative magnitude decrease sequence c display repeat confirm yield relative difference aside difference appear presence residual sensitivity group tr invariant restrict extend measure introduce exploit parallel order standard causality causality causality term correlation regression regression predictor causality covariance regression extend naturally multivariate rhs follow numerator denominator thus differ respective regression see always negative partial causality circumstance alternatively partial condition note section partial extent influence take explicit aim influence may influence strong relatively uniform measure partial affect predictor condition equal degree note express residual appear might understand proxy influence eq analogue trace partial covariance appropriately causality quantity develop trace refer problematic fail time f straightforward causal system previous define univariate condition dynamical system element element dynamic causal somewhat contribute globally potential various extension suggest various interaction principled scale size condition rest x r one predictor eq interestingly across currently explore could interesting predictor define average scale version measure progress density causality recently adapt operational system say g multivariate determination formalize along regression differ eqs multivariate situation element jointly self group activity group would g g activity macro extent micro extension consideration micro macro causal interaction ensemble perspective mechanic identification represent dynamical parsimonious extent dynamical extent candidate significant causal individually independence notion self independence useful description macro micro macro leave level description identify decompose causal within level finally iii characterize inter level standard causality measure originally totally residual term address residual covariance trace novel dynamical quantitative characterization novel independence analysis particularly biology may simple observed variable collection ensemble fmri voxel arbitrarily share similar eeg signal univariate meaningful fundamentally causality univariate important act influence cognitive behavioral since increasingly ensemble brain suit causal relationship onto principle brain operation important broad range decompose indeed acquire series approach multivariate propose measure residual variance provide determinant invariant linear transformation maximum distribute standard enhance fully
use say eigenvector e eigenvector cluster choice cluster bind distance incorrectly preserve span closeness canonical angle subspace span orthonormal eigenvector define angle affinity matrix ten eigenvector second affinity ten second canonical angle underlie fail coordinate justification figure coordinate eigenvector canonical maintain small underlie right affinity eigenvector eigenvector leave eigenvector difference angle still eigenvector local provide representation datum coordinate cluster compressed guarantee fraction entry measurement less ambient method distance compress completion perturbation eigenvector top eigenvector distance spectral define perturb first coordinate preserve original classification label membership match quantify misclassifie misclassification rate function handwritten project nd rd eigenvectors distance measurement often degree type perturb ambient dimension hide error matrix completion perturbation affinity membership unitary transformation compress sense measurement preserve perturbation sense measurement application collaborative computer wireless sensor network entry incoherence property reconstruct reconstruct distance preserve perturbation completion provide rigorous bound affect compressive generalize current perturb compressed measurement preserve span eigenvector close span show perturb unitary transform vi completion eigenvector cluster see previous perturbation eigenvector expand partitioning preserve small perturbation eigenvector column span perturb eigenvector eigenvalue similarly eigenvector write diagonal angle second require perturbation eigenvector orthogonal canonical angle space projection establish closeness span project eigenvector preserve form top unitary find unitary matrix find decomposition canonical space value cluster row corollary eigen perturbation thresholding incorporate eigenvector top develop perturbation coordinate knowledge amount error affinity many area wireless lose independent practice observe develop take inaccurate bound eq robustness perturbation let defined corollary probability give coordinate corollary technology lose corrupt costly incomplete fraction entry constraint solve nuclear convex task recover unknown know incoherence property obey resp resp sign rigorously property recovery completion completion local x kx obey high problem eq apply similarly multiply rewrite divide subtract simple ci proof combine corollary synthetic face handwritten digit distance frobenius datum synthetic dimensional sparse onto three nontrivial dimensional image compress sense measurement keyword keyword group wide balance feature create unitary transform level figure middle method compressive require perfect classification face pose view order experiment database head face fouri compression transform ideal capture desire difference transform level degree figure ambient measurement second capture underlie order often signal may measurement signal product fourier way image image face pose compressive spectral coordinate use range misclassification trial plot cluster image compress misclassification full dimensional standard spectral face addition grouping c image color classification label misclassification transform would level proportional show cluster handwritten compressive maintain make cluster first h handwritten digits data nd rd graph form distance random use varie stack row increase misclassification combination two completion compress measurement span completion traditional entry observe apply completion eigenvector cluster rank three measurement take spectral coordinate ph apply mathematics university california mining harmonic process diffusion mine ph mathematics spend university department california stanford technical institute interest harmonic digital signal theory book serve corollary spectral widely extract efficiently sparse partially combine cluster rigorous work spectral clustering incorporate eigenvector track compress sense affinity perturbation multi require affinity naturally number example mining become fast grow research topic mathematic spectral meaningful grouping similar eigenvector structure signal hyperspectral costly among work organization extract preserve compressed inaccurate measurement perturbation eigenvector compress affinity span eigenvector unitary assignment perturbation compress sense completion widely compressed noisy entry satisfy costly clustering compressed possible vector technique signal compress sense derive compressed turn lose numerous task portion understand impossible alone lack suppose stack contain use usually error propagate rigorous depict image cluster section h image range pose face
th trade suggest al european second structure comprise connect neighbor though area mid subsequent erm regard join common switch uk exclude uk regime lead party base erm adopt display market integrate possibility join european somewhat return exclusive regime use show minimum yield well return covariance assume per day regard show portfolio high changing thereby incorporate induce although mixture hide markov emission model extend nonparametric kernel improve rate high problem nonlinear generate outcome regression derive computing generalize regression implementation mixture model gibbs update group one computationally intensive slow plan explore near alternative particular move develop monte carlo algorithms feasible number however dimension experience walk inefficient base heuristic partition difference come graph predictive numerically recent development acknowledgment west provide ar support nsf dms brief auxiliary concentration parameter concentration auxiliary mixture l case lr full l introduce variable graphical independence lead turn tackle emission divide heterogeneous consider infinite mixture allow estimate illustration exchange fluctuation pre demonstrate provide extremely keyword markov inference size parameter challenge graphical deal type ill pose problem enforce sparsity graphical vertex node indicate covariance popular range finance multivariate conditional zero covariance relationship copula graphical model decompose joint pairwise copula however alternative copula countable mixture expectation consequence motivation investigate mixture often microarray encode conditional dependence information pathway implicit expression pathway individual different linearly study economic exchange economic block understand interact evolve membership mode interact mixture tool interpretable challenge implement determination estimation partition data grow exponentially mixture graphical mixture search specific determine number component address nonparametric mixture emission time employ since focus sampler integrate problem associate approach greatly avoid explicit representation state discuss outcome combination binary auxiliary begin bayesian inference graphical introduce graphical move mixture specie mixture markov emission illustrate section research multivariate mean decomposable graphical zero miss definite entry dependence relationship induce follow factorization clique correspond subgraph clique joint conditional mean prior precision g wishart lebesgue trace product clique wishart distribution normalize decomposable necessarily wishart factorize clique hence normalize subgraph associate explicitly prior variance specification identity result prior weight wishart likelihood k distribution jk become wishart distribution similar argument g assume decomposable calculate numerical approximation computation consider come model probability fully bayesian specification mixture heterogeneous involve practical practice know assign involve use reversible method inefficient dimensional alternative mixture q denote put define discrete say process dp distribute identically refer break dp appropriately useful dp prior integrate among sequence distribute distinct first model imply give kind grows increase cluster assignment also update follow neighborhood decomposable neighborhood connect graph graph differ candidate unchanged see improve cluster time cluster carry adequate mix interest n inference cluster predictive burden mean mixture easily sample conditional denote assign observation dp concentration parameter inference try datum auxiliary idea mixture gaussian graphical similar exploit analytically sampling model dirichlet combination extension first replace process species exchangeable follow specie distribute sample accord predictive weight satisfy species model include poisson normalize induce dirichlet imply give precision grow application consist exchangeable sampling baseline discount modify dirichlet modify weight reflect improve develop approach linear demonstrate assumption application markov emission correspond hidden hmms context hide trajectory trajectory evolve transition probability dependent infinite generalize hmms number process mixture model particular allow control evolve account nature transition state sub transition base dp precision update empty create trajectory kind auxiliary update simulation arise brevity result run representative run cluster connect cycle precision belong remainder compare dataset burn process run attention restrict cluster use full space decomposable full panel full observation incorrectly restrict graph properly dramatically correspond capability model translate edge exchange dataset consist daily observation five european european
slot arm index clarity algorithm logarithmic note feasible practically example polynomial computation traditionally armed bandit play expectation arm work turn bound regret consequently arm grow polynomial policy logarithmic like upper valid state chernoff hoeffding proof counter initialization period update follow slot period two happen play time pick increment arm pick element play arm equal indicator arbitrary eq indicator false arm pick get arm element h j jt chernoff similarly equation false l therefore policy consist binary reduce arm ucb generalize formation infinite set maximization least conclusion chernoff hoeffding need assumption optimization weight many associated matching constraint formulation weight learn regard propose storage regret logarithmic problem weight bipartite matching algorithm cognitive channel channel operation channel user due secondary potentially channel throughput spectrum channel period denote model mean linear ht minimization accordingly algorithm eq policy exponentially bad bellman short cost armed formulation path subgraph minimum cost constraint bellman time objective span section channel cognitive channel secondary throughput player show storage naive fig policy lot throughput I time high grow policy grow pt channel reward arm time slot total brevity policy problem modify pick play arm accordingly upper regret policy proof expect reward arm k expect arm play hold pick minimum arm note q arm play observe substituting regret consider reward unknown random policy ucb regret policy store alone storage combinatorial solve policy also computation short insight future question achievable conjecture intuitive rigorously unclear whether bind well context cognitive policy independently would policy open finally great tackle prove convex edu armed bandit dynamically operate yield reward accumulate player know mean arm accumulate policy policy regret grow multi bandit yield reward linear grow polynomially number exponentially policie storage generalization broadly applicable many formulate optimization problem span computation multi armed mab classic reward identically policy arm mab problem fundamental tradeoff hand explore order observation exploit gain immediate reward apply wide domain include network fundamentally combinatorial environment broad use armed argue barrier wider limitation basic formulation entity deal exponential setting exploit arm dependency handle provably well storage computation unknown action element obtain broad combinatorial multi armed bandit ucb yield regret storage approach focus maintain computing observation dependency base storage directly rather computation substantially selection policy linear storage essentially tight apply asymptotically time straightforward guarantee reward propose deterministic combinatorial np programming special interest thus property different policy readily extend regret well applicability bandit reward cognitive short minimum far work time allow random provably application combinatorial problem arise algorithmic economic finance operation engineer related armed bandit linear reward computation path span widely useful various combinatorial show extension large section contribution point paper infinite horizon arm bandit generate unknown linear logarithmic show allow compute policy logarithmic formulation work paper reward un restriction finite ucb regret rather exploit arm paper ucb therefore decentralized policy develop policy distribute player operate key focused arm treat dependency dependent case arm model assume present theoretical regret similarity regret difference reward show upper regret sum static across bind set work linear reward storage linearly regret regret bound among first combinatorial optimization grow polynomially restriction matching paper formulation system index evolve random restriction support normalize require unknown decision refer represent policy n arm component reveal instance discuss could eq reward arm indicate parameter arm
asymptotic reference think develop expansion sequence occur cf drive behaviour growth understand serial evolutionary describe overview phenotype stochastic model use feature age sequence dna relationship patient cell site patient examine problem part suppose record molecular member ask observer organization paper work homogeneous remainder asymptotic allele simulation asymptotic provide motivate sample dna patterns division switch site vice versa pattern obtain represent string binary patient cell two pattern neutral site measure cell cell pattern cell circle way simple consider information detailed body number allele sample third row number different sample cancer column distribution variation qualitatively cancer far cancer allocation among cancer cancer next section variation mm allele r allele identify mix collection typical pattern cell evolution mutation describe review provide population cell assume cell mutation assume cell division mutation sample neutral assumption express classical one neutral combine data allele count column depends denote write vector count cancer sample q cancer maximum solution give entire combine consistency see observer parameter interested assess goodness fit sample joint distribution count see observer chinese restaurant statistic relate multiple observer simulate sample cell allele count chinese restaurant crp individual individual type label copy individual assign individual low integer represent indeed subsample replacement sample count appropriate size sequentially remaining require say arrange crp run reject since allele frequency independent freedom determine observer sample observer course use statistic suggest see aim observer know answer aid understand evolution understanding variance number statistic average discrepancy use comparison combine th th crp simulation suggest anomaly underlie leave consistent investigate homogeneous crp homogeneous interesting suggest adequate attribute side percentile percentile share inconsistent homogeneity observer many start construct range empty proper rejection nan homogeneous reason mutation growth might apply mutation likely far functional homogeneity approximation variation begin nz sample label observer sample distribute irrespective let observer difference motivate approximation suitable poisson mean give combine sample observer observer number group allele observer observer present time sample component take approximate note allele black eq differ label independent interpret number assignment count distribute compare suffice prove poisson random approximation see proper th observer individual frequency combine observer individual joint need independent multinomial trial
without cover maximal cardinality code hx large n proof follow useful derive compare cardinality contain pm p support union see follow classic polytope every face simplex statement face minimal equal equal tight simplex hierarchical interaction fully accomplish believe provide well note set cover face correspond face see convenient idea remark cyclic polytope vertice distinct moment xt nk cyclic polytope k k cardinality satisfy rough bind set small ap p ap p p b parametrize r see span k vertex less simplex combinatorial cyclic polytope polytope cyclic tuple iff combinatorial cyclic polytope may criterion odd pair hence begin proof correspond r f f p let maximal bound cardinality ball radius binary hard sphere cardinality simplex x xy thus z grateful comment grateful valuable thank read grateful mis partly support fa question theorem mathematics university decomposition every exponential exponential face lattice write distribution combination number arise family measure parametrize partition family convex give express convex closure topology product random outer tensor record expressive understand whether count combination small connection treat mixture independent non identifiability largely variable equal soon negative sum generalize distribution large expect I particular model unit interaction smallest represent weak variable give bind small mixture distribution interaction set contain distribution aspect support family forward analysis special iff probability stand consider hull closure rise small set family convex support discuss treat interaction family call simplex variable define set vertex unit natural correspondence subscript paper hold call statistic distribution simplicity always sufficient depend depend row exactly strictly closure exponential distribution nest k kn choose hadamard realize sufficient image polytope hull point expectation detail polytope define affine combinatorial face together union face affine transformation preserve combinatorial type iff strictly statistic whereby line pair set except point probability two support red right support realize hull statistic example assess expressive different idea relate exponential minimal cardinality packing packing require follow whereby cover packing subset place exponential support within appendix lemma well implication distribution contain closure jx ks q x moment map direction px third item contain small simplex relate theoretical problem cover see polytope hull less simplex polytope face support interaction polytope dimension compute lattice g face face cover family call closure exponential context boundary point lemma
employ possibly parametric g gaussians researcher computer motion review geometric argument motion lie subspace algebraic method generalize principal sc affinity subsequent sec identify close classic local subspace recent affinity induce et al sparse subspace cluster ssc independently reconstruction sample rest coefficient regularize study theoretic ref liu et affinity employ nuclear minimization nuclear minimization surrogate semidefinite field effort sense generalize sparsity drive collaborative localization name cut solver accelerate lagrange multiplier alm see review learn symmetric reconstruction low without behavior learn affinity justify learn constrain affinity semidefinite lrr surprisingly connection canonical lrr formulation exactly characterize spectrum successfully uniqueness lrr lrr psd state lrr psd robust like lrr computational cost nontrivial elementary nuclear regularize semidefinite elegant aspect provide equivalence lrr lrr psd establish uniqueness spectrum lrr psd efficiently lrr notable difference throughout necessary framework affinity subspace next equivalence lrr psd lrr take briefly spectrum version lrr psd lrr proceed present tackle lrr psd robust lrr psd bold capital bold letter scalar g space norm induce value norm singular norm generalize vector concatenation sum besides inner induce alternative frobenius spectrum eigenvalue square collection real symmetric semidefinite cone say semidefinite simply requirement symmetry asymmetric matrix respectively notation choose subsection whole rank general invariant matrix unitary compatible value lie family norm apply norm singular fact place next duality nuclear characterization always achieve homogeneous formal dual nuclear duality take together characterization imply piece review np minimization envelope pointwise lower nonconvex convex envelope relate envelope pp sec envelope surrogate matrix prove mild optimization equivalent surrogate build lrr lrr tackle segmentation learn minimization surrogate incorporate semidefinite valid argue liu observation since affinity diagonal sort state favor segmentation reveal theoretic e exposition somewhat critical characterize lrr psd lrr obviously corollary immediately psd equivalence lrr lrr psd exactly minimizer semidefinite obey lrr psd lrr psd lrr psd lrr show objective classic nuclear get corrupt suffice spectrum property translate respective hence lrr psd however try employ alm see convert use generic form partial alm inexact alm routine fixing norm solution update update next show close basically asymmetric major cost decomposition counterpart robust partition hold similar solution take whereby ref remark theoretic proof derive convex unique minimizer remain admit spectrum cast unitary nuclear recover restrict since strict decrease reduce r programming obviously separable suggest form conclude proof translate since argue unique three ambiguity sign ambiguity freedom great sign cause value eigenvalue readily view I last problem repeat arrange act contribution rotation namely cause building devise real follow able thresholding unique take whereby element wise theorem ensure symmetry irrespective lrr psd either size svd convert eigen eigen eigen size include matlab evident eigen stick practice unstable problem solve systematically throughout experiment td I I dataset raw stack dimension heavily corrupt key equivalence lrr psd singular identical ref associate td hence simulate setting lrr psd lrr gradually increase regularization version intuitively tend clean present evident pass eigenvalue identically rest confirm empirically towards clean lrr psd please color psd lrr perturbation setting confirm although explanation produce thing evolve sense corrupt nuclear robust lrr psd spectra please argue noise essence objective version instead adopt totally random realistic variance collection percentage corruption total evolution corruption see well form speed psd conventional sc obviously c gauss sc ssc lrr lrr psd acc acc acc table present via affinity sc removal lrr psd lrr relatively virtue corruption removal affinity another denote lrr c lrr time lrr eigen place pursuit insight work affinity recently lrr psd towards understand behavior denoise scheme lrr lrr psd scale produce operational computational svd eigen large nuclear practical purpose work project office medium national helpful comment suggestion manuscript fact nuclear eq subset program objective constant formation solution readily definition interactive digital medium national computer national work employ high dimensional structural approximately lie affine affine subspace mention therein manifold considerably sc fail fundamental problem affinity advance one process affinity sc enforce semidefinite semidefinite
table regression use frequency ft motivation study reduction bandwidth cause increase surprising ar mainly absence short memory wider bandwidth quite control reduce computed contribution affect strongly increase spectral great extent bandwidth decrease bandwidth ft surprising nature ft correctly ft bias ar bias misspecification example misspecification end table estimate fractional correlation also calculate ft ft ft ft display show similarly ar counterpart correlation memory correlation increase coefficient ft method present superiority performance c mse ft ft ft ft conclusion performance c mse ft ft ft ft ft ft illustrative form observe box plot respectively table standardized claim fairly n mention property comparison would investigation model deal misspecification simulate part short misspecification relate specification memory ft estimate period misspecification ft ft estimate highly biased surprising bias non ar table misspecification non affect significantly ar stationary region part contrary contribution affect slowly decay lag ft method previously consider related coefficient c mean ht mean mse mse illustrate usefulness fractional third pm size two value display simple fractional estimate affect structure save fractional parameter artificial adjusted example analysis fit artificial request analyze short memory period present present request concentration daily pm concentration region greater comprise population million range throughout year average month period month month raw measure nd show autocorrelation autocorrelation figure plot series characteristic physical phenomenon expect datum since slow correlation lag lag multiple lag period process fractional parameter run period fractional estimate use tool section secondly obtain achieve ht pm display bandwidth use table estimate correspondingly nan reject table least fractional stable non large contribution fractional stationarity confirm max var var ar model ar infinite nearly scale autocorrelation adequate describe seem significant accordance memory identify suggest anomaly residual request correlation fall inside boundary request deal fractional parameter multilinear parametric estimator consider empirical general give estimate competitive presented implement sophisticated usefulness fractional daily pm acknowledge partial grant de suggestion part centre paris thank kind proposition ct department paper explore long property fractional one period counterpart stationarity series long accuracy investigate estimator pm estimation classification keyword fractional long memory common economic autoregressive phenomena period well know persistent long period become zero follow ar process also polynomial belong suitable fractional modeling short domain lag property frequency time non function spectral long memory model spectral memory ray usefulness model allow fractional paper gray france al flexible economic activity discuss test extend fit short contain introduce call publication property memory estimator multilinear guarantee carlo compare known method show multilinear regression estimation parameter however exhibit focus estimation one two period long ol fractional zero mean memory density generality even process next ordinary ol estimator example ols estimator normally distribute comparison estimator parametric approach misspecification problem ft estimator investigate final remark even difference binomial b ks long specific series fractional specific among fractional see ray return frequency filter odd term appear expression equation q fractional period process invertible spectral density spectral assume ij ij previously note filter order stationary process fractional satisfy property theorem prove deal simultaneous estimate memory et al reference well asymptotic unbiased inconsistent spectral series ol estimator slope otherwise integer ef ef k expression spectral variable center least run otherwise local center center noise constant procedure estimator introduction
respectively fr system bivariate case precisely reasoning multivariate fr hoeffding apparent obvious dimension presence variance bivariate construct first standard deviation correspond distribute random simplify notation construction present independently hx hx h quantity f ce also non decrease algorithm marginal correlation coupling note last paragraph work transformation preserve determination among know place perspective great entire distributional family remain marginal maximum derive follow dx distribute dx either double proof appendix valid choice marginal exponential worth many bivariate another recent concept constructive limit integer variable obtain output notice minimal correlation numerically correspond c density symmetry stand x minimum analytically minimal ii h minimum reader paper extension start identical pairwise correlation coefficient although applicable depend necessary matrix principal equal identically distribute correlation construction correlation choice dimensional determinant factorization requirement impose algorithm accommodate accommodate negative add restriction figure view plot distribution marginal region middle plot restrict give generally sign algorithm admissible number recommend reduce comparison concrete implementation algorithm thing chemical reaction beta describe exist compound copula correlate beta dimension form distribute illustrate fact cdf random scalar analogous quantity variable marginal bottom simulated note consider htp algorithm simple generation copula simpler require whole specification exact copula random generator desirable fast applicable accommodate range major determination range theoretical se emphasize low bound actually family commonly common distributional example straightforward ht independently stop semi one applicable produce w jx introduce huber li david discussion gm nsf dms exponential result section dx ij x xx I integer equal series converge add ii ii lemma section conjecture paper generation pre base fr simulation calculate example illustration detail implement dimensional positively beta copula algorithm beta simulation distribution specify margin back equilibrium fr evy fall hoeffde excellent overview al et al correlate pre specify play role stochastic encounter field finance environmental physics weather increasingly species dynamic far coupling copula become widely generation sample specify marginal dependency copula methodology rely characterized limitation surprisingly generate coefficient attention simulation notable work bivariate generation method present distribution mostly marginal common distributional gamma marginal
contribution monotonicity technical device idea yu maximization monotonically suppose increase objective choose call design general fisher provide assign design optimality seek represent choose log determinant view sample closure convert round chapter I use characterize maximizer generalize optimality mass example optimality general confirm satisfy mild parametric finite least generate assume limit maxima iteration establish yu omit question choice similar cost design yu carry treat iteration basic roughly basic slow al criterion theorem assign concentrated around adjacent design convergence slow potential convergence yu propose combine fast monotonic multiplicative et ingredient strategy exchange adjacent optimality investigate finding extend identity cauchy iw nonnegative coefficient eq write q multiply jensen apply completeness fix equality jensen proof finite finite nonnegative q integration yield form relate long hence relaxed inspection monotonicity finite remark yu layer prove partly grant university california author david computing multiplicative algorithm optimal design maximization principle multiplicative monotonic
project mmd mx xt dy sub moment probability class domain unlabele coefficient jensen one therefore k bx x allow keep separation invertible let vector onto space w tt pp pp lemma prove let ik x b set sub include index include b depend substitute assume line line g sub variable term k moment lemma l c l hold statement psd moment matrix orthogonal db q last equality fact ac il margin govern dimension conclude characterize rich tackle characterization rule learn source specifically tight characterization margin focus bound provide sample excess get dy understanding aspect rule compare upper bound often true sample tight lower bind exist source tight word concern g radius low support also precise characterization dimension determine actual complexity govern calculate rich light tailed distribution gaussian bound sample achieve bind concerned dependence desire obtain tight error contrast classical typically tight recently show learning example rigorously establish regularization discriminative novel tool believe work source least certain indicate sample complexity classical provide sufficient class distribution criterion see provide lower know learner specific learnable david set hypothesis line bad case vc balanced tight also distribution linear parametrize denote misclassification ds learning margin whose test concern size denote sample margin minimization characterize tail learnable restrict ensure tail direction sufficiently rich family distribution require restrictive namely coordinate far course multivariate sub extensive bound exist gaussian moment focus instance distribution bernoulli regard constant space mention introduction minimization term alternatively dimensionality tight rely respectively tight average dimensionality create converse arbitrarily high average option try trivially variance situation minimum dimension average integer dimension limit small example coordinate eigenvalue k k eigenvalue upper useful relate adapt margin establish follow prove proceed establish dimension term classic quantity prove upper minimization dimension column matrix space project project label similarly yy tm onto dimension ty ti tw yy w w j uniformly z I fashion corollary support sub variable decay appendix bind upper answer need complexity specific closely learn closely probability formulation precede dimension converse relate minimal complexity setting w dy involve existence equivalent iff margin margin margin correspond gram matrix condition denote th origin xx xx distribution sample apply independence hyperplane regularization homogeneous linearly require observation generalizing point thus use lower learn bind specify sample example distribute theorem let fourth asymptotic calculate provide low variance adapt dimension conclude lower bind quantity control conclusion derive however asymptotically highly separately family small provide survey sample distribution coordinate follow result also distribute proof find sub sub coordinate moment draw constant integer margin diagonal g whose small probability coordinate divide problem separately case assume draw theorem x x dm distribution draw hold depend conclusion influence conditional complexity highly easily interesting margin ng relevant irrelevant compare bound
marginal true probability compete section derive expansion marginal notational drop subscript glm vector response give regularity laplace taylor expanding exponent long bound meaningful derivation rescale design matrix n later determinant play simplicity take condition establishing asymptotic normality regression become align likelihood expansion satisfy r r prior n technical apply sensible normality regularity r bic compete bic misspecification second bic correctly specify expect introduce semi principle kl principle consider principle methodology compete fit glm index minimize divergence glm vector conditional principle proposition ig principle index divergence kl response true combine strength well drop subscript expectation nr expansion semi principle compete become become particular give negative penalty misspecification symmetric misspecification index natural interpretation regularity asymptotic glm glm approximate regularity n natural misspecification penalization misspecification expect ni implicitly refer cubic independent fit order comparison frequency regression aic bic outperform ccccc ccccc ccccc aic bic bic information simulate design independent simulate copy consider well apparent residual ccccc ccccc ccccc size criterion bic aic principle index well principle kl expansion bayesian principle family principle generalization natural penalty dimensionality misspecification respectively advantage correctly specify model impact analysis reveal covariance suggestion discuss take goodness misspecification possible introduce scope topic suffice da eq full proof easy uniqueness minimizer solve conclude q observe occur function maximum interior locate taylor segment n n nz schwarz lyapunov yield normality complete separate define n retain lagrange attain de denote kronecker delta idea expansion term imply order attain de along conclusion log likelihood easy concave attain value expand segment r entail key deriving list whose separately provide nj asymptotic expansion pick follow since fast rate claim q prove q give q inequality since remain observe n subscript follow proof entail complete definition variate distribution variate regression use formula least denote hadamard interesting true linear precisely involve due misspecification term maximum matrix quasi ignore attain condition remark university california university model classical principle kullback leibler lead criterion misspecification true true family principle combine strength expansion semi model new maximum penalty dimensionality misspecification directly demonstrate newly correctly specify principle model misspecification leibler principle principle model article explanatory desirable produce sparse involve subset one enhance interpretability fan variable problem compare predictor classical principle kullback leibler principle principle aic aic bic selection book account development aic performance study aic bic asymptotically true bic compare aic kl histogram asymptotic aic principle estimation measure absolute selection penalization wang li liu aic bic parametric priori fit maximum statistic wrong broad generalization aic bic model well among space set know predictor truly fan li fan family contain true neither aic bic misspecification discrepancy fit family true potentially helpful paper expansion several principle result semi sum maximum log misspecification independent observation extension classical rest organize normality estimator expansion principle principle present numerical illustrate propose methodology discussion implication necessary commonly foundation derive selection principle section white systematic treatment unknown function response entail ij generalize work z family contain glm set differentiable positive full two vector rewrite eq quasi concavity argument clearly maximum theory maximum white divergence misspecification observation kl density introduce upper kl divergence throughout paper specify regularity e follow divergence divergence unique solve entail f theory play quasi model estimator model include huber huber condition use misspecification extend maximum likelihood regularity condition n eigenvalue cn n small ny consistency intuitively neighborhood work far importance sampling target correctly outer product form normality row n norm specify normality conventional theory foundation propose selection methodology simplify technical presentation asymptotic analysis show asymptotic current g fan penalized grow polynomially glm estimate denote hadamard practice construct g bootstrappe treat square value study principle selection minimize kl show choosing maximize lead seminal compete connection
vector result experiment present middle panel already produce fairly vector believe remarkable another become panel cc satisfactory variance stable projection panel shannon size carefully choose good trade gm cc estimator projection right panel gm panel stable right panel stream strict technique entropy additive th accuracy study even entropy coefficient determine analytically know base skew recently th moment stream cc geometric geometric unfortunately still impractical prove cc estimator clean practical word improve stable geometric harmonic algorithm roughly extensive could accurate shannon even scale ten devoted stream tb database area g reach scale search typical source stream describe signal increment restrict strict suffice phenomenon relax strict moment stream q relax strict e hence strict e g web network shannon generalizations shannon enyi denote respectively enyi entropy converge shannon entropy enyi moment shannon entropy shannon entropy estimating numerically verify extremely let enyi clear frequency sufficiently perspective enyi entropies clear proportional variance course closely suppose complexity standard argument drawback provide shannon initially likely impractical know projection indeed exhibit traffic stream effective measurement network crucial anomaly diagnosis measurement metric shannon traffic goal describe histogram interested measure source service attack representative anomaly attempt computer computer resource service machine request attack typically site payment site attack statistical traffic certain since shannon entropy suit history anomaly entropy measurement detect attack low traffic time one traffic store stream recent devote shannon web search big analyze search use million triple particular representative stream history devote wide spread use shannon entropy entropy train approximate heavily computer science study algorithm space speedup processing note moment compute counter property maximally skewed random projection provide harmonic harmonic algorithm empirically another precise theoretical neighborhood neighborhood moment well fashion geometric adequate enyi entropy entropy contain harmonic provide geometric well note fix harmonic proportional harmonic adequate cdf increase g conjecture figure curve close approximate cdf dash curve lemma basically propose variance care random cumulative stable projection interested theory cdf see appendix compare mle addition considerably preferable prove asymptotically prove asymptotically use know complexity order normally tail least ideally really present bound tail estimator tail appendix complicated gain people even compute tail present follow formulate facilitate result bound write series rewrite necessarily word tail bind present together form lemma numerically k replaced prove largely overlap analytical expression actually analytical estimator actually fact intuitive small fact approach eq demonstrate sharp model stream stream shannon entropy application detect anomaly approximate moment even efficiently approximate frequency moment stream heavily science algorithm impractical shannon achieve bind base maximally skewed projection compress count impractical truly proportional previous algorithm entropy must note appropriate define j lemma algebra basically delta statistic careful make carry taylor take evaluate order moment eq property gamma eq monotonicity infer proof convexity convex suffice convex q eq eq tail minimize look tail minimize thus solution eq derivative prove prove propose compress geometric cc entropy algorithms estimation accuracy
train return reconstruct second spike fire hz firing course spike slowly presentation stimulus fire figure first panel firing depend spike history history select bit bit spike stay spike stimulus stimulus external immediately cause thus spike spike stimulus spike happen represent stimulus match stimulus history inaccurate panel fire stimulus presentation compare square discrepancy know external apply spike stimulus priori base something random firing panel panel wrong fact exceed indicate external stimulus quantify relative present post stimulus correctly internal stimulus strongly influence dynamic internal inaccurate us gain stimulus reconstruct second record iii discover state bit bit plot state randomness reduce resemble fire entropy low description suffice exhibit spike spike hz spike move subsequently spike neuron display spike triplet general spike move show neuron history predict spike spike statistic two reason neuron short due low firing second long present train emphasize parametric structure third respectively entropy spike entropy vary reconstruct spike train neuron bic select giving internal rate residual randomness bit fire history stimulus chain firing upon spike subsequently state represent external stimulus neuron spike experiment spike stimulus fire entropy stimulus average look however something complex place wrong stimulus entropy stimulus agrees drive complement different fire instantaneous external rate imposing want neuron encodes covariate encode identity existence uncertain pre cognitive state still use change population complexity examine macro entirely fire curve complexity array mutual apply calculate mutual spike train causal advantage causal represent behavioral pattern spike process spike calculate different neuron coherence reveal directly spike way spike train traditional analysis rigorously structure discover go beyond describe observe describe underlie spike also change thank valuable discussion valuable difficulty markov prediction time create filter strong dynamic linearly iid additive process amount maintain posterior update bayes call state whole unnecessary time transition remain update allow possibly chain circumstance go exponentially period useful place understand computational firing neuron considerable easily merely reconstruct cross part important check validity bootstrappe somewhat strong goodness test rescaling interval integrate describe rescale follow kolmogorov rescale kolmogorov ks cross neuron periodic firing spike set second stimulus fire bootstrappe stimulus fire largely fall stimulus panel rescale plot dash ks plot largely fall within bound stimulus stimulus show rescale stimulus worse surprising cause technique generalize reconstructing obtain perfect estimate inherent mathematically expand influence state mapping observable right statistic test distinguish set sequence idea state uniquely probabilitie produce input reconstruct future history spike g otherwise entirely computation statistical output train characterize spike train time quantify randomness show algorithmic content spike exactly describe minimal spike describe statistically randomness spike generate residual spike quantity regularity reconstruction analyse complexity spike train record device form one know activity inference spike train determine neuron channel capacity neuron give spike quantify randomness say produce throughout theoretically effective process reproduce train bit need reproduce describe noisy rigorous yet computational structure analysis spike train priori identify markov spike train train define computational letting quantify multiple group minimal conditional markovian spike train minimal capable call hmm splitting consistently paper use spike train history dependent familiar analysis analyse capable capture contain quantify spike relevant future reproduce effective statistically algorithmic information content average algorithmic content splitting complexity internal exact state residual randomness generative first quantify spike train precise functional version determine neuron require description method must spike quantify extent drive force simulate experimentally neuron train measure treat spike train binary divide equal step typically resolution structure present spike train program spike information need program quantify use minimal optimally predictive hmms reconstruct minimal computational predictor use available history limitation state group past activity spike train member predict construction ensure markovian spike train therefore like hmms graph node direct label symbol correspond spike average state markov chain idea figure simulate hz train second length iid train figure period spike period hz extra neuron hz state spike state spike equivalence past define period spike manner neuron proceed possible rest divide subsection theory consider understand spike train reconstruction spike priori discover spike notion namely statistical interpret reconstruct reconstruct response foundation causal sufficient one predict theoretic parametric predictive share optimality statistic actual statistic every statistic sufficient statistic summary retain basic statistic context spike train minimal sufficient statistic predictor minimal statistic always turn mean original homogeneous state causal stochastic minimal spike spike train statistically causal infer spike causal maximize spike train alone minimal empirically equivalence class distinct future sufficient statistic sufficient meaning minimal quantify observe causal recursively appendix find statistic spike hide state state cluster preserve length truncate infer building recursive long prediction rely sufficient future find save detail treat available program treat identically causal add state suppose alphabet statistic string map although contain multiple basis piece change conditional check kolmogorov ks fall right match wrong total l plausible irrelevant limit statistically reject sometimes complex nan start ii successive reach end ii sufficient state transition technical condition number discriminate traditional maximization bound use series knowledge structure reconstruct likelihood bic help causal grow increase datum aic bic markov chain class spike spike sequence spike state update recursively start state fix initial state use independent range grow fast next give symbol estimate limit less entropy output bin actual large see extended neural number hundred page inspection develop reduce spike probable remove appropriately transition probable train complexity sort find least probable state incoming edge remove keep keep state keep state stop transition merge reach repeat generate observe spike accept lowest iterate remove impossible chose gave choose show bic want check wise confidence checking coverage reconstruct check simulate spike train state probability forward inter spike pointwise pointwise often rapidly correct lie outside sort cross rescale algorithmic sequence comparison realization process algorithmic coincide entropy fact completely shannon statistic key determining algorithmic dropping term separate represent randomness spike quantify term intuitively hz figure zero state either probability contrast six need describe train period quantify high kind second quantify bit describe account spike capture train spike approximately two computationally represent generating process needs pick bit process always stay next symbol iid randomness total informative future bit randomness transition uniquely spike randomness stay spike need transition contribute bit reduce period spike less time fire spike iid bit rate update put spike fewer entirely quantify mean complexity complexity entropy quantity previous entire complexity entropy fire entropy variation stimulus fire probability vary spike stay invoke ergodic arbitrary integrable firing rate function randomness time entropy stimulus stimulus presentation analogously entropy calculation stimulus entropy time spike estimate appear definition interestingly outside average predict section external firing filter pass predict fire incorporate prediction simulation probability depend neuron rate generally external g currently formulate represent external precise spike train external
consider study communication sensor design sensor quantization rate scheduling good blue quantization decentralize multiple channel fc consider binary decentralize base communication communication receive considerable conclusion study terminal code popular scheme quadratic network access channel schedule transmission study show transmission efficient communication channel optimal orthogonal orthogonal law transmission result indicate source transmission channel access decentralize communication sensor code fc orthogonal quantization since quantization observation level general derive mle suboptimal local strategy represent various quantization transmission fc usually many receiver coefficient bandwidth fc symbol significance reduce provide summarize develop decentralize fc orthogonal serve low bound decentralize feasible estimator communication noiseless degenerate fusion maximal level redundancy redundancy sensor communication introduce various quantization decentralize transmission quantization quantization rest organize describe present introduce suboptimal estimator section analyze discuss codebook computational complexity asymptotic sensor fc deterministic inter communication among sensor sensor fc channel ideal orthogonal access protocol fc fc separate induce interference diagram decentralize system parameter transmission digital communication quantization communication digital analog sensor arrive fc channel use estimator digital communication extend analog parameter sensor independent identically transmission facilitate transmission quantization unknown write quantization nearest piecewise symbol much dynamic ignore transmission codebook quantization code decentralize estimator optimize codebook perfectly signal sensor channel channel period symbol receive I mean unit receiver transmission energy fc symbol special form receive statistically receiver fc px q shown give substitute omit likelihood simplicity digital communication constant symbol function mle maximize unknown fc function show received show readily substitute system symbol consist analyze define transmission upon signal symbol p regard minimum square error mmse gaussian mmse equivalent mmse ph upon mmse channel q l substituting mle fc model stage fc receive symbol receiver derive conditional function exactly unknown symbol implicitly provide pt mle show channel accounting channel channel contain correspond coherent coherent two simulation previous consider decentralized system prohibitive nevertheless practical derive low suboptimal estimator know unknown principle pmf lagrange interval satisfy interval expression pmf computing partial likelihood simplify necessary mle unfortunately explicit equation hand hand indicate regard receive stage complete estimator two stage view multiple necessary critical iterative good estimate stage mle iterative estimate criterion estimator probability uniform mmse first ip mmse unbiased inaccurate instead mmse hard system regard mmse variance mmse pt iterative follow substitute update repeat reach suboptimal differ applie criterion detect sensor true observation mle channel linear two stage iteratively accuracy perform simulation convergence study insight decentralize mle sensor communication perfect quantization resolution bit quantization derive mle pmf pt tell exploit signal enough perfect reconstruction communication level similar subspace follow ideal dirac delta term compute estimate mle obtain quantization detection perfect sensor receiver fc traditional technology communication meanwhile unnecessary substitute log recall normalize find quadratic max eigenvector fc estimator likelihood function observation mle communication pt eq mle apply centralize fc obtain raw sensor fc sensor error codebook show channel communication ht noiseless blue derive digital communication also transmission transmission transmission since rewrite rewrite normalization condition assume quadrature receive fc substitute log transmission derivation sufficient derive receiver part independent part h ignore real likelihood estimator transmission consider bandwidth contribution sensor local processor shift quantization equal uniform substitute cdf simplify channel channel digital communication transmission quantization scheme pdf show pdf pdfs correlation receive symbol optimal coherent rely real unknown ix ix symbols x ix digital communication transmission symbol transmission codebook ix ambiguity severe degradation decentralize auto codebook play especially unknown scheme code transmission codebook tn codebook cope phase ambiguity inherent codebook symbol transmission exploit symbol matrix enhance systematically codebook nonetheless preliminary result optimize transmission consider er rao fc second factor blue centralized statistically distribute among contribute equally fc signal long depend exploit infer know unknown bad mle take complexity mle mle searching order mse searching likelihood least fc term searching store conduct multiplication value get exhaustive mle estimator estimator consider simulation observation fairly energy scheme consider sensor low scheme cyclic code code codebook transmission also codebook denote code code transmission energy symbol ambiguity codebook symbol whenever unless codebook ts communication show mse quasi quantization blue bind practical comparison quantization estimator estimator examine sensor quantization rate bit rate codebook symbol simulation total identical use quantization total constraint sensor due example energy legend mark snr quantization bit quantization quantization blue low fig case extremely low medium snr level quantization inferior quantization bandwidth quantization observation consumption snr employ bit active similar conclusion draw consider communication consider communication channel convergence suboptimal estimator depict iteration unknown stand suboptimal communication mark depict transmission digital communication gain jointly traditional fc sensor combine final sensor receiver error discard error receive sensor symbol fc obtain quantization except use codebook codebook simulation codebook stand stand estimator stand bound communication suboptimal fusion quasi suboptimal snr mle transmission transmission channel transmission codebook exceed snr low snr cause transmission bad code reduce fusion base drop due exploit bad show impact observation stand symbol respectively denote mle apply true reference unknown symbol mle estimator codebook symbol evaluate symbol codebook communication ambiguity severe degradation symbol still bad
consistency compression promising alphabet set tuple borel sigma mean distribution dimensional define l b kb kb b kx nb bb b ergodic frequency occurrence generate priori ergodic distance set difference probability fine partition easy base empirical expression involve infinite consistent joint ergodic fall analogously grow whose already converge converge yet index moreover lb lb lx un thus partition cluster target target consistency sample partition asymptotically weakly consistent pf respect point next away assign second minimal already assign cluster cluster close assign iteration initialization design sample k jt distributional zero obvious complexity calculate enough last calculate statement let way stationary know consistent calculation requirement perhaps cluster finitely belong otherwise therefore c select consistency next pairwise distance second apart rest computation order calculate precise estimate want check sum replace partition subset increase q lemma integer infinity set increase enough estimate algorithm consistent cluster statement true condition cluster whose computational complexity think proposition possibility trade burden set take every unknown rate appear cluster know advance joint ergodic unknown impossible independently stationary ergodic asymptotic hold come make assumption expectation mix generate datum mixing without model assumption cluster bound multidimensional straightforward formulation informally process past one make assume stay sigma algebra generate strongly many process stationary irreducible markov mix underlie exponentially assign pseudo code distribution select weakly consistent let sample generate way joint satisfie satisfie depend parameter algorithm weakly process union sum I b eq every pair answer therefore together pair sample analogously theorem know bind multiplicative term make practically fact take individual frequency take union obtain realistic guarantee frequency cell partition coefficient speed strong without define cluster stationary process advantage framework make stationarity ergodicity simple check cluster initialization computationally implement distributional distance replace distance spirit direction concern rate finally consider length line problem length grow theorem claim net clustering parametric notion generalize statistical homogeneity consistency achieve stationary ergodic assumption neither mix give argument object object formalize work particular clear finding measure integral measure euclidean appear reason np hard concentrate certain numerous biological notion ergodicity define satisfied within assumption ergodic mean intuitively virtue ergodic select distribution observe time ergodicity countable sigma underlie alphabet tuple say sum
take ball true pt pt pt plus predictor variable support combinatorial turn replace cardinality envelope paper may knowledge many set envelope tool submodular norm algorithmic proximal operator support interpretation base group norm potentially overlap one factorial scientific signal process variable situation interpretable admit one look sparse low turn practical processing bioinformatic structure bayesian process prior dedicate focus mainly specific norm family instead follow support beyond cardinality limited pattern restrict penalty insight see function norm cardinality ensemble namely see submodular envelope extension algorithmic e proximal operator condition dimensional cm submodular recover give group potentially norm factorial experiment outperform relate absolute value vector submatrix define modular throughout fa cm refer also generality may monotonicity simply partition equal empty lead group norm extension extension note piecewise convex extension vector identify set sa fa submodular q component decrease order augment without strictly refer stable set cardinality set separable show submodular polytope soon strictly face p sa set play describe unit norm derive cardinality stable paragraph otherwise rely sa ga refer moreover look look lead bad fast practice equivalent algorithm solution eq compose absolute indeed envelope envelope strictly I norm envelope ball iii fa fa fa example submodular vice versa norm ball proof example point unit ball stable go possible sign example point concentration inequality supervise norm section novel submodular interesting example function function consider group norm overlap w soon submodular allow intersection strictly positive weight extend however zero go restrict sparsity give various topology group define acyclic element bioinformatics vision organize norm group vision group side figure empty gap already regularization order effect small norm come contiguous right side correspond equal plus contiguous limited fuse relaxation number jump extension semidefinite define nonnegative eigenvalue thus lead correspond entropy thus lead submodular see however dedicate submodular one applicable submodular minimum algorithm simple minimizer regularize induce pattern section lead induce norm rely submodular submodular restriction circumstance w w j set make assumption specification show one appendix simplicity situation assumption pattern assume lebesgue invertible minimizer unique one support assume support proposition decomposable property extensively j proposition proposition get similar jj consistency assume vector q concentration covariance set aa paper least square give unit gaussian signal function beyond compare proximal accelerated fista one fista fast n k three solve combinatorial w f approach induce norm forward ordinary method possible easy approach predictive greedy table factorial prior large support submodular multiply average replication deviation divide measure pair induce norm dedicate set synthetic norm worth current practice sparse enhance norm concept application link relaxation combinatorial study total variation cut cm partially project european author discussion positively homogeneous norm soon w p w w increase increase w w fa w desire face one happen get desire potential first prove subdifferential non complicated component magnitude e subdifferential subdifferential subdifferential stable contain subdifferential h w subdifferential norm zero unit norm desire nonnegative applicable z constrain nonnegative positivity equivalent constraint submodular apply approach regular solution negative submodular minimization unique zero may obtain minimum efficient regularize cost sa sa sa c fa fa result consider ii within interact include equal
subscript classifier change dataset minimize possible differential positive property perturbation additional cost compare perturb differential dataset p dataset previous due proof strictly convex objective function strong commonly machine technique huber adjacent perturb function convex minimize differentiable objective solution relation optimal perturb differentiable perturbation perturbation adjacent dataset obtain perturbation apply property calculate draw surface radius jacobian matrix mapping regularize rearrange definition square excess regularize classifier substituting prove regularize excess bind utility trade classifier parameter ellipsoid classifier intuition confirm proportional paper gaussian privacy technique add privacy learn directly trade extend work along technique theory arrive excess margin intuition large robust would insight design mechanism acknowledgement would anonymous within huber calculate huber instance eq algebra frobenius h frobenius huber one loss norm frobenius q lemma aggregate become develop mechanism process differential mechanism algorithm classifier perturb regularization excess perturbation year vast amount aggregate medical record database lead individual desire propose adversary instance instance instance observe new private able individual analyze propose modify add perturbation compare original preserve work binary situation multi class differential classifier classifier introduce modify year privacy privacy collection element randomize produce dataset say adjacent one substitution entry substitution modify dataset say satisfy execute adjacent query define differential dataset determine trade utility requirement think classification density algorithm differential dataset individual observe output algorithm already adversary differential oppose adversarial mining connection privacy study framework agnostic algorithm use create differentially classifier add estimate differentially private formulation modify laplace advantageous compute perturb introduce algorithm address problem expensive naturally class present differentially leaf theoretical analysis perturb investigate margin class training dimensional datum instance ellipsoid ellipsoid parametrize inverse offset decision mahalanobis scalar centroid simplify expand collect class also eq discriminative training involve semidefinite matrix rule training instance provably guarantee formally train centroid class least centroid one training penalty incorrect trace matrix label correctly ellipsoid prevent trade covariance upper identity replace surrogate hinge positive programming efficiently use interior
cause slow problematic carlo precision problem ht case q increasingly correlate arc eigenvalue generalize total square report carlo various bootstrap perform nan c correction bold test conditional test assess first sample generate alarm software package consider asymptotic fact log call mutual stein develop arcs acknowledgement ph school science article give many comment suggestion department pure constrain application frobenius matrix minimum maxima constrain test prevent interpretation statistic line strictly unique correspond covariance see valid use derive probabilistic direct let undirected configuration undirected graph present graph therefore also prove network uniquely assumption measure subset introduce monte test undirecte learn small bayesian bootstrap multivariate year bayesian successfully different include biology example rapid one grow score genetic hybrid one hill main need bayesian correctness assume size absence positive negative datum benchmark difference hamming evaluation real world either parametric apply large possible probability markov limit undirected graph underlie multivariate arc derivation exact variability network part network node variable article dependencie graphical dag distribution parent therefore measure specific presence arc particular variability goodness network bootstrap marginal arc direction original confidence evaluate dag confidence depend bernoulli marginal joint simultaneous every element result reduce parameter vector first result independence random completes extend ji turn sigma b c sigma two correspondence property normality second apply subset bernoulli variable also marginal bernoulli variable dependency uniquely identify numerical form element attain cauchy multivariate eigenvalue non hold complete sequence distribute preserve one univariate variate binomial closed guarantee therefore arc eq identify w parametric bootstrap several statistic variability direct graph simplify bootstrap case simulation two bernoulli several network frequency equal proof propose usually assumption multivariate normality three bernoulli call associate statistic structure bound generalized hadamard determinant negative definite reach reason convenient reduce instead frobenius network eigenvalue minimum behind instead correspond associate whose
complete hold prove regard plan q maximum submatrix follow simply generalization extreme singular permutation proposition value provide result well lemma along somewhat make develop union proposition regard lemma obey coherence therefore assumption elementary algebra hold difference martingale sequence inner independent gaussian let random draw every concerned complex random easy proposition let bound difference value eq real martingale simple union bounding follow inequality norm complex let independently distribute loss since follow rescale norm union event proposition definition remark linear structure graphical non model general regard generalize model coherence term bad coherence average coherence column design utilize measure coherence one term insight regard successfully carry fail coherence optimally signal average entry high key extend model use agnostic particular frame carry selection irrespective nonzero entry almost frame incoherence matching pursuit pattern recovery orthogonality process curse often break exploit live manifold vector p observe represent operate enable task computational fundamentally measurement compressed complementary nonetheless question need answer reliably reliably researcher year area known model denoise problem case enable objective problem among roughly case seem design coherence coherence introduce conference version spread within unit ball vector contribution area agnostic specifically selection term despite primitive optimally energy per nonzero notation equally nonzero objective regard contribution recent model recovery particular frame shift seed signal energy entry far generate coherence recover signal value entry nonzero entry away f solve rich context compress aic essentially attempt regularize square criterion seminal work well therein researcher year notable aic bic make procedure recent method lasso arguably become selection partly provide report correct certain result nonzero report asymptotic limitation verification selection result matrix plan correctly small nonzero entry singular bad coherence symmetric recent theoretical still agnostic propose thresholde use issue magnitude small nonzero know model plausible whether condition selection lasso beyond generic design arbitrary particular even tend demand much old complexity model present differ five completely agnostic order deterministic statistical nonzero linearly study design matrix influential report statistically around relate namely trivially achieve consistent model condition light rate consistent addition model selection also characterize partial regard probability cardinality hand study model matrix gaussian resp prior three result compress nevertheless conclusion long agnostic threshold enable threshold carry submatrix reduce third universal threshold report context recovery compressed setting exist problem bp selector variety ill suit complexity hand iterative match subspace pursuit hard pursuit fouri sampler show perform rip rip intractable provide design contrast sufficient condition entry characterize design highlight establish point signal bp nevertheless phase nonzero statistically uniformly lasso frames bp diagram arbitrary recover bp unlike letter scalar letter zero matrix use magnitude conjugate inner use denote index submatrix collect correspond agnostic extend result previous frame extension present use mathematically problem formulation begin unit white sake exposition perturbation norm make word intuitively speak incoherence quantify incoherence formulate incoherence coherence column coherence word roughly state somewhat superior incoherence key aspect simply exact proceeding selection word nonzero entry ratio small entry nonzero entry usual signal noise ratio worth point relationship see ready first concern selection selection coherence next quantity probability failure provide reduce measurement selection optimal opt notice quantify number need application fix specify successful selection suit satisfie p k failure complex model directly theorem important theorem easy threshold conservative reduce analytical tend might rely estimate magnitude entry characterize model design coherence depend p quantity therefore omit remark concern sort threshold cf preferred choice obtain next let complex result specifically far condition selection fix measurement circumstance still performance aspect even nonzero entry energy power average signal nonzero entry mathematically precise average nonzero ready performance coherence distribute k large integer failure respect rely great extent proof point never put study literature report devoted purpose assume selection literature average regard immediately bad gaussian reader bad case definition remain valid replace long appendix fix union fact establish design property long resp therefore correctly successful hand use maximum likelihood perform optimally matrix system high nonzero away average scale worth point also etc precede discussion regard gaussian selection thresholding prefer report bring lasso submatrix correspond away see require aforementioned part signal reconstruction whereas establish worth selection selection tight frame identify nonzero symmetric ii assume design satisfy early frame resp identify long suggest lasso succeed lasso nonzero entry away average equally attain nonzero certain perform optimally satisfy design devote construct goal first wavelet denoise design establish oracle like sense probability location noise thing present early specifically early guarantee location location nonzero entry energy nonzero recover require basis order agnostic signal recovery specify polynomial specify knowledge third impose entry limit exposition recovery sparse set noisy report study goal model intuitively note inherently model intuitive column strong illustrate coherence coherence property lemma design coherence suggest long regime gaussian satisfy coherence main satisfie nonzero significance consider approximately frame design satisfy recover sparse isometry rip guarantee sparse satisfy much weak scaling consequence see point order show slight variation sort difference theorem replace selection recovery three establish collection shift nonzero seed aforementioned geometry constitute design frame completely specify describe seed multiplication carry communication deterministic construction next finite hilbert frames constitute class frame time frequency seed vector follow emphasize block frame circular shift frequency shift seed ready frame nonzero subsequent norm signal frame prove concern frame coherence frame coherence facilitate mix w eq write shorthand follow algebraic specifically consequence fact nm likewise simplify nm cauchy schwarz inequality since divide expression coherence unit seed seed easy unimodal satisfy coherence long recovery suggests generate unimodal hope discussion design coherence selection frame unimodal seed mathematical research researcher recent year construction specifically unimodal j term autocorrelation decay suited frame bad recently frame seed check consequence
trace potential bold seven gray maximal bar height probability classify random manner perceptron symbol circle correspond choose pattern classify first denote reach potential condition approximate q threshold unless likely agree classification close present generation despite pair neuron spike generation summation spike within temporal window perceptron agree roughly overlap explanation cluster classification large connected error entropy yield calculate volume difficult ir function replica intra overlap domain inter overlap right compatible calculation enable estimate quantity ir character limit effect replica symmetry breaking affect correction behave sensitive temporal conclusion input integrate incoming spike generate response pattern adjust spike spatio fire despite simplicity architecture property superior performance perceptron temporal output spike support fellowship science de paris paris mi paris paris france computational neuron spike linear operation statistical mechanic extreme capacity task perceptron number per finite large size weakly constant concern static intensity neuron activitie spike furthermore stimulus system characterize suggest brain possess extract embed spike power importance decode embed spatio spike integrate spike spike denote temporal u correspond respectively except spike relevant classification fire output spike nan trace rescale fit factor line law circle indicate poisson letter computational standard classifying batch denote spike input neuron duration independently equal correct classification first numerical error algorithm capacity neuron probability correctly secondly important understand capability neuron time dynamic system complex solution arise computational optimization duration neural property easily limit sensible neuron fix capacity independent capacity require qualitative implication capacity bound exceed capacity spike arrive within single carry expect increase fig simulation behavior fast change significantly fast easier distinguish arrive learn algorithm solid large replica continuous mean perceptron capacity two different weight every secondly solution spread solution solution different overlap picture overlap probe walk find walk lead valid auto correlation drop fast
stationary optimum converge need minimal ascent maximization expense involve iterate expect perform maximum multinomial approximate solution cost slow implement per certain useful sparsity decomposition may active option alternative encourage posteriori conjugacy multinomial iterative map induce want responsible log value likelihood may priori knowledge relative component audio might expect map simple conjugate multinomial simple parameter lagrangian zero optimize original choose substitute large approximate optimum x
hard description consistent description description variety english case system bi gram gaussian obtain correct description correctly sentence comparison user average learn social need fuzzy cope ap integrate process perspective interact knowledge publish work movement semantic system language try reduce mean word context actually word learn several different syntactic case different context description user precise combination semi supervise description shape interactive website description human description description user green compound description dark circle green dark semantic learn conference description validation transform description set shape set shape multiple object shape section decide applicable degree label learn shape shape calculate match matching degree form generalize constraint constrain attribute value blue dark green triangle background projection corresponding projection list phase role sentence present description generate word belong project sentence phase new evaluation try sentence lexical phase frequency filter form word class generate composed class front light dark bottom blue green red htb analyze segment extract description compound description compound belong blue red green edge detector color group shape candidate shape pixel match shape feature average blue cb cr difference red bounding position orientation minor minor extension bound box height area pixel differently even size many label htb system relevant associated word share constraint aforementione decide robustness flexibility fuzzy tree classify accord fuzzy decision tree class class class class cr cb select class nevertheless decision frequent decision class else dark see htb calculate degree object obtain example color plot htb htb matching matching projection feature soft constraint fuzzy label calculate matching description ambiguity description degree matching could scene description calculate ambiguity scene description scene thus ambiguity description would description short scene short pattern word build ambiguity description return match else go repeat htb segment find object frequent label high system match frequent find degree ambiguity keep try ambiguity maintain influence propose degree ambiguity feature avoid description description generate light green circle front scene red rectangle generate description page user try guess describe description one come count description htb performing obtain description rank worse little well description obtain description bit rank far user allow system learn semantic class shape sentence correct sentence provide soft web description object allow guess complex work word context step word whose relevance highlight department science innovation program de european social mm mm es david soft word web describe scene guess give accurate user guess describe build analyze class detail model description soft constraint generate system description describe see cover range daily activity social phenomenon evolve complexity language link human capability converse environment lack information reality incremental fashion
fy north south fy north south north south z show determine node nod logical topology datum different consecutive train experiment different period path normalize approach path adaptive randomly rr measurement costly bandwidth measurement term resource potential measurement greedy active path probe iteration probabilistic nature accomplish random specify first entropy measure path path high assign select already follow relation normalize ensure probe call weighted confidence interval basis path path distribution region confidence estimate hand value hard use stop criterion encourage terminate normalize ensure available lie choose distribution probability bandwidth small maximize available hard decision efficacy model link path assign uniform topology probe outcome selection algorithm sequentially path seq estimate capacity lie interval satisfactory term average measurement stand rr seq employ naive outcome spread measurement software code although heavily author describe load prevent accurate load prevent precise train avoid receive train measurement train observe receiver obtain train perform obtain divide receive amount last delay probably consecutive receiver use inter arrival upon valid label follow online different path logical link low display transmission four disjoint path train second encode observe train plus probe clearly measurement estimation occur probe length reduce measurement collect trace use fraction rate measurement maximum posteriori calculate observe perfect chance available exception decay measurement topology path exceed confidence drop conclude determine avoid rate performance vary number network per train suffice significant use display indicate second although methodology overhead use topology sequentially confidence gain measurement second overhead path error estimate despite examine path probe train send highlight correspondence time algorithm bandwidth information probabilistic active informative estimate fundamentally different metric capacity connection briefly review almost focus path exception address path aim available bandwidth path year thorough self train great bandwidth traffic increase way delay receiver available inter approximately equal bandwidth vary converge variability available bandwidth delay employ deterministic inference hoc rule average liu input counterpart technique service service flow et provide end context bandwidth estimation propose elegant min case estimation estimation consume song subset observe load scalable sharing internet link near share behind core define quantity high datum tolerance methodology present address multiple active software software model format propagation marginal path take marginal require passive round perform encouraging many explore need trade take advantage relatively could formulate term resource budget also automatic learn network might need train short train suffice investigate employ scan measurement use informative link distribution currently adopt prior encourage sparse link accelerate software platform validate tool circle black fill thick thick picture know send high transmission streaming introduce define term traffic metric distribute multiple path share process dramatically require application beneficial offer specifically know send traffic induce valuable guide stream choose transmission streaming association content paper send probability almost closely path accurately involve path train rarely overhead acceptable path share load path independently inefficient resource probe information accurately path exploit bandwidth something available share link know share particular overlap estimation estimating capacity estimate large high bandwidth file transfer propose available path network goal multiple measurement approach base concept probabilistic path inference path also quantify measurement sequential measurement evaluate available bandwidth maximize gain path encode model factor well represent available correlation among connect path graphical joint bandwidth path formalism exploit perform inference select collect active learn involve quick measurement indicate whether iteration select path probe uncertainty probabilistic path bandwidth monitoring difference probe fashion significantly reduce traffic comparison sect probabilistic available bandwidth sect assumption sect active belief propagation sect obtain sect contribution relation sect conclude probabilistic multiple probabilistic available metric capacity train two cite estimation scheme employ delay al argue equivalent pass delay employ delay measure similar apply regard bandwidth available probe liu et bandwidth datum confidence stop percent range bind upper terminate satisfy termination stop reach meet secondary sequentially measurement past minimized topology network infer link ip address know incorrect ultimately available bandwidth actual logical topology complexity able limit operate logical topology topology stable minute moderate topology sect throughout different day topology sect link wide estimation employ measurement specify probe indicate whether rate triple binary difference test measurement fuse summarize maintain graph belief distribution easily new would network bandwidth use adopt active choose path probe already approach create probe probe new whenever path receive traffic service cause path ultimately low input employ behaviour trend great display bandwidth path distribution relationship topology link graphical
solely measure try objective unify image segment include contour recently description datum segmentation subject quantization agglomerative segmentation texture merging effective human image preliminary utilize seek image se window around group highly redundant overlap window adjacent entropy image encode pixel use smoothness boundary relationship belong texture entropy justify segmentation segmentation result obtain compression correctly encode texture code encode texture distortion distribution window shape texture image incorporate segmentation encode boundary texture carefully boundary adaptive code principle optimal minimize entropy image code purely yet regression proper quantization achieve optimal segmentation conduct extensive segmentation berkeley method conceptually measure purely objective extremely human compete segmentation texture segment one window around channel stack color inside window segmentation approach filter bank construct texture pixel take neighborhood pixel stack ease reduce dimensionality feature project principal denote principal assign empirically theoretically gaussian mesh model particularly texture synthesis consistent window fill texture nonparametric compression must distortion estimate empirically variance bad compression rate distortion code length describe texture region quantization code distortion vector sum length gaussian vector codebook codebook uniquely empirically exclude window window well model window empirical ht b windows r grid encode furthermore texture represent redundant approximate window ideally rectangular become rectangular code window region belong codebook generic multiple sample asymptotically code scheme inefficient leverage component efficient way group membership image pixel code orientation direction encode three along region image representation chain code code region smooth boundary expect consecutive consecutive code orientation orientation compare original code difference encode eight difference code image human shape web table tend human htb c c code angle change liu coding compression describe hierarchical deal multi texture window simple yet effective regression choose distortion set segmentation image represent boundary region find agglomerative initialize process pixel texture belong maximal texture window adjacent strong interior adjacent purpose find maximally decrease w capture difference boundary merge merge region ht measure denote intuitively small probabilistic well denote ht discrepancy label please refer infer image distortion extract distortion level statistic ki ki sensitivity intensity measure intensity insufficient accurately discrepancy measure optimal estimate around distortion agglomerative specifically either monotonically ground observation discrepancy square attain linear recover training program close learn distortion test ki nevertheless average discrepancy publicly berkeley comprise natural image cover scene landscape database partition testing set provide ground several human subject average five investigate human subject image seek determine color approximate utilize representation check validity segmentation texture perform widely manually berkeley dataset color bit encode texture information compute ground map finally code dataset entire volume thus pixel rescale mean produce range oppose achieve comparison normalize constant across average eigenvalue feature dataset representative examine code length therefore rest convert color metric probabilistic rand index information rand pair label great partition adaptively choose measure sum gain clustering extent nonnegative indicate great harmonic precision metric boundary segmentation boundary precision segmentation measure ground truth pixel adaptively multiple truth ground ground truth ensemble truth image performance segmentation segmentation ground feature vector convert rescale set rescale constitute observe I excellent result quadratic estimate step least segmentation namely compression texture merge good evaluate segmentation human six propose result therein agree ground metric treat ground truth computing qualitatively illustrate htbp human ms noting seem gap index g versus good mainly construct texture region contour edge category literature fail visually inferior thin texture choose texture algorithm fall behind human situation arguably shape texture texture appear break texture geometric pattern thin second enough texture texture region ill condition unstable middle segmentation problem investigate problematic able handle relatively well geometric slightly segmentation category thin poorly mean shift roughly pick point segmentation figure novel use principled texture respectively partition agglomerative hierarchy optimal minimize coding determine segmentation efficient distortion user experiment term region contour aid evaluation website novel segmentation boundary effectively code segmentation image short code boundary agglomerative cluster window texture feature estimate overall true publicly berkeley segmentation dataset achieve
degree freedom fit optimum still nonlinear nonlinear estimate reliably stem value practice show uncertainty may severe intend gaussian parameter value freedom compare trial assess yes normalise accord know estimate normalise residual show fact alternative definition however expectation value indeed approximately root severe comprise task fit well fit single assess true value consequently neither reliably clearly much increase true parameter quantify uncertainty residual mean derivation reliable assessing fit alternative beyond manuscript concern error thing order goodness indeed trivial residual already cf measurement distribution normalise residual need plot normalise residual histogram find residual quantify kolmogorov test ks statistic theory compare normalise fit well fit find may eventually start residual peak happen fitting stop guarantee similarly whose normalised match win truth radial velocity nearby presence claim circular claim additional justified table six every ks model six display normalise residual six distribution imply neither likely explanation discrepancy elliptical assume circular datum discuss normalise subject uncertainty difference blue red comparison gaussian large powerful usually computational give know fit goodness fit sample compute repeat goodness whole multiplying likelihood step require order goodness simply repeat make validation computationally bootstrappe cross validation likelihood bootstrappe bootstrappe discussion draw data replacement sample bootstrappe every th bootstrap contain predict repeat least bootstrapping aim prediction therefore argue completely sect absolutely nontrivial freedom quantification linear model cause become model number degree guess aware reliably degree consequently impossible sect see approximately within model assess convergence consideration popularity certainly apparent justified matter model degree severe use concern explain cross bootstrapping sect normalise model infer model concern want consideration concern correctness convergence thing gaussian acknowledgement david discussion david also couple thank helpful manuscript pm support measure error position one likelihood likelihood manuscript therefore reduce used purpose assessment bad fit whereas consider set ask close one fit stop converge sometimes evolves stop soon reach value one sometimes claim fit certain model error compute already case divide manuscript aforementioned major arise degree freedom explain affect application sect dedicate explain reliable sect degree freedom point I guess however explain degree split discuss linear model discuss freedom number piece concept next give linear parameter degree appear model mean superposition position example cause actually zero highly consequently nonlinear rewrite exist degree freedom introduce concept generalise conclude infeasible practice order concept give fitting arbitrarily short perfect three prior actually modify add influence fit two lose word fit constant
hmm ard ard lag dominant infer indicate turn lag rely lag truth matlab turn order truth variant dynamic instead distinguished base markov switching tracking understand physics section many mode force act specific form place jointly parameter refer choice normal prior detail independent volatility switching process switch underlie conditionally daily represent return stock interpretation non filtering cope log squared daily noise sometimes match mixture gaussian volatility stock stock exchange cite period event match order volatility model target sampler herein impractical training infeasible recursive infeasible leverage filter herein flexible dynamical phenomenon discover experiment dynamic considerable variability inverse wishart portion give hdp ar move wishart sec since time tighter unsupervised examine difference approximate walk raw create dynamic hdp iw sec iw degree freedom mixture measurement noise iw degree expect moment match al hdp compare fig place equal hdp use model table iw initial dynamical cause account observation accounting switching conditionally mode dynamical system utilize unknown persistent mode relevance infer allow vary switching var process develop combine dirichlet utility flexibility sequence stock bayesian method financial target country intervention national appear rarely observe possibility previously unseen motivate develop nonparametric dynamical simple hmms hmm markovian conditionally independent mode switch var process rely mode infer new dynamical paper one agnostic mode return previously prior hmms mode variant hdp crucial switching extend hdp persistent capture wide dependency explore underlie contribute employ ard realization dynamical mode possibly vary dimension switch var autoregressive provide survey recent approach switch dynamical mode ii noiseless switching var process algebraic rely cardinality autoregressive penalize identification simplify deterministic present assume dynamic output find switching subspace mode author also dynamical assume variational continuous sequence evolve dynamical mode state identifiability bayesian adopt incorporate mode cardinality prior complicated describe allow distinguish place simple aic bic manner dynamical system herein previous conjugate induce conclude model synthetic formulation time ss processes covariance process denote var observation var process form though ss model process phenomenon examine behavior model linear dynamical transition index drive gaussian hmm mode conditionally examine relax dynamical mode first analyze simple equivalently concentrated operate space emission parameter take define whose mode dp h space weight sample stick weight weight proportion remain stick prove many see examine draw discrete observation within representation take eq reinforcement extension prove useful define assuming variation global show measure et observation encourage expectation model dynamical mode persistence flexible nature hmm prior hdp add transition specifically expect transition original hdp et learn bias hdp hmm var capture switch unknown dynamic mode model illustrate var generative process table hdp ar hdp observation hdp distribute e hdp place generality fix matrix imply component choice state essence sec hdp ar hmm sec sampling hdp iterate state give mode sample hmm sequences ar hmm exist step involve straightforward hdp hmm construct involve capture underlie state sec prior posterior need condition sampler hdp explicit concept hdp hdp ar utilize definition outline table hdp hmm r ny yy x ar hmm dynamic lag form comprise observation available hdp resampling discuss mode consist k model prior infer single linear enforce mode condition stable matrix straightforwardly derive normal covariance inverse degree update tn k problematic grow identify irrelevant component lag hdp address ard encourage drive presence hdp ard place dynamic zero dynamic amount determine iterative become column whose insufficient imply mode examine look mode imply rank overall realization ard restrict attention mode dynamic must component assumption criterion fix identifiability issue considerably less ard prior may use switch entire hdp dynamic placing prior decompose lag give lag large lag block order examine useful ard q replicate hdp ar replicate recall hdp hdp hmm eq represent recall precision distribute regardless observation remain informative prior upon maximal hdp ar observation place inverse wishart oppose additionally measurement share mode measurement hdp ar hdp hmm sampler state conditioning resample parameter mode step term pseudo specify ar hmm hdp z dynamic ard move hdp additionally sample noise condition sample hdp dramatically improve hdp switch direct correlation variant forward backward backward truncation recursively condition sampling via sampler note encourage dynamical mode hdp dynamical simplify x first backward recursively message recursively tr slow mode analytically marginalization accomplish condition time sampling conditioning mode x backward produce pz forward backward local yx x sequence full derivation information kalman note update parameter sequential still sampler sequence sequential gibbs mode transition hdp compute pt sequentially hdp ar pseudo pt transition transition utilize hyperparameter pseudo mode sequence ard prior hdp specific transition pseudo calculate message initialize sequence compute count sample assignment increment likelihood htbp sequence construct pt associate k k pt htbp pseudo mode set dynamic iterate time construct switch var ard analyze power hdp var hmm hdp var hdp difference hdp hmm fig b display error true estimate mode hdp transition proportion informative hyperparameter generate five switch self mode transition two well dynamical comparable hdp var hdp contain hdp scenario middle ar hdp hmm significantly hdp ar posterior hdp hdp slow continuous scenario bottom neither hdp hmm hdp switching yield significant improvement baseline hmm effective use less rich switch difference ccccc ar hamming hamming ar hamming ar ar hamming ar hamming ar hamming ar hamming hamming hamming observation blue red mode switch ar middle th th hamming quantile trial hmm hdp hmm hdp top middle hdp observation utility ard prior true dynamical model two mode self dynamical equivalently white dynamic directly dynamical white noise contribute equivalently white noise original combine dynamical mode satisfy nevertheless realization still expect set ard recall prior informative size ard superior estimate state component infer fig e sample dynamic identify aa ccccc hamming hamming ard sequence blue mode mode realize solely hamming quantile ard e ard first value non dynamical within order location switch walk roughly straight rapidly body turning involve turn display tx head switch wish six prior hdp parameter sec synthetic datum process pre processing involve center scale dynamic range seq seq seq seq seq seq seq seq seq seq seq color blue head colored label compare detection achieve change number
close time obstacle many spin spin heat establish critical temperature phase regular even dimensional ise spin phase transition finite temperature study spin spin hard illustrate markov chain random dimensional dynamic interact lattice walk lattice integer site nearest periodic understand coupling walk lattice walk evolve identically couple eq coupling thus time also interact periodic particle coupling joint walk integrate yield meet evolve way simple choice walk independently site stay move site randomness mention particle dimensional line coupling particle evolution possible scale whereas behave lattice random configuration function coupling consist sampling walk coupling scheme preserve label meet see particle move entire maximum conclusion walk couple spin regime couple walk previously consider temperature evolve heat dynamic would correspond evolution configuration coupling evolve couple couple monte dynamic diagram carry spin heat choosing random configuration ise dynamic regular temperature preserve guarantee ise interaction although de heat coupling temperature show patch times agree partial coupling show realization coupling time vary couple time entire coupling heat spin phenomenon dynamical phase grow phase dynamical finite although mathematically single property illustrate point verify correlation remain constant heat discuss accept coupling spin update several configuration coupling place opposite configuration stay adapt regular number couple qualitative temperature metropolis couple temperature heat share qualitative confirm dynamic opposite well qualitatively couple spin key physics hard hamiltonian realize molecular dynamic carlo hard couple configuration critical fraction discussion efficiency concentrate couple canonical birth monte particle position life disk one diagram system birth death hard disk disk accept dark reject light configuration yet configuration dark produce diagram survey realization configuration horizontal cut diagram patch sharp regular couple birth death packing monte dynamic random initial life sample exponential dynamical packing density limit patch version birth death label time choose particle random move create coupling birth death closely couple similar choose define move disk disk example show hard consist configuration coupling use follow successful coupling place disk shape region radius algorithm hard pack disk center consist place disk create overlap particle balance generate move metropolis fig succeed coupling couple density conclusion couple importance nature ise spin phase heat coupling
solution literature new combination l evy superior almost apart population parameter subsequently optimisation powerful application problem parameter popular potentially mathematical analyze progress potentially insight optimisation call search cs develop present extensive newly cs problem cs far obtain particle implication mathematical optimisation problem involve design write range nonlinear maximum stress nonlinearity often multimodal landscape subsequently search hill solution modern search genetic particle algorithm attribute million important select good candidate sure search randomization call promise could outperform cs validate function include apply discuss feature study review outline specie though al specie basic direct elsewhere specie world specialized colour pattern specie al reduce species action et nature search effectively walk move state direction implicitly model mathematically example study evy landscape straight l evy style free search feature evy evy al subsequently promising capability simplicity describe search three randomly high solution carry discover either last replace new random maximization fitness simply form fitness rule basic search xx initial get evy fitness say replace fraction location evy good quality solution find current perform step size entry evy step draw evy infinite infinite consecutive walk process obey length tail pointing discover related difference good walk biased supplement reader relatively analytical test list extensive description optima occur landscape final marked figure optimum distribute multimodal optima number optima may become multimodal try population simulation also imply extent sensitive fine adjustment dependent need study literature new validate function de sphere minimum occur function unique multimodal also global hypercube function sharp domain global minima almost deterministic deal test design wave snapshot landscape value landscape multimodal global fact function stochastic extend draw stochastic due stochastic function deterministic hill fail however see recent study genetic ga conventional et al attribute partly lead evaluate evolutionary detail al genetic implement time meaningful less format optima stochastic stochastic generalise stochastic location iteration ccccc ga cs finding optima success rate evaluation instantaneous modern example second stochastic genetic
use tree newton terminal interestingly interpret average thus alg replace boost use sum formulate base class exhaustive strategy derive abc boost develop abc combine boost alg derivative alg differ iteration terminal np r bp bl r bf base number differ abc split procedure alg fit regression boost look hessian freedom fix diagonal determinant zero base base matter list boost concern small study fairly dataset moderate image image boost accuracy learner iteration overfitte mis report mis improvements abc also sensitive upper mis bold upper test mis l boost must exhaustive obviously expensive paper proposal abc boost base illustrate boost far computation reason regression boost iteration base insight step select base iteration introduce gap boost cost fast boost additional overhead boost experiment subsection obvious loss accuracy dataset moderate boost improvement especially large mis times panel mis error abc mis classification robust ratio boost original boost large task boost boost accuracy k accuracy boost achieve boost phenomenon surprising view test present loss accuracy mnist situation somewhat terminate accuracy mnist boost negligible compare boost may produce large small figure experiment present mnist test obviously propose boost abc boost serious boost class prior abc boost require base class base base exhaustive boost gap use sensitive accuracy boost encouraging fast abc boost tool boost exhaustive base boost report however boost prohibitive overcome serious limitation heuristic choice boost well encouraging
replication produce form namely virtue chebyshev inequality estimate upper bound guarantee large grow relative p estimator prescribe level optimality derive efficiency virtue obviously one mind estimator computational come effort drastically split elementary multiplication comparison generation single uniform incorporate estimator say number evaluation square coefficient variation closely relate algorithm back set linear benchmark efficient indeed direct network basically multidimensional walk constrained countable characterize quickly disjoint put simple transition lead subject negative system see transition specify arrival service time arrival take denote job open stationary condition receive job leave eventually th otherwise correspond encode period shall review next section equation gaussian elimination find fraction space state entry fall band show operation g possess efficiency work normalize sense run compare equation efficiency analysis insufficient analysis suggest evaluation solve previous encode two arrival service rate iii stability embed customer epoch system total precisely eq times x q mean deviation theory specify splitting queue motivate early basically constraint constrain random walk motivation deviation split embed discrete chain term type notation transition induce negativity state different region index encode origin empty boundary space empty depend subset eq represent arrival represent q boundary careful I dynamic queue admit walk type give constrain deviation network somewhat walk recognize non create leave aside simply describe large extremely important behind deviation play increment similarity suggest walk surprising increment crucial deviation deviation behavior scale scale queue length evolve eq queue process q negative characterize taylor formally arrive together previous equation characterize behavior game theoretic wang solution weak everywhere coincide calculus variation large deviation see asymptotic appropriate equation sign replace thereby obtain equation guarantee translate said surprisingly construct solution discuss use procedure equation equation use lyapunov inequality give mention rare event growth variation place initial position construct put n appropriately define q total particle reach wish constraint reach n guarantee p suffice ensure relate really particular picking nu suggest weakly efficient estimator precisely conclusion heuristic discussion development property sampler network heuristic method turn precise indicate xy j indicate run simply analyze previous fully branching splitting death zero think branching conceptually particle run sa particle reach update child indicate reach continue particle death particle iteration weight indicator l eq position refined efficiency estimator break particle part deal technique queue quantity reach split direct solving take markov turn queue network use give q x xx path satisfy also part correspond proposition follow consequence also simplify look expect stability one keep mind concerned evaluation involve measure sum particle weight evaluation bottleneck proposition v cp x c tv point measure suitable quantifying need account evaluation generate address expect effort particle remain effort th reach position intensity position particle intuitively level advance next move dominate expect inequality independence facilitate analysis moment add notation analysis exposition self contain generation denote particle disjoint grouping accord parent generation follow moment generation common technique combine term readily arrive take satisfie di order moment dominate turn asymptotic particle begin use particle beyond imply sum weight define stop time k q derive constant proposition expectation dimension increment constant bind finish follow negative since constant leave dominate side obtain finite combine reach lemma allow follow upper term use equality back second dominate equip ready evaluation evaluation sufficient achieve accuracy immediate bind effort per run along system conclusion splitting improve polynomial look network totally substantial evaluation total enjoy store vector encode system importance exist evaluation suffice sort meaningful comparison analysis resort bounding expectation point introduction room refine analysis insight claim conclusion criterion theorem convention splitting rare subset initial position suffice fact suffice relative bottleneck network rigorous splitting directly evaluate algorithm network subject substantial literature last decade specification open reference influential development efficient popular approach efficient rare event sampling splitting involve simulate consideration case network occurrence rare nominal splitting attempt behavior system idea occurrence nest occur occur keep particle reach course particle estimator popular efficiency analysis rare event simulation correspond optimality contribution discuss consider customer fix set reach origin queue length reach level whole intensity weak number replication suffice interest explain elimination system sparsity equation solve many intend exhaustive carlo exponential
evaluation aim diag expect taylor around decrease absolute obtain cf assume numerical algorithm care call diag imply critical order figure west always stay west might reduce slow west take approximately minute four minute perform fairly office pc take double diag recursion cumulative univariate mathematically normal evaluate numerically many lead lack univariate intuitive taylor bivariate cumulative normal univariate cumulative bivariate normal taylor axes overview discussion cf implementation variant although reliable also mainly book double absolute double arithmetic library straightforward indeed double apply competitive trade little often refer survey distribution book bivariate standard normal going use follow recursion convenience define numerically function recursion scheme run apply numerical q formula cf avoid q order favorable q apply correspondingly discuss derive section c quantitative finance field comment reasonable instead inversion avoid algorithm evaluation avoid without cutoff double recursion compute priori increase dropping condition accuracy still implementation always check upper bound sign summation double diag double double double lambda px px double ab px double odd double double odd ab double x double double ab odd odd even odd b odd odd odd odd return max provide comment check evaluate cutoff visual inspection optimize mm double help double double double b double else
difficulty coefficient increase htp avg avg positive group variable component variable hierarchical high choose group avg false positive avg negative avg avg positive avg negative logistic jeffreys diagonal parametrization invariant minimization seem serious example region little effect method regression run repetition respectively give penalization exclude map simulation lead htp avg avg avg false negative avg avg avg false negative jeffreys parametrization order repetition table hyperparameter average false positive negative roughly htp avg correct positive avg false negative htp avg avg positive avg negative reasonably practice bring together correspond ie tend contribution iteratively reweighte methodology resolve issue concavity assess utility estimate posterior mass work secondary contribution generalization penalize practitioner acknowledgment thank schmidt availability optimization code implementation technique popular amongst increase regression involve computationally attention quasi posteriori induce expand date bayesian art provide give hierarchy graphical adaptive recent sparse regression term signal computationally tractable regularization truly posteriori prior denote family involve log penalization penalization penalization maxima function become practice use component penalization induce convex increase difficulty result estimate lead literature reweighte processing hierarchical amount marginally induce hierarchy rise expectation maximization essentially iteratively iteratively reweighte independently suggest user incorporate coefficient flexibility grouping immediately lasso interested setting assume give conditional fy approximate point easier especially solve q equivalently think term construct prior low give ie exponential obtain ie eq prior become induce sparsity propose place inverse integrate call prior compute map prior logarithmic penalization introduction natural resolve difference size model come thin tail concave however one mode latent conjugacy inverse laplace distribution j clear enough value em algorithm mode weight penalization justify partially oracle select asymptotically worth increase remain increase quickly trade generally come exponential distribution laplace conjugacy respect scale normal still additionally concave gives scale adaptive contour plot plot lasso ridge give sparse together high example mapping group procedure prior never algorithm group machine wants solve related issue issue maximize density jeffreys likelihood parametrization methodology multimodal value reduce algorithm converge note still characterization mode open white black black none none e ann node state g node black text black draw none state node node j node edge h edge propose estimation paper interpretation flexible propose obtain minimization however derivation flexibility generalize family positive obtain marginally use prior ie prior statistical literature close whose single family generalization estimate return posterior logarithmic find form motivation smoothly absolute penalization induce prior slowly bias related reweighte penalization basic reweighte except limit case exponential family family penalization reweighte selection hierarchical differ place oppose difficult estimate although computation marginal prior member mcmc improper produce sparse explain prior improper unbounded density figure ep group visualize framework one regression group similarity clear develop iteratively reweighte standardize put improper integrate jeffreys improper
height box become large produce get meanwhile symmetry preserve drawing assign interval node coordinate could replace product although kx help generate graph manner number final box convention color row color accord thus diversity correspondingly generate increase like keep degree subsequent iteration achieve choice use box sized box expression simplify keep increasing increase exponentially characterize calculate measure cluster node row topology sub first sub give generate formalism follow show trivial coefficient near rather simple analytical analytic formula distribution significantly speed measure prescribe degree obtain average symbol plot together obtain show agreement show coefficient neighbor plot panel boundary capable produce diverse prescribed optimize generating requirement number node box actual box boundary self adjust shall follow optimize generate conceptually simple give case implicit way box keep write give note case actually degree like system sort plausible actual annealing also decrease slowly consist change amount energy small change accept procedure generalize principle well measure respect three choose target rather free result optimize respect show agreement adjust typical respect different degree circle mark symbol come law degree correspond symbol experiment bi modal respect different coefficient graph limit network alternatively ever network become would contrast graph analytically see si infinitely relatively simple structure extremely region infinite regime increasingly slow growth appearance si summary demonstrate construct characteristics degree coefficient turn graph hypothesis anneal small observe spirit song thank discussion science office generator biological os institute mathematics os biological physics new conceptual simplicity generate variety degree hypothesis datum suitably define potential allow variety topology infinite network increase present analytic iterated generating distribution parameter measure anneal researcher biology science biology complex enable create becoming include phenomena unit behind network unit connection turn realistic distance distribution modular understanding design control system line increasingly graph representation technique new year recently l dense adjacency et phase variable suppose continuous everywhere limit support develop grow size conceptual rather natural internal organization large school organization playing since complex principle hypothesis measure many successful interpret various limitation degree distribution graph attract remarkable method include systematic approach analyze topology specify degree sized subgraph spin symmetric matrix take value count hierarchy interesting hierarchy os enyi random distribution scale approach around adjacency achieve adjacency link element multiply version element real multiplication element obtain tail eigenvalue small diameter simultaneously kronecker similar point et inspire diagonal give link probability summary plausible generating procedure generate graph stochastic growth prescribe vary configuration stochastically adjacency simple pair connection relate construction fix increase iv assume infinitely limit singular infinitely
respective prox optimize construct multiplication analyze complexity accelerate homotopy introduce homotopy smooth obtain hinge warm next homotopy lp losse ix equivalently replace full saddle function subtract prox prox center project piece wise hinge smoothed hinge error theorem error hinge completely calculate use determine thus smooth arbitrary calculate lipschitz calculate svm define sum equivalently saddle saddle point smooth subtract prox smoothed project explain result smoothed piece wise approximation larger bound completely direction thus use definition nesterov smooth primal rate round construct iteration round represent solution svm auxiliary auxiliary prox whose prox guess gradient auxiliary step iteration round start solution proceed negative gradient round weight round gradient iteration round optimal theorem correspond smooth smooth guess input guess solution termination dual part round iteratively svm expensive computation multiplication step proportion increase addition simplification completely correspond nonzero element nonlinear replace replace require penalize calculate fw ix round reach let accord hence round complete multiplication homotopy lar parameter solve prefer start optimization accelerate start warm start prefer accurate approximation large induce rate accord expensive accelerated term categorization vision task e scene scene ghz processor gb scale solver light criterion classification accuracy perform respective cpu second svm solver test time mean svm solver census categorization scene scene recognition adopt rest wherein feature multiclass one adopt code available solution guess termination criterion census categorization repository census show train training set scalability slow svm solver search svm main irrelevant scalability solver short shorter batch method classification dataset axis pixel sample select group store home public space place color texture datum testing solver value achieve solver optimal descent matrix multiplication irrelevant light sensitive show scene g office include texture intensity scalability svm solvers svm take cpu event image bag split training fig four solver solver light second base expensive cutting svm less recognition event graphic randomly class bag extract accord scalability four svm less solve svms programming square complexity svm gradient combine descent determine lipschitz multiplication require round homotopy improve efficiency norm homotopy adopt warm start time cause solution homotopy competitive efficiency four popular svm svm light insensitive refined efficiency computation smoothed already accelerate future machine svms tool many intelligence sufficiently efficient deal gradient svm compare solve wherein differentiable hinge norm primal iteration round historical vector multiplication require exist solver addition nonlinear homotopy accelerate dynamically categorization scene recognition scene effectiveness available website assessment machines smooth nesterov machines svms prominent machine tool intelligence svm great feature work rapidly dimension addition decomposition sequential cost optimize iteration relevant support vector optimize problem select work set programming impractical complexity slow convergence optimum efficiency svm firstly compute violate constraint training cutting add current working qp structural qp efficiency difficult serious svm scale inefficient qp constraint primal available solution correspond vector plane reformulate classical non stochastic achieve solution convergence second newton method replace hinge differentiable sigmoid function expensive impractical primal second online step update rapidly reduce therefore apply
covariate c demonstrate among nonzero regression true know index fit cox predictor package survival statistic difficulty model correspond effect statistic estimate nonzero figure relatively hard oracle difficulty one datum demonstrate propose classification study solid even hundred united tumor patient nb diagnosis age patient clinical free h microarray include gene site overall survival five array array consideration patient survival information available patient censor survival summarize marginally jointly gene appropriate sis powerful van scad select gene probe cccc coefficient e try significance gene predict survival hazard function correspond log eight gene next eight cox proportional hazard seven log freedom comparison test gene select cox eight plus record repeat average likelihood merely eight gene reduce lot increase develop technique survival dimensionality large focus sure iteratively apply screen filter utility moderate partial select carefully study sure screen section corollary fan partially support grant dms dms grant wu partially nsf dms north state scientific advance huge covariate snp technology clinical understand information clinical survival extend cox study technique demonstrate screen survival biological death failure failure try censor termination study dependence survival covariate covariate goal hazard hazard depend nothing instantaneous rate covariate proportional partially due deal censor assume hazard function note uniquely one identifiability specify identifiability condition hazard introduce proportional reference detailed literature cox hazard need estimate survival also baseline hazard reader recent advance collect huge amount microarray snp information clinical covariate associate clinical outcome quantify contribution data predictor mathematically nonzero model subset reference therein modern survival consider scad concave adaptive event among consideration dimensionality exponentially propose independence sis marginal ranking theory sis predictor vanish extension iterative sis marginally uncorrelated deal sis method non asymptotic theoretical rate although extension cover explore independence screen hazard censor event cox proportional extend sis nonparametric additive carefully organize detail cox proportional variable give cox sis cox demonstrate sis denote time censor associated correspondingly denote censor simplicity conditionally independently identically likelihood fs ft conditional survival conditional hazard respectively failure time consider hazard become consider informative nonparametric observe failure consequently maximizer get censor maximize newly hazard coefficient leave final consequently important variable handle variable procedure accordingly handle cox proportional advanced penalization receive recently penalty function perform optimization capable penalty many scad penalty elastic adaptive penalty function penalize maximize sign front literature scad penaltie study event many sis scad quadratic spline origin recommend argument scad penalty convex scad convex work penalize variable technique great covariate performance penalization variable selection subset mathematically penalize partial lead sparse index sis jointly marginally sis compare sis base make joint sis begin sis penalization base refined true denote utility utility measure contribution include large small covariate top denote respect get update step idea propose improvement idea repeat reach adopt identify note generality two sis estimate screen include probability tend asymptotically due individual number covariate include new variant splitting asymptotic include exchangeability show dimensionality please want remark applicable study bind splitting define new variant original sis screening alternatively choose ensure variant explore compare cox proportional tune different covariate setting generate identically distribute marginally serial marginally multivariate marginally correlation case except serial decay coefficient response marginally dependent therefore expect challenge sis sis dependence condition implementation sis sis var sis select intersection size typically sis scad coefficient whenever necessary bic good censor hazard censor censor correspond censor rate censor correspond censor censor censor coefficient randomly independent though marginally marginally design repetition row median median row label report proportion repetition procedure consideration select application report median
incorrect deterministic purpose prove deterministic satisfy minor difference change fortunately valid compressed sense modify concentration rao achievable constraint valid correspondence organize definition indeed provide form domain rao bind sense discuss show er rao estimator sparsity equal measurement random discussion matrix noisy joint cardinality set denote sub index generalize generate matrix know accordingly lemma additionally assume expression hold unitary extract comprise proof part sake go try random decompose obviously addition show kk k k gaussian generality assume tail take bind say decrease bind inequality bind approach previous decompose unitary lemma gaussian assumption domain nevertheless proof free assumption generalize result area correspondence continue generality first eq addition operator denote element n sake n I kn I km ss eq mean rewrite assume proof use satisfying appendix take respect negative plugging introduce left reader first part far eq important apply upper finally come complete conclude validity valid accordingly theorem rewrite interestingly obtain obtain similar r rao I phase rao bind use two phase estimation process estimator achievable know want er er consider depict form fisher equal er rao er er rao consider one identically eq element gaussian go relation us rao estimator knowledge deterministic concentration seem interestingly randomness fortunately investigate combinatorial proceed joint estimator cardinality estimate unique show constraint mse proof assumption especially variant valid necessary generate additionally exactly minor repeating step apply bound term lemma easily attain upper additionally obvious grows additionally equivalently grow linearly exponent grow polynomially respect compare come asymptotically achieve correspondence er compress sense analysis sort concentration mainly focus build er bind rao bind degree indeed achieve unfortunately impractical er rao open reach find root solution substitute n expression expression obvious hand n prove complete thank ms mathematical department university technology comment author remarkable motivation grateful anonymous associate constructive comment theorem proposition proof height em depth noisy compressed er comprise building measurement randomness generalize drop fact matrix theorem family concentration measure generalized er achievable compressed sense cs vector propose signal usual correspondence signal indeed measurement measurement precisely identically gaussian etc compressive index stand model require present lemma correspondence recovery many theory estimation main compressive estimate noisy measurement effort find solution algorithm relate search existence estimator square estimation much rao bind er rao low mse depend amount knowledge estimator sparsity rao know non contain index location show correspondence structural least estimator second er rao rao kind al differ maximal show uniqueness et support case size er rao equal equal accord estimator limited degree proven state achievable effort estimator knowledge close rao low es factor interestingly use estimator know certain constraint er priori infinity remain asymptotically achieve asymptotically rao achievable bound generally check noisy vector concept compressive estimator solution er typical bound two mention probability event support typical denote second event jointly typical average q rao important mention element randomly distribution consideration
weakly possible make small reduce introduce systematic systematic fig accurate eps eq systematic default function fig eps systematic default straight curve approximately mesh develop code almost code make eps color statistical systematic default red calculation perform broken calculation average calculation average sample cross give use thick break blue curve illustrate average calculation arrange check undesirable datum remove point accuracy value realization average classical default systematic cross give illustrate large systematic fig calculation different realization red spectrum calculate spectra thin line scatter average spectra value slightly energie small eps eps eps optical sample show calculation sample use iterate default function curve iterate default calculation fig eps eps eps eps start default use systematic calculation spectrum default split batch calculation default batch calculation calculation increase average however reduce lead net reduction systematic b realization noise spread substantially vs systematic somewhat vs show different splitting calculation effective averaging reduce net unchanged realistic value actually dependence batch thereby individual calculation calculation increase approximately behave increase classical choose comparable perform calculation discuss curve traditional optimal would next exact opt almost flat curve fig eps eps color calculation est split several batch result opt use find iterate use model iterative cross default error case batch product q average average calculation batch realization difference systematic contribution default calculation calculation systematic two suggest systematic compare calculation calculation latter favorable batch result est opt try improve default iterative recommend calculation default spread large imply systematic reason default statistical calculation calculate nonlinearity result correspondingly case split batch default error drastically reduce iteration substantial factor reduce systematic compare result use close lead iterate allow lead logarithm result analyze behavior expand low deviation define approximation method guarantee spectrum analyze entropy iterate iteration q define contain eigenvector symmetric expansion eigenvector nc rewrite eigenvalue large many close eigenvalue default model show eigenfunction low increase one rapidly corresponding tend weight table expansion default model eigenfunction default crucial calculation close strongly contribution deviation default fail structure correspond since additional show scale make remove fig pattern differ calculation see fig contribution output mainly eigenfunction fig calculation perfect probably nonlinearity due logarithm expand generate function node two also component eigenfunction shift slightly projection operator state eigenvalue iterate eq systematic relative systematic choice linearize pay include operator nonlinear bit different expression expand logarithm eigenfunction couple high eigenfunction node depend component whether different add contribution high component often favorable iterate one favorable study low default expand default expansion unity small eps eps color eigenfunction correspond eigenfunction one rapidly approach define systematic default systematic statistical batch batch total calculation often bad statistical error serious reduction linearize formalism see error deviation default illustrate default potential hard use batch fairly value result sensitive alpha alternatively batch use lose thank want study divide systematic lead statistical splitting batch calculation average systematic iteration often bad splitting batch result systematic max universit energy depend monte carlo quantum possible response ill analytical treat regularize introduce spectrum default control statistical maximum perform analytical approximation decomposition stochastic regularization usually
maximize I application theory large recently quantum mechanic possibility attempt review incomplete useful system perhaps shannon carry state physical might associate explicit rational belief another drive good preferable enhance successfully idea extent wish rational belief information available put explicitly sufficient far capture driving incorporate rational mean belief everything go acceptance arbitrary rational behavior rational exercise piece reliable raise question implication accept belief useful though notion bit introduce acceptable information design handle describe method distribution new constraint value allow problem select increase preference clear preferred ranking assign real preferred possibly equally preferred distribution lead I entropy maximize impose I perform external posterior entropy must produce entropy call entropy redundant drop candidate identify number special prefer shall criterion adopt single entropy elimination approach extent selection applicability reason eliminate quite criterion violate justification entropy inference entropy prior ignore aspect update virtue maximize something keep summarize brief motivation entropy detail refer infinitely class processed refer particular condition update drop multiplicative ranking entropy function sum integral affect criterion entropy multiplicative affect drop function three argument argument locality criterion universe usually value whether coincide leave next turn family choice principle agree maximize entropy constraint multiplier yield intuitively reasonable turn known false straightforward give couple method generalization bayes knowledge relation limit uncertain long know q maximize lead correspond datum derive moment relation prior tell nothing versa treat relation elsewhere one possibility addition know seek maximize lagrange multiplier multipli q multipli corresponding marginal datum take bayes factor reproduce reach likelihood example I issue decide maximize prefer extent update give constraint define space acceptable label manifold write maximize lead preferred question extent distribution believe range assign I represent relevant new extreme product know tell versa retain use original choose assign probability volume choice hand frame coordinate volume fortunately space probability distribution rao metric uniform unnormalize crucial joint select preferred joint normalize convenient lie prefer maximize entropy tell degree result limitation application fluctuation beyond fluctuation bridge deviation adapt lead notion ignore concerned belief rational agent rational belief force agent belief quantitative main relative update I method single inductive inference allow old recent mechanic acknowledge valuable
obtain bayesian parameter estimate mmse sample full sample via inversion outline automate popular co variate numerically approximate target inversion sample stage suffer curse become highly alternative problematic become optimize weight instead sampler variate methodology curse simple adaptively rw rw move deviation tune produce via likelihood information fan variate methodology proceeding algorithmic markov chain impose restriction location proposal j initial prior inversion obtain perform walk element n move proposal j several adaptive distinguish markov sequence transition proposal allow depend many algorithm particularly markov ergodic appropriate stationary propose condition ergodicity algorithm develop metropolis history markov ergodicity bound ergodic prove ergodicity mcmc define convergence derive guarantee develop condition variate proposal kernel know ergodicity detail estimate use markov theoretical choice base mcmc algorithm replace follow variate sampler move proposal di bayes note computationally inefficient involve chain td approach require single factor alternative van van work produce rank compare rank question bayes compare bic comparison bayes factor eq comment numerical implement detail critical handle numerically numerical handle appropriately survey inference involve note average probable probable probability adopt one able reduce potential associate several choice probable turn associate involve popular algorithmic framework model average estimate direct knowledge probability probable mr typically mode careful demonstrate van popular choice proper integral quantity distribution quantity achieve model uncertainty rank bayesian mmse mean methodology develop contain synthetic methodology real data real datum true study p sampler generate conditional sample discard realization mmse deviation sampler point average intercept pre tune local walk perform use mcmc q nh mmse accurate mixture local move global estimate require effort discuss sampler develop actually achieve performance therefore superior gibbs automate gibbs sampler recommend algorithm utilize plot adaptive demonstrate rapid mmse initialize far additionally become sample uniform range identity realization true row first path high dimensional markov chain initialize true rapid sampler true take ghz ram study estimate consider series simulate realization time per run assess paper financial comprise mini mini mini series market daily price sep series datum present mcmc present run sampler initialization split increase datum datum rank model amount converge predict particular estimate suggest distinguish suggest application algorithmic trading repeat perform series comprise mini analysis analysis financial initialization give preference trend analyze evidence co analysis index period period mini mmse step integrate uncertainty bayesian uncertainty accurately study begin segment contain series bayes mmse square series proceeding day present square random posterior mmse perform clearly average uncertainty bayesian selection reflect averaging approach wide selection though assess confirm day demonstrated variate adaptive metropolis alternative local move parameter formulate rank estimation analysis real average unknown develop extend show perspective develop integrate trading trading triple make series p mini perform average select co integration mcmc framework automate two anonymous associate comment exposition university financial theory new york pp uk economic apply trick max l obtain c mmse mmse mmse mmse mmse mmse mmse mmse posterior mmse global ml move global learn simulation start away generate carlo utilize adaptive move represent additionally c index bayes markov length year log chain cm corollary department mathematics south sciences bag email capital mail com capital mail com capital mail com markov auto automate develop mcmc framework model sampler dimensional series block utilize sampler impractical involve full spline ideally suited reflect joint rank practically adaptive sampler also development automate consider make financial trading pair trade able framework demonstrate posterior random trading adjust potentially market condition coherent auto adaptive monte carlo analysis auto several variate improper care implication bayesian structure blind var improper distribution example paper aim admit conjugacy posterior dimensional variate dimension significant correlation random sample involve allow significantly increase typically adopt framework van conjugacy consider variate unknown covariance matrix contain rate mean unknown vector posterior rank dimension large justification direct inference var model set pointed model var model useful property widely use develop base follow utilize along model conclude range posterior triple trading typically note integration
find focus identification fundamental address assumption quantification uniquely associate writing explore issue study basic level close multi discriminate implement discriminant categorical quantification handwritten classification identify accuracy handwritten note quantification study derivative identification system implement laboratory describe document return short system short provide brief overview categorical describe propose leave cross predict unknown classifier predict pseudo simulation classifier discuss evidence long american computationally tool evaluation evidence possibility base need point computer suggest absence identification interested insight set identification comprehensive review statistical discriminant near neighbor appropriate neighbor unknown classified database study computational different together build necessarily document apply subset short different quantification document quantification use quantification select euclidean classifier near neighbor weight weighted segment write sample bin new investigate associate proximity normalize document recent contour white pixel calculate chi square well alone distance provide database correct return improved recent research character potential identify mathematical quantification handwritten convert image individual character manual automate letter character example make letter character skeleton skeleton every identify belong smoothly another shape different letter alphabet appropriate character letter particular letter count represent letter occurrence letter system describe extensive report write sample sufficiently letter alone frequency letter sufficient information individual letter information unknown pool business good mr news week go join col arrive nd letter address dr within five conduct whereby writing collect various year collect ask modify letter l letter paragraph letter letter character character ignore letter modify letter paragraph letter collect write segmentation character manually letter character text paragraph letter character individual segmentation letter association error character association less process another micro feature divided hereafter consist result document miss failure character write reason issue involve micro cause character present usage micro letter character per character per summarize character study facilitate letter th letter th write frequency th l denote place subscript subscript document unknown say unknown count use letter uv count letter let th letter multinomial letter observe parameter number attempt document assume letter dependence classifier accuracy compare principle bayes il suggest posterior multinomial shape procedure estimate conditional document write identify unknown know database plug combine naive rv employ preliminary classification class play letter word bayes rule tend extreme behavior plug literature chi measure difference chi distance chi distance measure suggest determine cluster character part feature single cluster proportion neighbor unknown proportion small document extensively effective range application document relatively chi chi letter combine chi square letter chi square take relative information letter pearson chi measure letter pearson chi square count freedom letter handwritten document degree chi square letter chi square evaluate freedom chi square tend score square document associate letter pearson square way table document database letter chi document number letter chi square chi degree write large reasonable combine pool pearson chi statistic author measurement exclude write author fit example chi chi type use study chi see vector define classification estimate letter il p analogously kl distance th eq write distance neighbor application reasonable document pool evaluate plug naive document classifier document leave predict validation classifier document classifier correctly identify scheme chi square document classifier incorrectly correspond estimate three effectively letter stress would classifier accuracy document explore subsample character write write possible size subsample would limited additionally subsample letter database study implement classifier entail leave entire author classify implement stress sample write sample document unable look simulate document construct letter left document occurrence letter letter letter proportion estimate letter letter left letter generate unobserved document document document leave letter effectively different original
patch texture texture error error use universal instead example texture pixel classified belong class convention even tell belong patch pixel assign majority detection regularizer outperform value incoherence adjust cross map precision show precision positive design model universal coding model approximately subproblem also show prior code patch well laplacian adjust additional flexibility dictionary size weight show practical impact active image bayesian burden several force reason conjugate demonstrate introduction address aspect future design nonzero overcomplete noisy acknowledgment partially wish provide fast toolbox thank incoherent helpful comment integral obtain precisely definition substitute constant mean easily du moment obtain nonlinear technique newton starting significant moment one function weight develop take naturally distribution value iterate function approximation symmetric approximated regularizer ta da da jeffreys exponential evaluate plug function plug desire however close central trivially yield solve part e I moment order one possibility solve u possibility fix conditional jeffreys jeffreys proper assume model plug fisher n last derive result distribution considerable attention lead art regularization critical minimization coding compare presentation classification call learn contribution theory practice learn collection many processing reader sparsity control parameter challenge automatic natural assess describe length framework term interpret bit describe coefficient reconstruct coding develop work denoise wavelet description coefficient compute work previous designing code obtain encode natural universal lead consistently process desirable robustness improved recovery decode signal compressive sense compare practice yield correspond sparse code turn lar regularizer dictionary improvement aforementioned task organize derivation universal present propose denoise conclude j cardinality goal model design dictionary solve problem column usually divide sparse among try commonly close certain condition coincide formulate respective jointly approximate alternate dictionary alternatively sparse code step efficiently update turn code compute good sparse interpretation summary interpretation insight provide norm see regularizer commonly constrain lagrangian form maximum posteriori estimation scale contaminate additive consider term previous regularizer special meaning signal task image compression coding generally response zero patch phenomenon reference subject suppose image reconstruction obtain good compression consist probability stage bit assign known provide approximate shannon modern finding minimize encode distortion reconstruction assume encode choice coincide error encode residual combine obtain l p lead sparse code characterize reconstruction coefficient instance enough interval pa fa pa fa sequel treat density course needs tune interpretation framework offer compare different introduction coding already study early consider shannon solution describe actually sparsity include magnitude uniform coefficient description furthermore actually section size patch well case arise decomposition result careful image coefficient form generalized justification observation heavy tail hoc available possibility close perform family numerical technique derive close coefficient empirical log heavy tail fitting encode patch dc component image histogram variability region correspond scan order contiguous block obtain value atom coefficient explain previous regularizer coefficient different row vary greatly empirical underlie figure occurrence value range associate lead weighted regularizer modeling perspective adjust poor thousand parameter learn model new bad estimation problem unweighted regularizer weighting failure signal atom position estimate impose hyper obtain sample code example compressive approach expensive something choice lack proper theoretical justification derive flexible avoid burden solve sample tool successful theoretic extension compression summarize several possible scenario deal code knowledge secondly identically underlie scenario yet would almost scenario fit original universal code code encode compressed coding assume lead concept class kn find fit simplicity arrange single length sub describe single probability optimum relationship define correspondence scheme assign class parametrize omit write include reconstruct know precisely negligible way universal part optimal value bit model thus quantization p letting require bit write derive complete uniquely equality satisfy l q regret necessarily something often previous assigning coefficient minimum mixture via evaluate exclude result mixture appendix refer evaluate obtain get convention approximation explain code note sufficiently regularizer tail laplacian happen determined fact could bad minima aspect tail bias perform typical use column regularizer induce leave regularizer thresholde shrinkage large coefficient much tune possibility fall model limited dynamic basis exceed true quantization practice advantageous adjust solution exist estimate moment obtain detail deal case jeffreys improper improper jeffreys first sample key observation jeffreys prior integral result appendix conditional jeffreys explain furthermore moment sample give practice converge delta jeffreys approach universal code view jeffreys loose flexibility deal lead prior problematic trade degree flexibility model yield regularizer code regularizer sparse problem convex sparse coding suppose j convex discard term point meet regularizers limit function kronecker delta laplacian therefore mixture point code time sparse code warm begin iteration course j choice choice na code laplacian regularizer discussion around lagrangian code formulation give error since actual term give compute regularization distribution regularizer separable coefficient expression appendix lie total showing efficiency optimization comment estimation exception principle influence need cause dependent laplacian issue compute zero deal properly work patch draw bit channel converted image channel scale unless otherwise overcomplete atom subset seek test sparse dictionary lead show additional encourage atom advantage ability fast incoherence empirically sparse code coefficient prior single matrix compute restrict empirical plot along good conditional jeffreys use moment yield moment model mixture kullback leibler fail improve sense model sensitive hyper estimation hereafter well prior jeffreys well figure vary atom confirm k fit atom globally fit laplacian critical fit atom improvement practical base pursuit ground contaminate regularizer application see measure support fall figure respectively model coefficient image patch dot well fitting laplacian bit clearly properly almost fit tail desire fitting laplacian dark laplacian light dark red fit index atom outperform per atom active define improvement range highly clean project subspace square true recovery
break distribute denote dirichlet process become cluster prior mixture atom provide distribution component partition develop mixture utilize heterogeneity focus couple dirichlet process hierarchical concentration dp hdp aforementioned linkage assumption change dp framework stick stochastic process co infinite proposal suitable problem functional close author introduce dependency dp mixture stick spatially vary variable flexible work focus mostly interpolation infer locally group introduce dependence distribute allow assess cluster extract arise generally index centering multivariate local formalism global dp use dp realization atom group global induced global refer nest yet broad dirichlet induce covariate summary hierarchical specification incorporate dirichlet go fully framework propose distinction hierarchy hdp rich hierarchy take hierarchy bring model place hybrid dp fact serve dp curve curve hybrid satisfactory despite functional require pure pre specify utilize switch random focus prediction instead worth note simple exploit graphical dirichlet spirit quite dirichlet admit independence fundamentally goal brief background dirichlet process hdp define explore graphical dependency sampling characterization also offer global group distribution recent literature brief dirichlet proceed hierarchical measurable dimensional dirichlet distribution concentration around center distribution probability constructive draw concentrated stick break independent beta iid measure positive integer viewpoint dirichlet show dirichlet process perspective accord exchangeable atom likely induce chinese restaurant utilize component study elegant account application dp modeling give brief formalism set group set index covariate let assume exchangeable suggest use mixture identically specifically formalism endow refer index factor prior factor conditionally hdp formalism statistically couple measure conditionally dirichlet base fully distribute base hdp share countable within distribute accord restaurant restaurant statistically couple fact distribution group mix hdp assumption coincide assumption lie product endow correspond borel algebra dimensional whose index modeling specify distribution relate distribution enable cluster associate basis group index nonparametric linkage among govern stochastic spatial measure conditioning vary center amount index graphical index summary collect specification v u draw turn local factor center global across hierarchical involve hierarchy level distinction hierarchy operate nest space hierarchy relate say probability distinct vector explicit place probability obtain hdp provide draw suppose collect choice spatial well many location available graphical also field undirected offer computationally finite collection undirected graphical independence relation connect representation h p h relation dimensionality distribution crucially wide exploit graph express stick break mutually necessarily k g follow satisfy interpret integer stick break atom support aggregated distribution local center local dependency induce dp distribute specific ccc model location independence relation u share q independence relation long among provide interested infer turning factor moreover among integrate partition h e k provide prior quantify finite index ab variation measure define simple derive distinct location due turn local exhibit variation factor extra govern concentration dominate turn stage vb vb across location factor vanish correlation increase either ratio vb dirichlet fully retain factor introduce vector I multivariate kt kk need taking support total factor ik give realization mix increment condition u claim local issue identifiability factor additional u function global nontrivial true identifiability also depth observation specification base inferential behavior essential sufficiently tail domain place factorial job put shall describe nest hierarchical sampling marginal approach dp stick break former characterization conditional leave detail approach dirichlet addition issue dirichlet mixture aspect membership integrate reader notation turn local exclude leave leave denote leave also standard distribution local factor integrate directly reconstruct mcmc construct unbounded finitely represent quantity computation density combine generate previously new value likelihood value use f fy dy tn always proportional become local exchangeable last collection change change mixture membership tf fy main stick break instead likewise measure dirichlet k conditioning equivalently stick break distribution associate location kn explicit g sampling involve variable atom mixture count suffice markov chain end factor disjoint subset conditioning collect item group component dirichlet concentration correspond form let index example tractable long computational conditional density exploit conjugate computation conditional alternatively independence model problem computation readily available tf uk latter still alpha markov inference cc cccc ccc cluster dash solid marker ccccc entry depict cluster average figure set spatially cluster population factor spatially vary mixture normal generate kind encounter tracking problem covariate snapshot particle point move switch identification know move smooth cluster move path illustrate variation number cluster location generate simulate longitudinal draw relatively gp exponential split value previous global normal generate observation clear cluster location give different essentially specification take gp set fig datum fig b specifically support local cluster wider credible band specification factor perform sensitivity weakly across factor despite still estimate somewhat hybrid dp encourage expand concentration extremely reason robustness global second nest equal robustness record logarithm subject cycle day interest day assess time global identify global pattern specification set cluster close addition match group variation day elaborate subject cluster average day interval find indistinguishable sharing range last separate distinct regime cluster sharing drop hybrid process dp perhaps literature global hybrid dp curve subject reveal sensible measure illustrate grouping complex specifie cluster functional curve observe example curve switch two fig probably due overcome proposition consistency worth note hybrid practically directly describe nonparametric global locally nonparametric solution dirichlet virtue global dirichlet center support moreover cluster process local cluster spatially whose dependency canonical dirichlet rich behavior find particularly adapt manner mixture direct local cluster probability atom lie take condition regard hold fact density choose incomplete value gamma probability interval overall away follow west obtain equal variance standard metropolis step prior specification characterization posterior integrating rather index reconstruct think thus markov space principle explicitly play role give data th location conditional previously value takes previously calculate possible f uk fy u dy take tn future always proportional may correspond within treat last index group set tf fy auxiliary draw equal gamma prior
table row average middle row proportion bottom sample package specification w rv scale axiom theorem condition theorem exercise notation summary edu rv tail inverse heavy tail cumulative formula author usefulness introduce implement publicly theory link I error pattern often many white model practice exhibit wind internet traffic datum particularly notable signal heavy develop necessarily tail cauchy stable forecasting long process heavy attractive perspective viewpoint successful assume understand transform rv rv vice versa thus knowledge software still optimally include normality special testing parametric transformation estimate efficiently heavy tail transform choice model convert heavy illustrate researcher make avoid whole base tailed statistical tailed fig modeling nice subsequently technique fig semi parametric suffer drawback sample limit meet transformation restrictive identifiability condition three fold meta tail tail implement statistic author pdf introduce many study tail useful unimodal heavy confirm also benefit remove heavy tail exploratory ray density estimator detect remove finally methodology proof computation simulation publicly wise version jointly still well behaved deal skew cox mle limitation negativity many limitation shift show box cox transformation cauchy fail half desirable underlie process discussion box cox cox lower variance contrast framework tail remove heavy difficulty heavy tail rv rv cdf vary infinity characteristic basis rv heavy tail parameter skew respectively transformation tailed tail tail great inverse express pdf fall specify numerically approximated parameter match empirical quantile recently analytically tractable essential spread long ease result strongly skewness skew adapt skew fig generalize rv rv parameter tail parameter gaussian rv define rv input rv continuous rv rv transformation w rv shape away increasingly tailed value far heavy tailed tail necessarily fig lead property transformation dependent unique remainder state otherwise e implement recently appendix eq transformation available view popularity tail heavy tailed compare deviation see kde pre bandwidth use likely also kde kde kde figure transformation heavy axis remove axis operate degree tail map generalize closely skewed w ease location heavy rv yy f g yy w w z family allow bayesian statistical various different equal solid black tail dash color cdf quantile standard estimate equal family see quantile compute quantile software package useful education tail statistic cauchy stable yet transform via previously method transform tailed world e quantile straightforward tailed cdf equal cdf normal theorem functional student degree freedom often heavy equal student rv scale always match student tail heavy tail w distribution opinion return inference regard moment easily heavy student identically pdf quantile inefficient replace fast usually introduce size use reliable quickly accurate size yy maximize specification eq decompose transform q necessarily decomposition decomposition mle must calculation transform trade transform versus extreme figure show contour transform increase transform datum close input equal likelihood penalty monotonicity red green monotonicity imply black numerically existence uniqueness assume remain form continuous twice etc usual let case mle z principal real satisfy say enough heavy student problem unknown contrary get truth support disadvantage priori specification heavy tailed estimate distributional assumption present base back e analogously skew heavy skewed entirely supplementary back comparable estimate heavy tail rarely numerical confirm fact magnitude property classic standard low sample good mle clearly outperform heavy prove tailed inference limit quantile ml show usefulness demonstrate cauchy sample heavy w excellent daily section heavy tail pattern law know poor location go symmetry cauchy heavy tail two extreme ml fail summary statistic gaussian average excess normality value version fair comparison use well influential affect relevant good already clear location approximately toy work nice use observe heavy tailed transform stand min rr lot financial negative skewness financial model skew student generalize upon implication avoid derive heavy complex far beyond scope direction unconditional figure log return daily table confirm heavy sensitive skewed skewness zero double tail versus computed ratio freedom likelihood double equal double pay lower transform twice give significant fit reject b autocorrelation top right kde normal plot make decision trade negative significantly ignore heavy reject table heavy tail heavy mean respectively essentially scale lead conclusion exist case moment financial literature finite fourth study financial datum actually fit light heavy fit ccccc est pr est ccccc est ccccc est se pr ccccc est pr est se pr study would back indistinguishable gaussian von successful adequate log return trading close truth non even small treat optimistic observed tailed heavy tailed joint heavy tailed optimal researcher improve tailed datum preserve ease usage previous focus remove tail believe underlie gaussian convert assume might interpretability observe unit become helpful exploratory detect tail reveal improve inference image cutoff peak count rate rate datum package approximately day background gamma ray heavy tail make inspection lie drop end drop decrease ray detector sake comparison figure last observation cut amongst visually heavy tail underlie trivial heavy insight optimality cut heavy last heavy tail reveal mean cut equal transform standard fitting component cut tail would gamma ray rate analysis intend gain interpretation statistical ray count adapt skewed output heavy tail contribute unimodal gaussian skewed often assumption research direction perspective distribution view distribution literature discover statistic transform approximate versus tail show direct tail tailed practice I package available facilitate acknowledgment thank attention give detailed suggestion supplementary material list property definition z one relate simplify decrease remove heavy tail gets linearly hold line equal derivative use w gaussian evaluate rest maximize loss multiply yield since square sign conclude likelihood r z sketch stay maximizer occur
aim scope reasoning incorporate programming applicability metric logic long start observation triangular composition area find symmetry hierarchy dissimilarity precision tool mining build generalize ultrametric lead power tool algorithmic relate begin motivate hierarchical general geometry come play metric ultrametric induce datum datum implication conditional analyze conditional spherical ultrametric inductive especially computable shrink sequence generate master program real interval ultrametric systems section ultrametric dissimilarity set define map positive measure dx dx dy dx dx dissimilarity metric map ultrametric need endow metric instead dissimilarity satisfactory hierarchy term dendrogram define embed subset index subset totally require subset ultrametric embed subset constructive induce hierarchical pairwise property I iii q h positive include ultrametric dissimilarity dissimilarity viewpoint triangular metric triplet relationship particular hierarchy innovation capture understand notion anomaly take document close query target unlike query focus ambiguity record appropriate situation query situation illustrate ultrametric triangle small take query sort triplet define define ultrametric study equality side away sort explanation provide query novel treat raise subsequently handle dx dy reading hierarchy inequality ultrametric hold distance close distance ultrametric dissimilarity construct hierarchical newly criterion either agglomerative criterion e connectivity near pair reciprocal guarantee reciprocal prove agglomerative criterion near neighbor article journal les analyse des package information find dendrogram relative rotation alternatively rotation group introduction image case group cyclic shift permutation permutation cyclic group cyclic structure generative level simply subtree define denote product subtree alternatively look shift group amount root us space constant component component vector move hierarchical replace cluster disjoint couple denote invariance discuss haar dendrogram spatial dendrogram successive work build something figure vector inverse determine exactly read terminal detail signal coefficient wavelet dendrogram datum wavelet hierarchy regression entail hierarchy show wavelet input data haar dendrogram give approximation term observation I hierarchical hierarchical way haar characterize read give example give vector supremum sup chain clearly root partially order complete partial alternative closely relate domain endow ultrametric space motivation come monotonicity relate completeness set chain chain ultrametric space rooted ultrametric space consider ultrametric observation pairwise agglomerative hierarchy follow haar dendrogram embed haar dendrogram allow us chain point singleton haar transform call member increasingly approximate correspond two respectively set observation rest note subsection ultrametric ultrametric ultrametric ultrametric partially order generalized ultrametric distance value generalize dendrogram associate ultrametric dominate node could design whereas read dendrogram subset reasoning monotonic rigorous conditional sometimes relation program kind mapping monotonic reasoning describe ultrametric mapping critical usefulness ultrametric ultrametric arise logic force monotonicity monotonicity operator metric study problematic program idea examine I partially order set negative finding monotonic operator logic database introduce imply enhance monotonic long applicable overcome technique analysis argument latter include metric ultrametric ultrametric discuss subsection ultrametric join comprehensive background hierarchical relaxed often object real object dissimilarity attribute characterize object individual etc etc notion join set presence dissimilarity object attribute get distance value prefer treat get three consider
tie inverse draw way define dirichlet random construction draw form increment nonparametric stick break draw put construction considerably connection specie reference readily context bias generalise direct modelling density formulation weight component specific take algorithmic ji j limit joint dirichlet subsection strong statement independently view integrate conjugacy multinomial integer b b atomic joint equivalently draw set extreme preference equation number neutral population factor account allocate lead together familiar formula page partition imply derive consequence sufficient also context integrate take cluster specific methodology derivation dirichlet exploit make notation use label partition lack thorough methodology exploit bayesian highlight procedure mix respect thus dirichlet terminology dirichlet process formulation start implicit locate notable apply rapid power researcher routine computation example visit far exploit bayesian fitting margin substantial dirichlet aspect modern development throughput capable dimensional quantitative genetic biological sample gene expression gene datum replicate typically condition interest specific although variant gene dependence gene condition exchangeable hyperparameter strength efficiency nonparametric counterpart instead describe variation across population gene prescribe model dirichlet consequence atomic share probabilistic cluster expression profile normal conjugate explicit posterior marginal multivariate continue section real essentially univariate unable handle flexible drive consideration mathematical point numerous exploit limit behaviour component rank limit law rather establish limit law see rich volume take poisson pd back stick stick break define gibbs kind effort dirichlet alphabet rich sometimes area perhaps stick break representation dependent covariate book excellent development result surely discrete effective achieved impose probability branch set smoothness approach literature markov mixture without reversible jump demand handle reversible jump obviously appealing integrate chain solely go well reinforcement generalise term p show side far concerned particular computing inference arise define method aim target conjugacy posterior dirichlet begin sampling later hyperparameter together cluster conjugate probability partition partition likelihood expression interpret allocate cluster inverse mixture explicit form limit item require available conditionally possibly independently necessarily identical c sampler likelihood partition multiplicative proportional dp c multinomial er dirichlet ease motivation dirichlet model equally restrict form probability item propose study kind need either fitting demand expect shape methodology level purpose cluster consideration common one foreground imagine cluster belief represent exchangeable would parameter section kind aim exchangeability stress information item draw purely describe class henceforth colour variant cluster colour dirichlet dp independently colour pair mixture different measure colour cluster identify observe cluster stick colour segment stick break I define distribution share content leave atom within colour generate expression simplify significance ignore degenerate ordinary clustering remain analogously draw availability partition gibbs item accordingly colour n py ik expression simplify many gene cluster exchangeable think view k leave label background two regular mass colour p readily adapt regular illustration background vector measurement distinguish wish ss profile j particular probability heterogeneous measure include mass gamma hand density covariate mean variance univariate scale joint eq j km g h tt ks density methodology expression gene record period development system point day stage day obtain totally cluster phase development take represent linear separate phase singleton discard last partition theoretic pairwise negative
increase small mr iii monotonically value hold satisfied right side r meet exist treat r rr give eq take account whereas equality second factor q tend factor tend replace argument tend result numerator converge proposition cm estimation inverse slope interest motivated significance use asymptotic low asymptotic allow number meet prescribe quality irrespective minimax asymptotically estimation sampling asymmetric trial arise branch engineer measure use rather normalize square normalized analyze confidence associate require parameter advance sample sample bernoulli sequential procedure estimator variance procedure efficient rao sign multiply asymptotic different plus observation minimize observation error appeal stop many necessary obtain exactly random observation sufficient stop useful namely function regularity whose arbitrary guarantee exceed asymptotic error associate prescribe irrespective paragraph equal specific costly example nevertheless time second assign accomplish generalize slope weight positive normalize version ratio situation proportional large symmetric name motivate inherently subtract minimum unbounded error symmetric represent scale risk loss symmetric ratio normalize dissimilarity generalization allow multiplicative side natural representative incur production certain device production presence result pixel systematically incorrect expensive discard adopt produce sensor accept correct part process camera camera desirable sensor classify produce sensor depend merely acceptable advanced whereas type production type deterministic sensor number control primarily line sensor require need advance make actual great sensor either resource camera inverse binomial case turn discuss minimax asymptotically proof success random inverse binomial normalize incomplete define follow relationship binomial distribution similarly function parameter pi ip q express estimator risk take identity seen analyze arbitrary follow consider see take follow stem strict hand greatly simplify case apply single namely reduce addition consequence account seen see case close justify worth establish already generalize ratio prove desire exceed irrespective suffice choose illustration depict certain natural consider I risk guarantee criterion minimize estimator address value n include one optimum achieve arbitrary estimator necessarily risk monotone unique determined condition substituting make easily monotone expression computing theorem degradation see far degradation furthermore minimax minimize possibly ba determined tend follow establishes estimator theorem approach asymptotically consequence approximately sense fact commonly mean absolute minimax theorem minimax immediately stem cover comprised curve vertex ensure hold reduce iii strictly increase function increase iv defined express decrease thus j imply prove strictly decrease
parameter mean simulation discard first burn monte burn extensive testing period value summarize accurate parameter parameter mcmc delay rejection observation due fast speed delay produce computational equivalence run monte carlo ratios ratio monte method compute minute hour mcmc therefore expect monte initial benefit univariate quickly monte em simulated call observation method iteration delay value load matrix dataset n dd identity mcmc dimensional delay mcmc contain draw discard monte summarize delay rejection delay high equivalence score delay delay almost monte carlo consistent mcmc vs k replicate plot suggest em converge take hour summarize estimation k low optimization method ratio almost method note jointly exact article find delay stage acceptance table delay use decrease acceptance ratio acceptance high method case benefit sub iteration converge iterate replicate converge delay converge similar delay rejection around iteration iterates generally mix plot iterate initial delay rejection immediately delay rejection benefit block em start method delay fit example likelihood copula sub block agree sub block reduce univariate compare several delay rejection gibbs optimization replication volatility moderate unconditional quite optimization delay mle persistence parameter reason poor could wide suggest may far restrict range allow delay rejection factor score gibbs base delay compare term factor score fast become model apply discussion approximation closely suitable factor several delay loading univariate factor model student delay density tail parsimonious variance series factor economic portfolio asset pricing risk management determine particular volatility expense determine factor expensive propose delay fast optimization determine article simplify approximation estimate apply compare estimation considerably financial market cm cm delay monte carlo report diagnostic replicate acceptance hour minute delay stage ratio loading loading mcmc stage delay stage stage c carlo report replicate across replicate include stage delay stage report replicate monte include calculate replicate replicate replicate replicate replicate marginal mle cm delay rejection mcmc report delay rejection ratio panel ratio replicate univariate exp middle replicate second univariate example right column replicate factor p ratio iterate leave factor cm cm cm pt monte method delay rejection metropolis factor towards stage choose method computational particularly likelihood estimate marginal markov propose estimate stochastic especially useful monte method delay monte use financial economic series capture pricing pricing build existence asset capital asset pricing pricing asset second market asset west asset pricing multivariate volatility excess asset generate portfolio decision way become financial market quickly portfolio estimate literature bayesian considerable limit applicability factor hence article enhance likelihood dimensional difficulty dimension variance covariance time govern consider multivariate stochastic volatility whose distribution student allow asymmetric multivariate govern model model latent coefficient article estimate extend estimation chain univariate transform model hasting proposal parameter block scale mode adjust asset volatility govern autoregressive ar persistence ensure writing unlike error kalman freedom gibbs sampling computationally inefficient normal degree approximation metropolis hasting step write variance article component write introduce latent conditionally univariate equation various summarize initialize jointly density build target maximize density kalman negative inverse hessian accept prior reject current retain next filter step build block carry scheme avoid parameter burden iteration severe iteration consume optimization order proposal h form similar monte carlo univariate treatment right walk proposal likelihood ratio bridge focus feasibility carlo univariate factor model literature latent volatility identify follow factor mutually uncorrelated structure govern first estimating factor mcmc execute mode freedom accept candidate sample space indicator note bf decompose jj jointly sample separate jointly simulation computationally slow delay delay optimization find proposal mode delay rejection build proposal use walk stage value reject load mainly sample discuss dimensional although build two rejection stage efficiency chain monte adaptive stage reduce suggestion first five delay rejection method second split block contain number sub block last update optimization delay delay rejection univariate generalize evaluate represent univariate monte carlo use gibbs burn expected respect f ff ff maximize since conditionally equation substitute practical issue determining choose marginal solution available necessary simulation need given factor marginal choose article use advantage later posterior median mcmc median numerator sequentially integrate auxiliary filter illustrate detail prior decompose marginal alternatively discuss estimate posterior necessary accurate
conclusion couple sound cross comprise coupling detect wrong strong coupling synchronization measure qualitatively result base entropy deterministic multivariate autoregressive var delay distribute lag model dr model well test identify correct depend determine random independent lag form frequency embed show table length htb driving regressor realization add criterion bic shorter demanding absence drive identify direction drive slowly couple detect one two embed vector linearly realization correct embed realization subsequently datum coupling coupling examine significance example positive driving exceed drive lack analytic coupling coupling use simple technique shift surrogate couple time surrogate bivariate series contain chance regardless strength embed patient patient patient b top left panel panel difference channel channel complex differ purpose test system order decide introduce large bias variance distance rather stable far turn simulated termination strict strict need lead quite contribute would detection noisy couple nature information series embed enough embed estimation able detect coupling adequate sufficient scheme purpose invariant cross mixed capable detect measuring demonstrate system multi eeg though channel channel right component measure use measure different coupling significance restrict bivariate apply indirect coupling acknowledgment ed co national general technology european thank pl eeg channel investigate multiple time derive continuous discrete information criterion past purpose causality detect evaluate couple system eeg publication embed setting noise dynamic delay one projection theorem one relate embed context spatially system simultaneous variable location implementation reconstruct later extend series multivariate embed widely strength couple parameter series projection independent thus optimal nearest create vector couple dynamical uniform embed delay neighbor basically extension account characteristic different series univariate embed reconstruction criterion select parameter univariate building embed allow delay simple sound modified purpose multivariate analysis bivariate series scheme sec propose scheme strength result patient summary dynamical forming series pseudo reconstruct delay embed delay topological equivalence original selection goodness fit prediction criterion local delay autocorrelation lag n autoregressive false near embed series dimension reconstruct also choose unique even value minimum uniquely theoretically system thing reconstruction due finite reconstruction marginally observational reconstruction elaborate concern delay range reconstruction time dynamical include series investigation opposite often redundancy address use series prediction reference series optimal consecutive non consecutive component redundant information pair constitute redundancy investigate treat usage lag vary lag inspection combination determination computationally large moderate lag case evident practical need method embed lag collective select reconstruct property dependent dynamic system two time future represent horizon start empty step x I represent maximum j augment embed vector build mutual step inclusion even augment information explain previous univariate series modelling purpose create want mixed preferred quantity mutual vector delay lag univariate mutual appropriate dramatically million embed accurate information mi dimension base entropy near neighbor distance th joint space neighbor dx dd cube metric entropy joint denote immediately mi heavily setup use explain time joint entropy neighbor formula entropy onto substitution distance plus expression estimate note independently mi estimate project embedding rely examine mi eq data estimate bias bias since expect mi systematically reduce truly bias reveal time zero mean series regard fix dimension correlation reference embed accordance time top right denote last next dimensional range setup effect correlate degree comprise series study presence setup series vary identical expect lag second dimension mi fig case htb figure mi reduce fig mi correlation neighbor decide pilot study show though embed form time small mi denote mi theoretically criterion however formula perform well another serious mi criterion reconstruction term stop estimate one mi contribute inclusion component bias negative may theoretically impossible balance bias see bias comparable embed scheme variable xt xt xt contaminate observational white series interested predict variable obtain mi cycle result embed component htb explain mutual iteration solid lag dot represent lag highlight select lag exceed panel embed mutual information cycle panel mutual criterion embed left axis value respectively identical similar embed fourth completely give embed cycle decrease gradually gradually non multivariate embed couple information causality well implement compare idea causality contradict embed accord delay matter couple causality model causality aim embed depend adjustment may threshold increase series different span test quantify coupling strength future eq set near find prediction embed compute increase thus couple contribute varied ahead replace information give measure explain embed mutual opposite direction coupling series strength take uniform embed realization large frequency delay ahead detect effect threshold variation realization necessarily produce select detecting drive also weak coupling realization couple show htb map measure vector monotonic contribution shift small observational repeat select embed occurrence couple map select vector either component strict conjunction presence equivalent regard presence embed information transfer similar c well monotonically system couple identical map strength drive opposite apply coupling frequency occurrence map criterion strength predict expect fact embed conclusion strength turn quantitative measure table detect substantial coupling equal direction take couple respective measure increase weak coupling see couple give x ty use drive account significant delay frequently realization occurrence couple embed vector fail transfer realization frequently embed htb couple accordance form modify prediction monotonically slight start synchronization system coupling comprise component dimension simulation large iteration drive series enter detect coupling vs htb couple r dark gray detect whereas couple large variability change time enter embed form really system even give spurious produce zero system direction regard vector investigate future horizon I r embed either similarly vector weak coupling form make presence small
range code figure scale optical identification peak galaxy distribution intra distortion advantage distinct proxy detect ray candidate explore numerous ray dr survey follow optical confirm likewise arise ray always lie peak scatter ray furthermore contain nsf department contract ac sf theoretical physics ii foundation science foundation max high education web www american natural history university university advanced group university institute nuclear chinese sciences laboratory max institute institute university university university united david david particle laboratory il department university ann mi ann university il physics department university california physics berkeley national laboratory berkeley physics il institute national laboratory stanford stanford ca california department physics il laboratory york galaxy galaxy release identify red sequence galaxy feature galaxy field galaxy correct run dr large ever optical rich range completeness ray cluster range cluster member website decade universe confirm acceleration explain modification gr component pressure adequate perhaps simple possibility challenge explanation retain something dark possibility history growth central physics one growth abundance galaxy peak abundance encode universe galaxy cluster cluster galaxy detect determined ray emission optical galaxy impose galaxy rely physics though often mass ray emission potential high mass consequently relatively free optical require cluster search optical identify optical much dark serious projection optical detection volume survey existence uniformly old galaxy remarkably energy include strong break galaxy shift optical create galaxy galaxy varied sequence prominent galaxy remove field galaxy red galaxy dominate exhibit narrow scatter color sequence galaxy refer reference therein galaxy develop cluster galaxy gaussian identify sequence galaxy spatial galaxy around digital rich galaxy extend completeness test efficient cluster dr single challenge optical detection demonstrate red sequence color outperform step introduce construct dr use match know ray publish cluster test conclude summary future optical survey convention omit detect galaxy cluster precise uncertainty detection therefore optical effectively de calculate plane position galaxy along limited technology year optical galaxy cluster mainly roughly classify de projection cluster algorithm decade de l type projection magnitude smoothing kernel band adaptive band hybrid band cut enhance friend friend band band color band band band galaxy make galaxy method detect massive unfortunately maintain low intermediate contamination cluster create scatter derive technology greatly optical galaxy decade precise survey magnitude galaxy spectra color effective red galaxy narrow scatter color history determine position galaxy basically way de multi color obtain project detect directly color space principle machine color magnitude galaxy train reconstruct reach galaxy sense perform galaxy I compare different well cc cc color compare typical velocity km galaxy small precision possible alone projection insufficient remove cluster population magnitude color dr full right scatter tuned alternative stay color galaxy display color sequence cluster field galaxy red plus provide powerful follow section unique galaxy find galaxy wider include foreground galaxy cloud show galaxy magnitude represent fit purpose measurement proper modelling red traditional therefore correct gmm effectively parametric analyze span range galaxy red curve red blue background galaxy member line deviation gaussian galaxy component intercept sequence panel color color break across informative vary filter red color near band therefore detect cluster span wide color search adopt typical galaxy determine examine require determination candidate cluster broad chance low modal even reduce occur broad window member galaxy search adequate addition member application apparent adopt galaxy cut simplify structure around galaxy addition right band apparent concern define color galaxy color band degree contamination background limit relatively effect detection produce need color mean adjust definition filter galaxy center algorithmic physical motivation focus galaxy potential galaxy potential theory center simplify comparison theory galaxy galaxy center dark potential sense cluster center galaxy choice somewhat uniqueness act determine phenomenon detection search galaxy dominate motivate factor identification play selection galaxy give color list range color wrong choose inaccurate serious problem usually place near filter filter filter apparent adjacent color locate fall band combine red sequence either ambiguity impact estimate detection consider size mass scale keep consistent ideally computationally substitute approach attempt metric radius measure fix everything scale size member circle candidate galaxy error relevant fit color overlap galaxy around gaussian analysis conclude give well hybrid select red candidate candidate foreground galaxy belong sequence red red sequence lie deviation next quantify spatial plane radial kernel important analysis kernel bias shape member galaxy radius project choose regardless essentially peak smoothed field position introduce another weight galaxy galaxy band magnitude band magnitude correspond introduce indicator whether important double check minor negligible quantity calculate straightforward basically step every evaluate galaxy identify red candidate identify cluster finally search list scale procedure summarize essentially peak smoothed height peak find cluster galaxy quantify identify galaxy peak merge criterion contrast previous procedure motivation peak merge peak avoid star importantly peak indicator structure probe internal follow radius candidate fall inside merge set avoid merge galaxy foreground star identification algorithm color optical completeness test match inclusion filter addition vary grid testing select maximize filter statistically motivated feature radial serve filter less biased cluster priori detect advantage algorithm execution neural near polynomial measure color reason easily apply release digital construct optical rich cluster quality cross match ray create dr completeness detail construction digital color imaging dedicated point mapping dr include area unique object identify paper survey area calibrate degree north south input galaxy http server galaxy band magnitude less clean galaxy galaxy neighbor galaxy bad band great principle galaxy candidate subset list color additionally band candidate cut galaxy take color false project cut keep galaxy search effectively color worth note galaxy dr relatively contaminate star galaxy weight reject range color determined galaxy within change background see measure make across whole range relate two mapping however simple color measure narrow range percentile bin bin scale scale figure relation scale affect rank strength unless note cluster strength rescale scale scaling remove much measurement band generate full galaxy dr search reduce effect weight strength much weighted image cut figure contaminate star remove star pass star datum processing mask star star mask add galaxy dr mask fall inside mask full release refer coverage tag public cluster cluster figure cluster b tag name unique galaxy dr gm galaxy inside gm recommend use galaxy public cluster previous section apparent bi situation dominate impose modal potential long dominant modal narrow width color separation vary change overlap overlap project galaxy cut galaxy appropriately figure color deviation sequence cut impose member coincide red equal get select galaxy galaxy contamination weight automatically count cut always count demand leading recommend public tag finding criterion completeness quantify quantify whether however calculate ideally dark matter issue high resolution color interaction dark create simulation prove factor resolution galaxy color term observational realistic cluster put addition check completeness ray cluster uncertainty full accommodate realistic background widely similar step realistic dr rich galaxy input remain keep color property unchanged create whose range match ray visually member galaxy pick number member position color remain unchanged select corresponding galaxy
property bayesian social version segment explain ph distribute say lot region simplex segment simplex iv line set condition statement nice belief outside belief lie outside numerical example iv set twice show iv figure right policy decision maker alarm delay take account decision maker cost pick continue delay operate incur decision social choose decision signal pick event equality scalar actually choice similar constrain optimal bellman hyperplane lie decision assumption decrease continue hold theorem sufficient optimal polytope assume states polytope polytope proof establish omit ph decrease reason even though characterize example consider suppose lie change ph transition satisfy optimal characterize interval ph increase hyperplane preserve structure stochastic obvious dimensional parametrize linear parametrize ph line define appendix lie stop belief iff ii iff increase linear policy leave include sense policy threshold curve threshold nice triangle require threshold example threshold threshold threshold line show hyperplane lie polytope h cc linear policy close resort stochastic aim linearly problem eq initial convenient solve algorithm deal constraint local optima necessary several iteration sophisticated base applicable solve stochastic problem likelihood specify completely protocol multi controller network previous section index act agent act choose stop early false alarm operate mode micro mode obtain observation mode equivalent sensor time state time instant achieve agent micro management detection view macro operate belief micro micro interact decision micro determine decision macro sensor mode avoid solution sec social micro pick sensor convex agent depend mode observation conditional integer unlike view mode belief measurement dependent py probability depend state polytope mode constraint assumption base micro aim detection view subject belief evolve sec agent pick macro bellman theorem detection proof dependent observation observation yield mode dependent consider follow assumption sec recall denote lie c regard decision macro relatively inequality hold j accord appendix present early section present illustrate multiple mention sec ph prove theorem geometric example change change social global choose delay check hold comprise triple compute construct point implement phase change illustrate threshold curve learn ph belief simplex triangle social local cost choose discount model chose geometric since indistinguishable markov transition plot ph transition ph geometric distribution type decision solved imply converge theorem plot hyperplane polytope segment hyperplane lie thereby ph ph therefore hyperplane phase depict example motivated understanding local perform related agent scheduling unlike optimal multiple result detection result threshold behavior simple ph sufficient hyperplane simplex give switch nature straightforwardly underlie current work social order agent market background reference stochastic proof theorem iv optimal social monotonically monotone order restrict simplex order since preserve ordering let stochastically dominate denote suppose increase ii state state order ordering segment connect summarize useful least great great comparable chain e submodular function submodular increase e tp order tp multivariate mass p multivariate say scalar th dominate function associate classical associate kt induction iteration complete induction value pointwise step induction note positively I bellman since concave function preserve concavity therefore concave concave condition filter pa jensen detailed version lem yy c intersect namely p tp b b p increase c b see imply b b definition iii immediately e p e single verify bellman read threshold state update e imply need partition namely introduce interval lem furthermore imply recall straightforwardly verify b b tp imply symmetric claim hold statement straightforwardly statement return proof piecewise four interval vector crucial social since linear concave piecewise result interval follow define proof comprise statement I ph every belief segment belief segment intersect hyperplane straightforwardly establish since statement ii increasing ac decrease appendix statement imply learn belief hyperplane formulate belief polytope statement vertex clearly normalization imply I proof induction algorithm suppose imply pointwise initial step bellman trivially iff two part preliminary public update bayesian filter tp need fix zero non assume say either case equality identical suffice b lr l tp course condition say tp tp prove tp imply ia ia bb iii row first iii ia iy iy appendix tp straightforwardly belief sensor management tp arbitrary tp ph mathematical induction iteration clearly decrease polytope choose increase inductive decrease polytope kt v q k uv complete finally pointwise decrease polytope polytope definition decrease need show decrease hold equivalent iff title title lemma global interact stop decision via agent record private noisy process optimize maker four base detection general decision detection threshold lattice hyperplane curve sensor detection view prior optimal scheduling decision detect optimize alarm delay vast application finance team involve countable agent suppose agent act receive compute posterior subsequent process stop decision maker monotone belief exceed understanding decision consider decision maker achieve change global outline multi recent learn classical act sequentially distribution change decision subsequent agent choose optimize utility local observation agent public decision set decision multi underlie decision stop detection adaptive sense sensor equip local sensor controller multi system act underlie sensor sensor belief decision agent management tracking surveillance case individual sequentially fashion dynamic classical detection agent example generalization decision determine state determine decision continue determine instant decision lead belief decision threshold stop region fig visual optimal policy illustrate threshold policy horizontal posterior continue multi change thus local fig b programming unlike show fundamentally policie global stop continue local decision characterize triple value last decade social study economic financial market social see numerous paper limit excellent exposition learn important decision agent irrespective cascade motivation understand interaction global learn recently several information global paper address utility system design behavior give behavior theoretic involve equilibria ph ph ph geometric ph distribute since ph find ph ph change time social detection change analyze systematic investigation distribution deal characterize policy make perform organization paper present protocol problem formulate social classical kolmogorov classical sec stop term concave theorem measure although make nontrivial need cost entire trajectory classical policy start analyze belief theorem posterior sec probability small explicitly optimal decision policy change zero ingredient proof result fix social update region cascade social characterize policy maker detection ph ph state state systematic markov ph geometric ph distribute bayesian lie multidimensional policy ph kolmogorov question sec sufficient global decision characterize hyperplane multidimensional threshold kolmogorov sufficient characterize switch multidimensional lattice structural result involve stochastic agent belief respect threshold curve monotone ratio partial order policy linear sec agent detection optimal detection agent constitute decision framework sec incur decision maker detection sec sec estimate underlying act view instant act local agent denote local agent define sigma comprise chain point ph ph distribution approximate change time markov state evolve jump jump distribution occur one assume matrix state change appropriately choose ensure state time geometrically agent observation time indistinguishable observation belief agent belief past hide hmm filter denote public local formulate denote non incur agent pick state agent expect cost indistinguishable decision subsequent public belief agent public belief difference belief call likelihood likelihood filter local underlying filter implication belief due explicit contrast explicit aspect pattern protocol private observation instead decision take agent private likelihood probability explicit belief aspect social sequential public belief proof hard argument belief proceed briefly describe public belief interval ph belief simplex simplex simplex course formulate maker eq measurable define policy cost since actually difference social learning filter social expensive classical social access thus compare define policy incur bellman equation recall bayesian social detection belief versus social formulation say belief social incur higher consider time detection classical initial incur small social explanation lose symbol proof appendix observation value concave establish jensen bellman prove optimality policy function detection problem apply incur social comprise list partition convex likelihood specify detection policy rest ii sec consider change geometric double threshold sequential public characterize decision likelihood probability though decision let exclude recall polytope row matrix characterize global recall tp negative tp negative false alarm e submodular sec ph states example classic book distribution etc example iy py iy iy iy trivially numerous example assumption sufficient assumption geometric distribute time view maker need ph straightforwardly obtain matlab local always dominate standard decision yield state e cost submodular important appendix rest partition polytope belief theorem polytope denote decision matrix insight assume matrix define possible decision hyperplane ensure hyperplane intersect simplex ensure intuition hyperplane imply mean hyperplane intersect nice straightforward otherwise hyperplane multi simplex iv appendix hyperplane vertex lie polytope fig social double alarm cost change geometrically proceed behavior theorem corollary plan eq observation optimize satisfy bellman equation interval belief theorem filter characterize state point composite implication interval region dynamic characterize global also cascade salient feature symmetric right represent evolution public sequential hold trivially global q value cascade iii symmetric decision concave implication comprise three interval second public irrespective irrespective public observation subsequent agent notation cost incur matrix denote define interval recall associate optimal transform deal prove existence threshold policy optimal invariant characterize incur global constitutes implication social note cost region region simplify allow tight discount theorem piecewise value iteration element piecewise apply structure also cost optimal double threshold learn policy cost policy almost distinguished policy
precede separate split separate separate independent n nk bind cell constant probability completely radius entire end interested appropriately p I give ball hence radius intersect ball large disjoint cell less know level observe child success second variant covariance definition go guarantee small expectation point cell contain randomization albeit whereas max level dimensional riemannian radius contribute present trace fraction trivially set neighborhood manifold concentrate nice logarithmic improve packing room packing aspect ball radius aspect cell pack lemma moving result demonstrate constant attain mention packing application search however require max primarily right exist alternate space partitioning split partitioning split alternate split forced fraction child thus ensure depth point depth maintain packing space seem theorem leave max structure simultaneously reduction factor level lee point usage version thank research department usa institute technology lem lem lem lem structure adapt notion dimensionality new result level reduce packing structure low manifold local covariance consequence manifold curse direction lead etc almost useful assume turn euclidean freedom available dimensionality several automate capture actor marker although recovery frame dimensional body exploit extensively result example typically assume knowledge intrinsic dimensionality adapt assume structure offer guarantee reduction cell bound two intrinsic variant already numerous recognition resolution structure new family space partitioning bar structure one go cell side box shape cell radius cell ratio length shortest shape ratio etc radius number disjoint intersect prove ratio cell guarantee play approximate neighbor search result max type present bind aspect nature difficult aspect cell instead pack specifically aspect small cell max completely structure datum show dimension result robustness notion brief datum present effective aspect arrive packing cited present intrinsic dimensionality ambient max dimension datum hence take dimension ball cover split lie cell radius direction project assign child z child approximation prove property cell construct subtree root section consider go case start level expect go result great away level radius follow extension radius radius boost level least c radius radius power fact number level solve recurrence typical partitioning guarantee size property build cell dimension construct subtree root every example repeat cover cover ball suffice consider ball center split separate contain ball cell center cell bad apart say child max inside ball child propertie ball inside max split separate split cell split proceed project onto length choose ball get interval getting split choose randomly interval radius interval simplify dx e pair ball contain contain split least completeness pick lie randomly real line let choice random onto projection etc ball project hold apply lie distance show probability projection ball lie far projection ball choice ball separate split let pair describe dimension split work require observation p follows exponentially go center separate radius ball denote cell level contain ball follow k require attribute level cell go split frequent indeed tell contain way inductive split cell depth bad represent notice thus give sd
occur intra differentially locate cluster gene position present differentially arise experiment individual array genome plus publicly yield array probe classify probe take value status test quantile distribution status discovery classify present status around infer site forest minimum decomposable result produce display network complexity infer decomposable model drastically apply around classified figure small contain b three cluster encounter around central cluster examine b figure turn intermediate level water holding stem water hold capacity report global assess array contain probe publicly gene expression analysis way ap absolute pearson correlation logarithm expression probe value significance pearson coefficient accounting criterion classify characterize expression decomposable bic vertex cluster procedure result classify ap large cluster examine uncertainty turn low uncertainty apparent display ap ap calculated model decomposable likelihood yield arise sample genomic probe phenotype group present publicly available expression distribution differentially decomposable present vertex procedure representation cluster present differentially representation display figure total differentially display proportion differentially scenario ht evaluate high variance co procedure describe decomposable model set perfect denote nn evaluate simulate index average improve increase argument indicate essential account intrinsic structure level discuss association expression effect represent infer structure use correlation correlation avoid redundancy propagate spurious gene connect information expression level contain chapter moreover essential construction separation two separate conditionally correlate c separate connect element element essential group separate knowledge expression branch carry expression vice clearly gene special graphical model period adapt minimization aic maximization decomposable graphical act independently b j relevance step towards find genetic microarray profile phenotype trait r evolution establish g mixed aic forest cope probe level york exponential cope probe l visualization paris national university trait correlate pathway candidate water capacity core team foundation statistical g aspect elimination reduce acyclic green decomposable neighbor decomposable scheme analysis molecular diabetes de technique characterize differentially gene co model redundancy information avoid complex relationship allow make thousand typically occur throughput study take internal structure compact identify gene differentially gene less differentially gene locate region notion short relational pattern phenotype simultaneous expression several thousand recently produce gene work technique characterize gene redundancy idea value advantage informative illustrate level determine sample g microarray statistically difference produce naive search association level correlate association spurious might exist change association disease status level gene mechanism disease expression level group gene group association status spurious usually determine express therefore suggest take account level gene experiment I co differentially association association gene gene locate expression whether involve association involve differentially locate less region potential association position gene expression affect interpretation differential informative locate central might aspect perturbation expression discussion summary aspect interpretation differential assess position differentially express take account differentially large gene expression thousand gene possibly form network co expression graphical study connect graphical conditionally gene network similar proposal conditional step graphical expression rich decomposable co expression compact well differentially expression network differentially express informally involve gene expression datum gene graphical way bic criterion restrict decomposable making package aic bic yield optimum condition briefly report sophisticated characterize take internal classic exist enumeration enumeration perfect recursive procedure several sequence perfect respective assume independent realization level mean nuisance concern give graphical decomposable clique greatly simplify calculation instance write clique perfect enumeration clique respective eq cardinality statistical graphical inversion gene individual imply singular decomposable graphical minimum bic consistent estimate ji span forest successively bic take calculate decomposable algorithm stop find stop else find consume technique decomposable cg ic edge set describe representation express co differentially differentially co network connect sub say term exceed pre threshold produce call k adjacent exist adjacent call graph expression graph component clique graph classified algorithm find gene output proportion find ic j j clique dense label red contain label clique clique directly solely clique connect form form cluster drastically idea construct establish differentially express argue
variance estimate carlo correlation cause statistic variance statistic via simulation var variance million replicate statistic behave array use method permutation microarray statistic permutation permutation array permutation column array behavior include study nan commonly need thousand variable create testing procedure control dependency commonly false fdr fdr fdr relax assumption slightly broad negative correlation step arbitrary dependency control fdr dependence dependence step permutation adjust direct estimation account correlation one estimate local fdr fdr average tail study effect row correlation study matrix model four fdr control discovery proportion four base empirical compare package repository structure matrix two row block diagonal simulate three reflect respectively size diagonal block j hypothesis realization conservative estimate proportion number hypothesis reject average ten also report test test test test test decomposition repeat several dependency among fdr besides support correlation simulation c fdr permutation perform similarly conservative control fdr gene reality around dependency conservative hand perform seem confirm dependency among column problematic present methodology inference remainder covariance key model simultaneous row close relationship take covariance covariance share eigenvalue correlation population among make accounting covariance problematic regularize accord model covariance model allow singular row variate place covariance concentration context induce penalty follow decomposition remove ij motivate discuss consideration column datum usually position placing encourage one dimension row penalty partial strong correlation row diagonal secondly gene penalty leave remain establish inconsistent estimate establish multivariate covariance estimation importantly norm eigenvector result reveal covariance de use estimate form accord correlation row remain correlated noise symmetric square let root adding correlate n remain noise row examine example look process datum favorable minus fdr un conservative fdr close fdr estimation datum pre step advantage power gene microarray translate top improvement simulation study nan inconsistent dependency microarray datum thus repetition gene microarray center empirical covariance simulate restrict variate mle mle observe result r test test test repeat time value fdr without study improvement fdr estimation microarray simulation fdr simulation confirm give however still close ten repetition mean accounting appear significant correlation array diagonal simulation among drive pre false discovery variate normal decomposition surrogate datum coefficient estimate nature previously notation account datum covariance whereas additive model respective change ranking unlike capture row c test effect fdr reject method take repetition simulate batch follow zero give ij iid follow ik iid k batch column fdr method standard pre use available package simulation order test statistical power substantially pre specific test reject rejected change true reject problematic standard false positive however fdr close conservative variate processing processing demonstrate use statistical problematic method solve problem technique de approximately row reveal simulation disadvantage fitting covariance penalty may currently propose un gene signal gene detect researcher interested approximation filter component testing outline property consistency remain direct statistic array examine conclusion several correlation major simulation permutation inference conduct despite support many statistic estimate covariance framework fdr focus prove useful variety taylor helpful also partly inspire test test test test test test test repeat ten fdr estimate compare vector scale independent distribute random trivial n w corollary trivial proof since proposition ij k x ij characteristic trace function let consider random nz n independent proof dependence consider form often column interest example array dependent commonly covariance distribution result problem solve covariance simultaneously give statistic scale microarray reveal offer area discovery variate normal often ease pool inference assume test allow assess testing gene simultaneously procedure theoretically measure dependence among able incorrect meaning row significance correlation row account differentially microarray genomic dataset complicated structure pathway correlate gene act negative correlation array many suggest arrange form matrix dependent microarray scale inference testing protein protein array generation sequence testing significance functional significance take subject several begin introduce paper first microarray disease array two microarray filter gene compare nan compare nan could nature thousand another cause could array array study effect separately column generally consider column lead incorrect turn result much problem dependence restrict normal paper devote effect interestingly column dramatically lead fdr seem fdr solve correlation covariance regularize sphere row conjunction standard covariance microarray reveal important test lead conclude discussion study present restrict go microarray gene differential nan correlation section propose study correlation simple matrix variate multivariate distribution independent identically array flexibility either independence correlation structure restrict variate introduce propose variate normal n mean array marginal meaning gene array row decompose eq meaning restrict variate microarray array two notation first array
univariate finally notice also situation interpretation observe element category size appear conditional permutation element sample among reproduce reflect selection speak equally draw number common unit across scalar distribution obtain express practice equivalent population random multinomial variable regard family hyperparameter model express count poisson conditional specific association put characterize product similar assessment variable also sum px b direct acyclic display beta discuss simulate distribution implementation base update step pt pt pt interest draw sec thus quantitie one draw assume population category x computationally difficulty directly simulate easily conditional simulate draw however associate justify mixing cycle order approximately standard update pf f b n k j draw first distribution play multinomial parameter either get two typically former uniform multiply complete obviously simply substantial rest fix eq total value coincide one easily truncate draw beta specify quantity correspond column conditional couple q set configuration eq conditioning draw key method able popular record linkage literature basically consist homogeneous group record inferential strategy record linkage usual record linkage represent create set theory produce match plug first statistical calibrate match ratio false match wavelet currently record linkage procedure pose paragraph record linkage interesting perspective formal posterior expect appear linkage context mean match formal decision theoretic seem necessary characteristic action select optimal decision loss translate measure linkage give define consist zero notice eq loss calculations ab ab ab ab ab ab loss optimal ab b g show easy minimize adopt expected stress theorem consequence additive say decision perspective control type loss account match reasonable linkage actually error miss true practical fact denominator practically data file consist record census population record post enumeration survey example population represent category block represent gender education category total key focus example allow illustrate hyperparameter assume support use supplementary trace distribution measurement probability area record linkage estimate people census outline code hierarchical gender education age category variable distribution match new box plot implement illustrative namely hybrid population plot panel posterior wide interval pattern remarkable fact correction reduce clear conclusion aim estimate quantity rough end panel obtain sum least record survey fact introduce concentrated estimate accounting match large panel evaluate hierarchical study often linkage computer six scenario different population always scenario key category case frequency equal b ib ij j category contingency ib c c variable scenario pt pair hierarchical discard constrain model focus group pair mean close process misspecification element hybrid trend observe fact produce coverage expect dramatically level estimate wider provide partly comparison interesting size three scenario scenario table match comparison method particular criterion well performance follow however consequently criterion prefer constrain conservative perspective favor single measure record linkage criterion produce low linkage pose perspective framework within comparison record methodology statistical also play extra inference framework analysis approach answer strong assumption situation allow assumption inference perspective soon size intensive simulation approach problem popular linkage value error computation reduce computing build actually categorical finite boolean place stress provide survey naturally record linkage model example association important benefit incorporation matching process record sample confident extension capture use advanced incorporate handle multiple exchangeable record addition measurement file may also develop block separately evaluate allow strength extra layer extension future approach handle aspect record linkage address use list provide problem scope believe idea generating might context note linkage entity resolution recent relevant paper statistical model present marginal sum block matrix f f j j f b j bn f f j p f b f f b tn f j pt b b pt f n tn f f acknowledgment comment suggestion substantially version manuscript material file contain file cat file pdf show section di hierarchical statistical linkage size linkage statistical actually available uncertainty plug correct uncertainty size linkage motivate simulation datum source ease lead great popularity merge record linkage refer relevance record linkage highlight significant paper hierarchical record capture size population interest capture current population remarkable mark abundance propose unified uncertainty naturally estimate approach size respectively comprise census census comprise census enumeration datum report gender education perform also number census unit file agree perfectly observe might agree second record true moment summarize distribution produce dramatically different posterior pt n matching beyond relate response analysis perform link linkage entity two file different context happen unique analysis available furthermore may create set survey considerable lack different variable henceforth connect record miss record na common describe paper put kind problem far advance paper among introduce record linkage idea exploit paper record file comparison assumption fundamentally illustrate state paper propose bayesian observe variable always record concerned bioinformatics problem configuration match label situation bivariate sample record get consequence information represent unknown permutation arise material collection describe material come source record linkage attempt suggestion seminal paper structure record linkage monte method need loss methodology evaluate illustrative realistic conduct finally section discussion extension improvement record configuration categorical set data configuration sample whenever avoid subscript notation key variable element set arrange indicate sample population b probabilistic linkage perform assume conditionally py ab ab ab py ab u ab abuse independently comparison vector identically bernoulli assess gold available maximization analytical setup decide whether single posterior pair formalize approach opinion several pair threshold however problematic illustrate record
accelerate carlo carlo sample inverting use monte seem limited place control large demand either accurate importance absolute proposal approximate probability seem overcome frequently encounter proposal tailor nevertheless show correction importance sampling approximation overcome accelerate monte carlo confidence interval correction negligible p test confidence interval approximation guarantee conservative monte carlo still combination turn crucial many application variability control amount importance algorithm nan specify compute carlo analytically distribute support distribution function density approximation extremely importance size dimensionality reader detail importance simple correction importance observation correct approximation value p hypothesis distribution carlo correction reach fail properly ability sample valid generalize discussion become derivative discrete case mass simplifie ratio allow choice statistic permutation mathematically write test argument first argument invariant want sequence way statistic permutation center everything desirable context improve balance precise probability nonnegative version nz nz z z z proof special among nan validity correct practical validity well direct chain generate approximation hold different say distribution generalization marginal distribution random drawback correction randomly increase randomness already generalize theorem proposal require additional marginal place fact valid nan q confidence calculate appeal invert principle weight interval point create clear counterpart essentially practice concern correct behave choice concern target value approximation correct careful express interpolation respective non value shorthand suffer since alternative problem space typical avoid goal correct behave quite bad look closely affect correct interpolation valid validity see generalize abstract strongly conservative interpolation effect versus alternative keep setting often accelerate however play determine hypothesis take advantage ensure rejection favor sensible ensure event heavily weight turn interpolation always correction little power substantial ratio important value classical importance heavily case however functional design whenever proposal range confidence contain matter detail il perhaps dataset interested dependence multiple perspective give tx tv nan permutation test composite nan nan conditioning value become permutation use I single particular ensure probability compute exactly prohibitive monte carlo sampling accelerate sensible value real case label I standard cauchy plus shift tend poorly alternative sensible statistic associate size increase refer likely approximation case final refer correctly c c c incorrect guarantee valid detection drop acceptable furthermore determine large fail error correction useful unless validity correction work properly regression treat nuisance hypothesis remove nuisance sensible q detecting detect combine tail p define direct design sensible normalization validity combine usual way invert test confidence inverting need increase create confidence sampling suffer proposal might practice mixture conservative importance conjunction permit proper practical importance proposal detail hypothesis neuron control family report confirm one qualitative remain scientific conclusion result improper hypothesis practical benefit clear p correction false high variability correction behaved importance advantage interval construct inverting monte always diagnostic poor approximate standard issue p correction approximate sensible approximation correct p estimating permit interpretable avoid indicator appendix additional example demonstrate approximation close converge extremely software generator unable modify software include quite diagnostic purpose expect correction example proposal improve importance budget combine much diagnostic surprisingly maintain significance level perhaps target importance continue gain fit theorems sophisticated monte diagnostic completely ignore rely assume statement trivial assume exist th th begin nonnegative function permutation inside lemma complete two surely need permutation measurable second equality distribution fu wu complete nonnegative variable proof expectation wu wu theorem conditional inference target observe make increase choose binomial leave observed value permutation target assign higher tend label value section inference neuron depend event define set time temporal neuron probability case event case time prefer time lag test matlab execute pseudo generator logistic avoid record nu approximation cdf quantity monte use importance valid valid hypothesis line total plot panel dash cdf experiment involve around p cdf dot plot close since cdf diagonal drop indicate cdf exceed indicate hypothesis correct always valid seem although small computing important time none job even conservative strongly conservative desirable mse describe situation importance extremely slowly sum nan distribution class configuration tend row sum index distinguish nan alternative distribution sum suggest proposal successively sample basic would symmetry value standard although normally particular square mean weight extremely much reasonable direct agree heuristic solid line change seem end take report dash black excellent value plot value importance approximation fix statistic choice thin figure thin line gray thin test case special symmetry sampling choice row remainder equally line despite identical estimate reporting report times sample importance correction computationally demand simulation assertion paragraph illustrate less demanding computation approximate much extremely correct valid correctly preserve interpretation approximate repetition dot line cdf visually indistinguishable second
load negative therefore thousand simple matrix costly furthermore contrast lag involve non factor moderately large identifying contribution reveal loading result one rate loading error exhibit curse dimensionality offset via common factor improve asymptotic estimation strong characterize explicitly index depend relate manner forecasting organize method strength exist section extensive simulation imply technical proof time unobserved factor stationary moment loading matrix white otherwise combination example lag relax correlate past white noise appeal economic know body unchanged loading uniquely uniquely accordingly advantage particular convenient explicitly section measure strength factor instance condition k root smaller introduce small integer practically asymptotic remain constant go full include imply strongly weak factor link explicitly factor slow weak factor facilitate loading column orthonormal follow unless condition obviously negative ready specify loading estimation note unchanged tr definite matrix rhs q lr use load orthonormal eigenvector eigenvalue specify li estimate factor convergence estimator precision original convergence k pn l pn pn l x pn auto loading explicitly k op explicitly strength affect fast strong curse dimensionality offset factor implicit essential introduce presentation sense deal convergence directly two introduce variance vector away assume uncorrelated fix maximum different pool see k grouping look variance model constrain covariance specify convergence hold indicate difference matrix estimator even factor estimator frobenius factor advance numerically however perform sample counterpart provide inverse unbounded factor weak spectral irrespective fulfil error surprising hand study frobenius transform frobenius norm transform frobenius little strength rate estimator investigate presence factor may treat view group jj unless sequel continue estimation much eigenvalue typically practice distinguish presence circumstance strong factor weak propose justification method factor strength ignore obtain ii remove derive assumption ik ik l pp pn pp ik op loading may benefit fast op practical implication residual initial especially frequently method p theorem indicate normalization reflect norm estimate factor derive eq manner similar bad strength differ substantially hold method property via simple illustrate loading element hence list similar htbp cccc indicate theorem sample table column consider level factor distribution adjust strength normalize column th vector component either properly factor replicate calculate standard deviation use estimation right factor display replication show strength good well optimal instead factor fact quite factor strong factor simulation much similar similar pattern show show covariance estimator yield illustrate develop dynamic microsoft imply volatility surface obtain question imply w tu display period imply cross delta htp fact imply stationary amongst unit root reject course perform still favor instead p since perform window day satisfy methodology loading forecast depth well forecasting take advantage lag report htp window ten figure display eigenvalue window side large microsoft right hand second procedure graph apparent eigenvalue number company choose dotted procedure take display cumulative rmse window treat ahead except marginally consistently outperform benchmark microsoft triangular express upper rp lemma factor let factor relation r pp covariance sample hence similarly main q span sep orthonormal basis op definition function since large hence conclude exist hence order specify first ph n pp consider ji bound infinity pi ph op q finally use look rate definition note spectral theorem rate since r pp p need formula j noting note q op eq proof procedure want first find large small note size r op op l proof theorem l hold op also rate proof thus omit form unit eigenvector eigenvalue third note lemma conclude similarly order proof previous idea find use get
directly require sample approximate estimating become slightly context rare random fit chain approximation base asymptotic differential equation equation scheme method system consider change family change minimize time occur sample simulation suggest expectation expression system expression importantly fact basis formulate entropy execute execute change measure retain transition suppose kullback change substitute cross occur program apply lagrange multipli transition notice e direct fact denote start ease times return follow transition follow b iii follow return iv follow reach q hence transition term expression repeat update refer ce ce ce ce opt constant approximation entropy opt ce condition importance asymptotically opt ce opt opt ce upper use third equality illustrate service associate continuous process rare return probability eq transition probability reasoning proposition calculate numerically entropy iterate uniform transition size increase update event relative ce estimator estimate ce ce ce ce opt ce strong efficiency bound opt ce ratio construct simulation key consider rare event problem chain engine importance coincide variance sufficient estimator strong acknowledgement would visit algorithm assumption correspondence approximation simulate entropy asymptotically deal finite fail form set internal internal probability markov good start formally notation denote markov jump state
start dual problem condition entry flip sign modify scalar set vertex arc arc negative arc capacity constraint arc flow arc capacity incoming flow vertex flow optimization uniquely characterize one group ii arcs capacity cost iii arc flow arc iv arc cost canonical variable problem sum cost lead equal arc match show solve include canonical simplified edge edge capacity simplification illustrate quantity present formal reduce speed present draw fill minimum thick blue fill place thick draw black mm bend angle auto xshift g var xshift distance node right si edge xshift right bend g mm edge group xshift mm var xshift edge node edge pre leave var var u pre node pre u bend group h node cm yshift mm var node h node pre u right node pre right si node xshift u pre node bend edge node yshift yshift h var pre cm right si edge node xshift pre grey square group group capacity zero infinite arcs graph zero capacity group hierarchy flow problem flow study simple single solve ball machine research tree overlap group difficult solve cost lead take specificity dedicate share nonetheless perform dedicated parameter flow solution projection uv j v tv u informally return flow proceeding solve replace single sum low bind flow try flow match reach bind reach cut define potentially flow arc arcs arc arc point arc remove yield decomposed solve recursively call formal correctness guarantee bad however problem empirical despite fact guarantee adapt see efficiency arc cost problem look call sequentially use heuristic implementation high graph concept initialize enable warm graph step balance modify step linear dual norm induce proximal duality proper f duality argument equal f duality compute find flow formulation associate arcs problem arc prove graph explain dual gr j uv e v e experiment weight set appendix namely sg regression sg implement run core ghz overcomplete dct organize grid family contiguous generate percent select sg interior method cast qp cp formulation size note qp sg obtain solution whereas sg hour reach gap background try segment foreground object combination plus error w p recognition fact neighbor likely foreground foreground put square image effect regularization maximize pixel channel figure improve remove lack structural neither consistency ccccc cm cm cm method foreground pattern mask image another foreground leave percentage ground bottom use hierarchical learn dictionary signal combination dictionary express vector structure refer admit subtree element idea tree prune irrelevant word follow regularization operate w norm correspond overall penalty combination overlap alternate fix variable denoise patch study whether hierarchy dictionary denoise compare I learn predefined tree patch impose extremely problem sufficiently many rigorously mostly I select validation set hierarchical problem heuristic perform improvement present structured sum norm overlap literature network flow cast min flow propose accelerate wide problem formally equivalence summarize lemma equivalence weight share vertex equivalent follow arc arcs arc path cost conversely vertex vertex arc arc cost graph depend arc cost vector arc direct graph introduce easy arc build flow equivalent arc amount flow carry flow also easy build satisfied flow conversely flow decomposition exist path flow sum unique arc flow arc amount flow arc build show compute proximal operator dual algorithm find proximal introduce essentially condition termination optimality dual variable feasible g g min flow solve min equivalent flow capacity constraint correctness cut find part construction set node conversely arc arc follow arc value would infinite arc flow go property arc arc go thick draw fill blue fill mm fill draw black fill minimum bend source group xshift pre g xshift edge right part var g xshift dot right distance leave distance cm si pre xshift thick leave mm thick gr gr gr arcs bold arc dot scalar negativity constraint converge split process require onto ball max procedure sufficient procedure compute suppose classical flow min cut theorem equality term definition flow min theorem contradiction existence proof hold graph solve correctness induction node v prove initialize simple simple correct compute ball case optimality yield compute scalar situation q rewrite previous positive max flow solved prove next remove see vector respective canonical structure gr gr graph combine optimality arc cut possible condition show relatively arcs arc bite j v gr gr w u j gr gr gr gr u arcs go flow nan flow imply node flow provide gr condition imply u contradiction gr j v detail gr prove see give flow step graph another flow dual definition introduce additional inequality dual g take derivative variable simplify lagrangian correctness compute norm polynomial similar proof require finite max converge correct canonical proceed induction next canonical flow
pac bayesian analysis trade graph practical benefit regularization suggest model quality trade large reasonably practical formulate weight comparison pac co optimize state provide theoretical foundation suggest provide good life problem derive reasonably cluster tool include bioinformatics approach example sm term objective functional minimize whether cut theoretic approach formulate weighted analyze weight rational behind weight specific validation bind address finite order prevent resolve trade co suggest review adapt pac bayesian bind analyze widely row matrix good illustrative co movie rating predict miss triplet movie rate discriminative movie pair expect whether co collaborative space rating star absolute existence px x sample consider predictor along conditional assigning assign label cell c information eq x divergence discriminative clustering prove snp parameterize trade suggest alternate fix minimize substitute back alternatively collaborative adapt node edge generate unknown know node edge weight generate goal build formulation specific enable work immediately belong share edge prove minor cluster edge weight quantization divergence easily numerically co tune substitute validation suggest qx qx measure consider demonstrate superior mutual theoretic inspire distortion ct notational problem without low sampling matlab derivative maximum minimize delta px let variable index prediction reconstruction correspond delta norm equation way appear power repeat projection empirically even remarkably try initialization obtain comparable within follow value fold increment iterate random increase reach note analysis edge interval end round edge projection quantization increase quantization w w w continuous weight kl divergence quantization name edge weight similarity similarity second pairwise protein web edge protein deviation indicate curve coincide subset validation remain train observe parallel minimize substitute result bind perfectly meaningful almost dataset cluster cluster cluster dataset due appear twice node another analysis copy value slightly considerably beneficial edge global cluster first benefit cluster prove measure independently train preserve effective clustering illustrate indicate note
directly constructive strategy bound appear develop round appeal keep expression develop computationally basis development reader state von fan theorem banach space nonempty weakly let nonempty weakly minimax complicated include expectation bilinear equip variation see banach weak continuity semi result compact set measure compact accord example borel banach tight precisely norm example scenario make need minimax see application minimax brevity inf f proof every couple understand move respect infimum equal fashion scale depth cover center calculate invoke substitute arrive upper conclude long follow cube separate note norm apply obtain step also upper finish base bound vector give give q conclude use sequence observe give strategy attain give expert weight w randomize upper bounding constant theorem dominate leave expected indicator loss indicator crucially leaf reach infimum split minimizing let remove priori countable expert randomized manner proposition cl tw lipschitz lipschitz function trivial bound lipschitz universal next low whenever imply ambient next small class number tree analogously cover put piece together rademacher bound game consider exponentially expert countable proof closely include completeness need proposition countable expert thought produce history countable expert initial weight expert play receive update ix forecaster enjoy sequential prediction minimax associate several complexity precise learn particular online supervised learning framework concern probabilistic regard generate well sequential prediction procedure apparent sequence underlie viewpoint analyze value set useful somewhat abstract set well repeat game model choose adversary pick suffer loss cumulative loss fix pair say learnable algorithm time adversary employ origin compound decision algorithm perturb sum loss idea influential follow seminal work lead development foundation code compression information literature science connection economic partial reader prediction researcher include science economic excellent synthesis presentation different refer technique name perturb online gradient may unless tool learn precisely begin underlie abstract develop theory develop rademacher complexity relate note control online turn lead constructive proceed repeat learner adversary need technical separable weakly compact every henceforth notation stand distribution adaptive adversary base history game exchange dual easier analyze choice adversary adapt player measure mixed strategy reduce online learnable require iterate infinite formulate contrast regret possible obtain develop also extend allowed compute especially term prediction information f fx setting study scenario limit paper aim shall keep sequential complexity show characterize uniform law number handle briefly notion mention key sequential unlike I possess dependence dependence classical notion basic look tree capture rooted node identify label tree indicate length rademacher depth value sup stem martingale deviation control sequential complexity matching hold supremum martingale statement hope notion sequential show generalization classical notion necessary picture completely theory key definition class notion check cover analogue eq extend beyond cover eq tight covering analogue q sup p previously description combinatorial depth value depth function exist depth dd sequential generalization binary recover notion definition restrict hence dependence crucially growth covering definition combinatorial class analogue satisfies establish online class nevertheless useful property rademacher prove essentially therefore skip proof next rademacher class complexity individual familiar contraction property immediate corollary g binary value note classical contraction hold logarithmic factor logarithmic remove worth point ahead lipschitz otherwise relate supremum empirical I view supremum rademacher upper sublinear irrespective adversary establish purely base subsection early paper scenario pick adversary pair loss easy classic observe improper equivalence easy verify pre specify alternatively simply observe particular move minimax need apply hold original simply rademacher theorem remove include pass minimax contraction sequential rademacher lipschitz theorem scale supervise complexity notion logarithmic learnable follow learnable complexity function learnable sequential rademacher integrated additionally theorem martingale number property paper binary write investigate control absolute classification universal derive notably bound non cover argument ask able distribution example two framework g learnable online indeed interestingly closely slope supervise online online learnable statement consideration example demonstrate explore decade unify view mirror associate abstract say online online also recognize lose suffer possibly unnecessary nevertheless recent scenario move banach space adversary lipschitz online instead without algorithm trivially extend randomized reader try play norm example rademacher crucial duality smooth let subset banach norm sup z p conclude recover descent usefulness tool develop rademacher bound fairly play big theory classifier space study decision make seminal minimax static expert term rademacher rademacher classic multi layer learnable neural bound transfer function sigmoid hold q eq constructive guarantee neural online efficient statistical margin bound theory margin see recently margin show easily base sequential idea randomize suppose exist sequential logarithmic factor crucially consider tree depth class follow root value decision value decision read leaf tree membership conjunction decision path leaf choose leaf run leave decision denote tree learner reach leaf correctly classify tree minimize
approximately frequency switch p switch probability random follow direct rule dense average contain number one kind bayesian describe happen alarm call whether alarm similarly straightforward estimate full bayesian represent derive chemical reaction addition abstract operational semantic deterministic refined semantics semantics intuitive execution computable sense strategy long case probabilistic inside probabilistic execution em pt pt pt em either singleton fail specific transition execution strategy em em p class set execution firstly execution intuitively say two build equivalent execution get result state equivalent observation strategy pt em unique follow query execution strategy start chance chance execution different c program observation execution refine semantic program strategy correspond program I em em em em fp semantics strategy w every equivalence class final derivation specification execution program define unique distribution depend ambiguity refined semantic program bx cx execute first result accord two outcome ambiguity program every derivation program coincide example consist vice versa consider regular program derivation query end b program execution rule execution consider get therefore execution implement system currently prototype mu naive pure current http people advance logic language build switch exp possible outcome experiment exp value assign pn program part rule fact predicate rule clause allow body clause serve interpretation assign hmm simple hmm next state end state hmm output symbol either b clause indicate fact say value clause string hmm probabilistic event hmm generate string generate bayesian alarm follow yes alarm alarm learn illustrate rule translate graph translate nb true probabilistic simplification suffice remove store soon put rule expect fine computation way implement explanation search explanation create try cause involve fire chance take translate rule code adopt head translate branch actually version variant every probability program implement head tail conceptual point view semantic operational derivation however semantic confusion weight coin example head chance otherwise interpret rule answer head chance weight normalization runtime consider word heavily program localize mean head program tail propagation applicable weight abstract semantic allow execution efficient semantic semantic applicable rule fundamentally refined semantic applicable rule active occurrence semantic differ derivation derivation derivation semantic chance localize depend execution support execution I numerous logic family cp logic cp etc encode compact modular logic language inspire bayesian network give allow detailed description offer language namely thing instance evolution fact might represent logic time elegant apply range field largely extremely suitable example computational valuable many application domain extension exist approach uncertain example refer scheduling reasoning semantic web verification another automatic analysis music past analyse music set strict combine application music specific computation viterbi computation music exploratory formalism combination operational implementation relate language opinion advantage include wise completeness multi rule plain ambiguity relation ambiguity although implementation sufficiently explanation limitation current support would interesting transfer automatic program e obtain support far ground argument add support non query certainly winner probabilistic termination explore program always terminate go since remove program keep never end probability still compute make sense terminate handle infinite loop search stack ambiguity relation ambiguity implementation explanation search consider semantic chance rule applicability rule abstract operational semantic refine operational semantic semantic practical many support probability example feature essentially free challenge yet exploratory corollary definition institute technology mi ac cs institute technology mi predicate build support maximization handle programming rewrite paper formalism level rapid complex language operational semantic notion program distribution probabilistic logic language identify potential handling language originally language implement year purpose language language call statistical probabilistic sampling formalism chance statistical combine like expressive probabilistic implement translate clause early mostly subset bb operational semantic rule rewrite mostly cx like chance conjunction keep conjunction head satisfied body omit call empty call recursively conjunction constraint goal intuitively mean chance store head substitution furthermore head ignore actually rule store execute store one regular drop apply expression ground runtime occur evaluate indicate name execution theory build predicate program transition binary annotated rule start root walk acyclic reach program state probability probability path em em ik derivation semantic semantic partial
approximation select great remain great marker level remain therefore true relatively frequency general positive add toward end sec criterion select cover reconstruct principle selection prefer particularly preference dl entirely express trace node order list compress reporting marker node marker identify parent child identify nd rd require parent since possibility specify subsequent report sum description marker trace describe practice marker define network marker parent every first specify parent marker specify report potential parent specify report third characteristic code explicit define high expensive parent report create describe marker therefore short marker trace require minor covering turn directly network cover edge along explanatory cover add knowledge calculate include along baseline bind positive false positive first consistent covering result search maximally see exceed naive naive never positive away everything marker trace sec would figure verification match true obtain circle fig description point majority true positive limit inclusion naive coincide zero circle indicate cover line reconstruction various clearly positive remain increase contrast presence fig add proportion false edge fix exact network marker demonstrate reconstruction causal underlie branching spread entire consideration way able np efficient greedy two version consistency datum likely control setting propagation version common noisy marker observe reliable tend early good excellent plan direct cost stop another order life datum media propagation acknowledgement thank analysis university benefit centre sciences award partially grant ep fp ac reconstruction node visit branching algorithm crucially consistency local infer neighbourhood optimisation solve reduce cover np hard approximated extend noisy demonstrate sir interest recent reconstruct complex network produce diverse field measurement spike datum field generally nature inherently chemical share challenge causal dynamical stream address challenge reconstruct branching process occur direct discrete infection lie field infection begin stochastically infection initial report site pick reconstruction generally fundamental propagate communication notably medium optimisation set concept description reconstruct contain lose fully nature optimisation present sec introduce address concept outline sec orient node reconstruct infer denote branching occur transfer marker marker network propagate stochastically adjacent along terminology refer node become marker infection point infect infection marker refer marker marker trace marker trace index marker order carry marker infect notion marker marker trace w w marker trace define reporting marker trace clarity future path marker trace approximate make generating assumption generation marker node previously infect notion globally g besides ensure ensures guaranteed reconstruct reconstruct optimisation make sense require consistency involve impractical equivalence neighbourhood consistency reporting marker incoming marker early lc r w e lc demonstrating define construct early demonstrate path node hence trivially incoming trivially path every node incoming edge node claim induction lc formulate optimisation concept consistency I crucially consistency immediate neighbourhood turn optimisation subproblem total subproblem establish minimal incoming explain marker particular node unless specify describe discover node use local consideration consider incoming edge report marker time explain marker incoming explain marker report relate optimisation universe subset minimal cover define universe universe marker income incoming potential explain marker therefore every family define incoming early marker require incoming incoming reconstruct b set follow cover subject v b e repeat allow reconstruct greedy covering cover cover great ie reconstruct cover subset marker trace strategy ensures reconstruct refer positive edge positive fp positive exist incorrectly reconstruction simply find true reconstruct tp give achieve complete coverage marker marker must include reconstruction count number edge determine edge assess performance useful fraction positive successfully recover lower exclude false quantify false positive fp false order make set cover cardinality edge cover finally ratio obtained specify positive cover time useful size subset cover logarithm alternative member cover order cover relate ground cover report way
pattern group vary much sequentially maximally correlate small reference component match wise correlation acknowledge general satisfactory practice correlation match pair ica multi subject functional experiment keep record precede subject inform image whole body study volume slice resolution acquire acquisition experimental paradigm ten subject horizontal visual sentence processing occur stimulus ten occurrence trial give inform functional acquire slice resolution comprise four discard mr paradigm acquisition report acquire group subject acquire successively dataset department cognitive http www ac uk interpolation motion volume template isotropic voxel mask voxel procedure well movement flow cca em map stability ica em ica subspace match cc cc subject cca cca cca cca stability ica map average level level select study ica dataset sequence smoothed extract identify brain cca identify component brain movement bold feature functional dataset functional network fig equation without whiten pattern cca identify material table thresholded subspace fact conversely suitable study procedure stable subspace change tr one thresholded thresholded maps hand back reconstruct quite unstable population split one thresholded map perform component extract number volume result thresholde cc c thresholded rs matching maximal another assessment map give percentile dataset thresholded thresholded correspondence scheme compare supplementary material compare finally subject dataset table improve group cca marker fmri number identification previously activation pattern correspond cognitive study metric subspace run mode linearity give measure dataset metric cca use procedure consist preprocesse sub selection procedure cca implement principal whiten level cca whitening analysis group level component well score canonical perform table size material publish description implement package group subject perform successive average filtering group svd select select level apply impose criterion dataset simple concatenation material subspace canonical correlation less critical retain opposite cca group subject thresholded identify salient thresholding heuristic thresholding heuristic heuristic validation score different seem whereas supplementary material amplitude thresholding depend strongly histogram pattern component interpret functional group yield identifiable extract identify detect effect group two significantly group network understand activate whiten cca sub network encounter ica related visual structure statistical power resolve structure split separate region posterior dataset region show consistently separately default mode eeg measurement form differentiable identify associated display considerable finally state dataset language network map rich network comprise may expect occurrence cognitive vary appear component separate right task paradigm correspond low figure across visual area stand dataset different part visual consider observation also apply appear network level area process sometimes comparison activation topic consistent long produce interpretable much contexts activation correspondence spatial alignment perform preprocesse step subject difference variability smoothed metric correspondence induce score limitation method difference non correspond map validation matching technique purpose marker non nuisance ica ica criteria subject threshold identify one limitation subject lead subspace outline select component consider cca canonical informed ica non less tractable variability extract infer individual component purpose dual extract representative group model rely solely algebra ica loop optimize memory cost dominate group inference cca scale intel important fmri extraction ica step step costly principle component retain subject pattern present multi apply fmri non calibrate automate way meaningful fmri cross associate metric ica unstable extraction mix thresholded one exploratory use pattern marker whiten subject principal remain basis volume acquire wish estimate canonical select resample generate drawing noise correlation thus draw realization give access observe nan cca ica map matching map ica match fig de france team le france france france component increasingly image fmri set region functional marker disease open road paradigm subject group model ica inter comparison propose multi fmri introduce ica probabilistic reduction correlation ica method ica level control state brain gain imaging fmri derive activation fmri correlation distant region distribute activity context fundamental study paradigm bold dependent correlation study identify distant establish connectivity signal fmri correlation signal shape structure bold brain cognitive activity across assume brain function brain connectivity specific useful mechanism disease serve aid clinical diagnosis brain adequate disease study correlation brain activity comprise various voxel yield million correlation seed study potentially network brain correlate seed limited signal spatial identify meaningful pattern prior ica usually easy interpret cognitive often context seed base cognitive salient consider drive constrain datum remain unclear finding dataset sample context statistically seed bold pattern experiment extract exploratory ica comparison however overlap ica coherent report rejection goodness establish ica ica exploratory subject contrary level specialized adopt extraction pattern form challenge correspondence map assess merge statistical along apply ica group extension pattern share experiment novel group fmri volume procedure extract map model component subject canonical analysis cca resample automatic cross compare pattern sub population compare state art method concatenation rely subject thus formulate spatially group cross method ica data base signal base independence blind separation source eq pattern ica notation shape possibly observation depend extract ica guarantee rarely check component pattern solely movement far suit blind fmri right brain functional system pattern acquire fmri volumes ica often acquire ica principal component determine extract context fmri analysis pca basis subspace voxel fluctuation brain group analysis effect ica strategy far group fmri pca group procedure additional individual component inter consider relate level component generate level shape loading acquisition frame pattern load write eq q word matrix factor specific term specific level different ica residual identify mixed effect formulation glm source drive rely procedure extract generative model note pattern successive fmri subject correct slice acquisition motion extract mask brain center index pattern successively hierarchical separate subject pattern principal component pca principal pattern spectrum constitute loading component observation extract describe signal setting retain article select dataset theoretic initially pca identify source fmri many component study conduct influence ica method resample method shape necessary presence assess stability principal principle reader share dataset group interested sub subject purpose generalize compare multivariate successive univariate canonical correlation canonical cca subject variance ic account retain svd row canonical correlation retain canonical form pattern level keep canonical correlation level instead subject interest consist material level amount stability svd step span activation ica pattern lie result keep tail intensity nan thus unit estimate
study undirecte pair suppose denote typical problem interested multivariate particular undirecte value induce lie partition element graph value estimate two use splitting penalize hold minimization theoretical strong assumption optimization attractive cart result value effective covariate sample may estimate glasso one covariance correspond definite develop solve study prove mention condition glasso pt parametric glasso yx glasso smoothing g bandwidth apply glasso appeal nonparametric smoothing require global computationally partition finitely many region glasso take find recursively know leaf node cart optimize cart cart devote detail nx n px value sparse sparse graph covariance cart glasso graph dyadic split dyadic partition dyadic partitioning construct orthogonal dyadic split associate give denote tn kk side denote indicator piecewise mean define definition induced let matrix specifically penalize may always dyadic responsible suitable tuning element way discuss formulate practically split hold log hold cart evaluate require dyadic partition tree dynamic programming computation propose yet greedy hold empirical greedy generic easily penalize minimization form dyadic precision precisely correspondence run glasso large yield small select hold model reduce glasso enforce yield graph greedy start compute decrease hold large hold precisely split hold risk form split increase indicate partition split cart apply element record dyadic implementation pt hold integer estimate determine good splitting l r partitioning classical decade assumption theoretical go cart work oracle note might risk inequality assumption arbitrary induce bl l risk go cart excess r j rt input leaf reasonable inverse matrix unit two enyi maximum four small middle figure c inverse cccc ptc associate distribution size hold base dyadic present node leaf node carlo run partition plane split irrelevant dimension range moreover graph obtain highly dense immediate node simulation tree list term precision score true easy region contrast inside correspond hold significant pt furth ground appendix c deviation value analyze contain span location equally spaced grid month include temperature temperature cloud cover day normal uv detail location glasso connect contradict domain knowledge factor include panel reason edge pool correlation correlation dyadic greedy partition dyadic tree california adjacent suggest moreover factor direct accordance report south central location validation expert graph cart undirecte high dimensional dyadic using finite oracle excess risk consistency relevant partitioning computationally attractive classical advance technique go cart indicate cart several direction denote analyze least assumption bernstein hoeffding probability get analysis give subtracting least q enough subtract side minimal partition proceed different easy sequel need follow achieve plug inequality far conditional glasso lie one space os enyi vertice maximum degree output sure node graph guarantee hold greedy cart dyadic tree structure corresponding display cc compare graph glasso entire term precision score score glasso well glasso graph entire datum false positive cart
straightforward gaussian e adopting convert em maximum estimate finding adopt subspace sensitive effort robustness characterize nevertheless problem bottleneck factorization reveal robustness modify formulation extra nevertheless modification usually heuristic style algorithm get performance corrupt lrr regard generalization lrr solve generalize present subspace describe polynomial polynomial success segmentation certain restriction subspace method due polynomial cause algebraic segmentation resolve robustness difficulty polynomial subspace datum subspace firstly final spectral exist ssc spectral curvature fit lrr possess type method clean sparse sr could sdp within sr sdp ensure segmentation recently zhang multiple exactly even contaminate outlier propose lrr space provably determine lrr matrix capital matrix horizontal resp concatenation resp block vector norm norm use nuclear sum trace support symbol support complement I column obtain resp denote denote belong subspace svd svd appendix create subspace ambient small sum dimension subspace strictly nevertheless simple affinity spectral svd data clean membership independent form entry term shape widely segmentation simply subspace segmentation presence inaccurate lrr contaminate block indeed nonempty intersection pairwise independent still interest segmentation roughly want address store union observation sim analyze success sufficient goal difficulty assumption clean corrupt clean corrupt contaminate characterized fig unlike subspace highlight recover lrr recover cluster deferred order indicate e norm adopt characterize original formulation formulation structure draw actually treat sample much well inaccurate well suggest linearly minimizer obtain z fall lrr use basis appropriate dictionary rank reveal lrr ease exploration case clean rank minimization problem easy practice problem result segmentation minimizer general lrr strongly fortunately minimizer uniquely form summarize feasible problem feasible minimizer also problem low reveal segmentation namely sample sample subspace without subspace sampling basis span minimizer diagonal coefficient x low sr worth property sample group membership generality indice true membership lrr sample replace relaxation norm characterize sample choose show appropriate choice obtain tb inexact alm fix fix update multiplier e z update augment lagrange multiplier alm convert alm lagrange tx unconstraine minimize update lagrange multiplier alm call need solve please base step close thresholding via optimal I smooth alm inexact alm alm present still difficult convergence inexact alm three easy convergence fortunately actually ensure theoretical necessary converge one iteration monotonically decrease k resp ideal lagrange simultaneously easy convert condition satisfied subsection monotonically lagrange function guarantee validity moreover inexact alm reality alternate guarantee adopt nevertheless please boundedness problem size major algorithm svd consume fortunately lrr easily follow conclude subspace span advance column transform replace b recover number row solve lrr quite provide rank dictionary assume converge versa efficiency optimality always iteration utilize lrr address present identify solve nuclear prove lin minimizer clean reveal connection counterpart pca refer pca nevertheless outlier contrast prove lrr recover contaminate outlier subsection imagine datum column support characterize seem contain two choice problem subspace draw lrr support index away one part draw subspace member support denote total fraction lrr succeed minimizer produce space importance cluster notice lrr property refer verify conclusion outli fraction threshold ensure performance magnitude consider affinity matrix corrupt subspace affinity corrupt corrupt treat generate way randomly corrupt experiment carefully determine support identify away member corrupt happen call specific still valid lrr recover empirically conclusion support index corrupt handle deal case similar outlier sample heavily treat lrr illustrate treat non something else experiment create pairwise subspace choose corrupt large error finally total include specific recover corrupt support sparse still relaxed handle support contaminate unlikely exactly recover near inequality prove demonstrate lrr quite notice somewhat loose fig exist alm invoke thus explore affinity perform segment svd affinity assign multiply clean ensure affinity cluster segmentation lrr laplacian nj although generally subspace e resolve due produce strictly affinity firstly normalize singular block reality suggest subspace value laplacian near integer soft thresholding summarize whole estimating lrr detect possibly fraction clean learn approximately support datum affinity possible outlier discard affinity threshold type principle strategy characterize outlier affinity degree advantage easily extend prior lrr art segmentation segmentation experiment paper shall lrr subspace outli dimension std extensive extract total sample manually remove sequence notable level summarize large take resp test lrr effectiveness create database database image extreme condition namely view direction small light degree low rank face non dataset close lrr baseline segmentation subspace segmentation svd utilize difference estimation sim detect outlier improvement pca outli work introduce characterize adopt detect refer subspace case solve parameter lrr appearance sr implement sr compute affinity minimize sr enforce avoid trivial minimizer lrr subspace consensus agglomerative compression ssc curvature fit segmentation receiver characteristic roc evaluating outli detail evaluation metric please appendix segmentation always good part error datum slight evaluation result sequence range segmentation error almost unchanged phenomenon mainly two reason easy choose arbitrarily lrr imply minimizer always satisfie lrr partially stable analysis importance largely segmentation actually sequence overall impractical parameter especially lrr sr lrr sr lrr subspace segmentation give list sr lrr methodology recover segmentation pca design recover design reduction notice use segmentation error illustrate lrr fig sensitive example achieve increase lrr pca theoretically computational regard lrr lrr cost iteration predict rate absolute database also provide good estimate subspace underlie datum lrr illustrate resolve lrr ssc sc c result discard comparable lrr improve use datum dense reality learn space choose dictionary consider unobserve hide z lx extraction achieve subspace long difficult g estimate trivial contain outlier goal face segment rest segmentation outli acc auc investigate segmentation evaluation outlier obtain sr auc lrr see lrr well pca strong sr behind check affinity produce sr even absence unnecessary notably reconstruction sr handle datum contaminate outlier image lrr lrr fig leave original middle visualize lrr size lrr worth note error salient decompose matrix low rank rank lrr extract discriminative salient region low lrr structure correct lrr generalization recently establish shape interaction define different sim row corrupt experimental effectiveness lrr determine lrr recover illustrate choice ensure whether technique matrix issue lrr select parameter segmentation lrr detection use block diagonal transform block whenever two k iy denote k collection subspace subspace ambient dimension assumption pairwise disjoint j decomposition orthogonal singular value keep rv uniquely column resp span column orthonormal basis resp orthogonal since resp determine sometimes refer resp affinity affinity th proof follow lemma dimension column nuclear matrix compatible dimension proof compatible orthogonal feasible constraint orthogonal orthogonal matrix unitary nuclear minimizer second minimizer another solution orthogonal u accord equality allow horizontal concatenation denote partition definition q calculate u problem
whether aggregation mechanism independent perturb mechanism prove theorem order issue majority inconsistent result mild whenever opinion opinion aggregation mechanism consistency result absolute aggregation mechanism preference exist aggregation generalize result characterize independent consistent aggregation mechanism preference mechanism think phrase question aggregation global property section present testing binary classic prove specific aggregation family conjunction exactly constraint technique proof conjunction measure aggregate aggregate consistency independence relax constraint describe describe aggregation mechanism present functional conjunction state motivation deal aggregation preference describe aggregation property theorem similarly individual opinion denote issue opinion profile member vote vote issue vote individual consistent aggregation opinion profile profile simplicity opinion opinion aggregation independence mechanism profile aggregate independence notice generalization aggregation social iff wise comparison aggregation aggregate aggregated aggregation mechanism mechanism measure usually satisfy iff aggregation index inconsistent q aggregation mechanism j section dependency context define index natural multiplication satisfie iff exist aggregation satisfy mechanism opinion profile distribution pick consistent proving prove format bind preference successive relax prove hope aggregation take distribution profile opinion independence sometimes life scenario attack remove reason intra issue dependencie essence accord criterion complex issue without regard aggregation show contradict accept think quantify claim case criterion case change collective secondly aggregation opinion profile vote aggregation mechanism independence strategy player justify independence return easy represent justify vote public justification mechanism decision different place vote voting aggregate vote vote issue definition return deal aggregate notion framework ease presentation logical operator bit return member function pareto sometimes refer follow pareto return definition influence vote eq game define getting normalize measure distance df notation binary formulate two class truth divide type conclusion opinion attain might choose analyze opinion seem family later truth functional conjunction conjunction issue decide issue contract valid making contract valid issue decide consistency answer odd think represent functional study equivalence describe preference aggregation framework individual strict order interested function order framework issue aggregation consistent regard set aggregation mechanism tell reasonable aggregate preference extend aggregation result constraint root science long suggest aggregate mechanism try stay reasonably independence general tailor suggestion mechanism truth aggregation procedure aggregation well vote vote systematic contribute pointing solution leave independence aggregation candidate state aggregation exist aggregation mechanism candidate independence aggregation consistency field deal follow object determine g possibly randomly select whole small failure allow think view aggregation problem test highlight special term testing field case global try test mechanism current separately independence consistency pick uniformly check aggregated opinion issue profile opinion change opinion issue change mechanism distance satisfy main follow consistent box similar ask property property see study question define test one property similarly sub mechanism introduction systematic property deal family aggregation deal truth issue either conjunction b aggregation mechanism exist mechanism direct corollary define either conjunction several far aggregation functional conclusion get well g dependence ff aggregation mechanism affine represent truth functional conclusion lemma issue mechanism mechanism consistency section technique proof theorems approximation aggregation conjunction aggregation mechanism independent case induction framework change get j let independence insight issue aggregate influence aggregation nj nf I read vote member vote member outside return frequent assume issue mechanism aggregation conjunction characterization include h characterization aggregation mechanism proof theorem tight case consistent aggregation technique issue aggregate function aggregation xy df linear permutation exist satisfy aggregation mechanism independence mechanism follow mechanism independent aggregation mechanism mechanism deduce close issue aggregation study aggregation mechanism non conjunction characterization calculated dependency class probably include dependency work preference question inherent conjunction relation depend think assume distribute immediate extension extend complex functional preference open one constraint consistency class mechanism trivial true em proposition conjecture criterion proposition theorem theorem conclusion fact sketch mail il aggregation vote proposition aggregation relax constraint input relaxation notion fit main result case aggregation term truth functional constraint involve boolean influence protocol term test generalization linearity truth economics university author thank participant comment truth dependency deal scenario aggregate independent opinion suggest protocol stand criterion security attack network opinion independently cast vote protocol criterion think violated think enough third think scalable hence although separately protocol pass discrepancy majority vote security scalability conclusion later several logical
therein however concrete require level error quality model height appear denote jump jump simplify exposition error term identically distribute classic normally distribute finite cauchy tail double exponential possess chart applicable change control sharp construction control chart chart control shift standard base exist large simulation expect range underlie specify chart shift let review property chart roughly introduce since binomial variable sequel shift occur long large small positive negative detect testing alternative trial equal binomial arrive chart repeat obtain version chart control limit chart modification item sample either mind chart substantially large delay detection purpose chart count individual move control observation form buffer contain past exclude buffer length replace replace instant buffer whether buffer eq buffer chart form sequel buffer observation lie limit large state chart main chart production count buffer length moving buffer buffer present explanation superiority chart jump slightly version chart signal result extend general outline model jump simplify stop transformation integer small sequel current equal statistic center scale version control section exceed correspond choose value sample decrease call buffer length ensures asymptotically buffer length buffer available series buffer satisfies chart superior buffer strategy example put obviously chart mt consider buffer length suppose buffer run ensure fair let consider choice chart historical beginning classic small well jump q specify control alternative model shall purpose briefly relate underlie give thus sequel limit detail refer work general require martingale ni ni conditional array space martingale array x martingale deal series treat work dependent image consist analyze top origin low corner I array noise neighborhood l ij ij defines follow error equation variable I ij difference pixel neighboring pixel analyze ni ki ni p array respect concern limit theorem modify chart martingale difference hold change weakly time converge eq weakly notice identically bernoulli success say chart stochastic dimensional behave smooth theoretic result benefit modify chart cf case approximately brownian drift yield strictly large classic chart chart jumps detect random drift drift summarize indicate modified chart detect jump right point notice yield distribution jump dominate beneficial unlike chart carefully average false small level experience chart justify theoretical result control practice often chart buffer reasonable buffer fact figure plot buffer length minimize practical select chart desire buffer simulate control reasonable determine small demand reader summarize ensure integer prevent control chart select give jump height buffer length optimal account limit buffer exhaustive expect perform extensive firstly identify buffer secondly investigate normally third chart normally performance study chart table run buffer feed pre control fix chart buffer control provide chart considerably equal rl qualitatively pattern fix large mention handle alarm chart jump double behave quite difficult table c jump rl c jump rl jump c rl jump rl c jump jump jump laplace rl c cauchy jump rl acknowledgment anonymous constructive remark presentation visit technical support grant range science education point array bernoulli ni martingale bt b denote brownian motion satisfy assumption clearly conditional z ni eq set yield third term converge pointwise continuous handle martingale central theorem bt term equal define linearity triangle obtain tend mt imply ensure case obvious put thing height width em pt minus pt lemma proposition remark chart institute engineering technology institute chart exceed financial study modification chart use recent observation chart chart superior shift central limit model explain often arise whether true martingale array applicability firstly time series secondly image primary p produce tool large propose comprehensive review article property chart
contrast approach generation principled combine make reliable walk limited prediction require recommendation detection rank part nsf fa yu foundation microsoft yahoo fundamental china predict occurrence fundamental prediction snapshot would study effectively combine network attribute develop level attribute achieve use attribute guide walk formulate learn edge likely visit node learn facebook extraction descriptor management application term world exhibit interesting property model predict reproduce network research seek develop predict global many highly dynamic quickly edge node study identify mechanism social evolve edge understood motivation snapshot seek accurately add future time predict new problem view link recommendation link scientific facebook friend responsible significant fraction add facebook useful facebook member two point extremely sparse node case million unfortunately predict new near million prediction subtle link social use intrinsic network similarly user gender home edge consider example reason connect party facebook party age probably live link people meet party close people friend circle despite friend party social interact interest eventually principled profile interaction information simply extract common friend short path node combine profile address recommendation supervise principled combine characteristic network unify develop supervise walk bias walk nod often strength walk weighted visit prediction new create node random weighted link score create technical way directly strength function strength formulation view network datum show approach outperform extraction additional extraction combine network evolve addition offer insight network formation relevant social like predict future interaction direct business consequence broadly large directly benefit interaction member link organization research security recognize prediction suggest link form future link suggest work directly beyond predict link protein interaction give suggestion relevant page link network easily next previously un predict include previously come problem cast link moreover walk predict link unsupervised link prediction recently detection link relational challenge primarily precisely link come degree feature edge add together feature heuristic information link appear rich attribute combine use walk edge probability link create likely node walk network recommendation like recommend link link create create link clear create setting appear give positive learn distinguish link recommendation click link anomalous generalized recommendation general classification edge node go create approach imbalance create fraction total hard high imbalance second extract task like age gender edge hard however less clear consideration graph attribute two example might count adjust proximity degree neighbor go give two path thing centrality degree possibility extract trial approach become hard annotation know edge combine get pair link rank idea score walk random walk jump walk walk rank take impact age gender combine random walk powerful tool node edge visit create source aim visit often assign walk edge visit set make overfitte learn assign edge address principled walk set candidate create call edge link label candidate example generalize instance describe node gender interaction attribute edge strength parameterize take compute edge transition exactly learn strength edge walk run stationary walk assign top rank predict probability strength thus edge likely visit set assign visit optimal strength vector strength idea want node node prefer hard constraint practice unlikely satisfy constraint soft regularization violate assign violate establish derivative combine attribute parameter output walk transition jump back seed conditional give currently node parameter walk minimize respect derive gradient recursive introduce eq commonly loss hinge take recursively still compute rule compute arrive iteratively partial score k p kt free intuition weak parameter absence gets find add drop ignore strength continue green regardless vector descent converge validate link social ph mat ph facebook avg four complete facebook predict seed reason person connect facebook triangle walks give facebook thousand incorporate user million co name co area ph matter mat energy th physics ph every compute time create co create span closed triangle attempt make try create give source paper similarity paper co paper number common friend facebook facebook point friend people country friend user randomly node show individual mutual friend become friend facebook neighborhood speed computation mutual unlikely friend practically demonstrate figure create link friend annotate facebook network seven pt cutoff create request communication period common friend half train algorithm test consider curve auc top node actually receive appropriate suggestion describe dataset four co facebook evaluate several aspect strength choice walk weight type choice play optimize correspond metric top optimize pt square q huber margin window q auc function show function sorted iterate sum huber quickly evaluate around evaluate relatively primary perform indeed limit reflect auc order significant translate improved loss fact huber loss obtain unweighted use remainder uv certainly function scalar desire logistic pt uv uv w choice significant impact slight evidence perform version double seem comparable avoid parameter think extreme graph undirecte approach score proportional simply random walk eigenvector score become score add notion strength walk back value short walk away play unweighted ignore strength assign strength see significant unweighted ignore strength weight find overfitte relatively set walk capture idea strong type might friend edge capture idea weight slow divide significant benefit seed node label type type link node six increase move examine somewhat seed could connection path connect capital link end could set make jump correction work friend common apply graph node great small practice introducing help facebook graph facebook formation co long tie connection write people people edge may opposite social capital trust show social norm try fit friend tie evaluate walk examine estimation walk unweighted bfgs gradient notice auc iteration auc random friend degree dt feature dt path dt lr lr node lr lr feature edge common friend dt dt dt dt lr lr lr lr type multiple edge r r co mat co ph co facebook next baseline along machine create test show
context determine row experiment direct seminal sense discrimination vector center target apply algorithm word generate sense word matrix word opinion category word context generation tool lexical intensive whether automate automatically automatic cluster classification specific kind see section several researcher word word typical representation vector token word correspond annotate tag apply frequency unsupervise sensitive kind error semantic labeling semantic role sentence role sentence connection sentence role show word reliably predict lexical refer good level lexical important semantic role narrow lexical expansion query google yahoo document semantic query pay click google yahoo pay display ad query make ad context occurrence extraction field extraction ie entity recognition name entity place relation extraction extraction al frequency facilitate supervise context name pair suit measure pair see similarity angle correspond pattern cosine angle word angle matrix approach examine multiple analogy question college level college high pattern measure similarity construct text pattern infer phrase language retrieval question answer clustered word represent pattern cluster pattern representative pair automatically analogy analogy relational pair strong assess review classify noun class classify cause home classify home locate weather report weather search satisfy search engine cause cancer one relation cancer task search engine relational conventional engine candidate answer relational incorrect answer manually simulate task task attract system automatic automatic arguably relevant generation relation individually use pair distinguish word merely mapping mean analogy proportional matrix proportional select relational however atom contain mass atom complex application approach application alternative section measure semantic query alternative probabilistic model probabilistic retrieval measure similarity probabilistic language information retrieval approach idea view matrix share measure similarity idea represent semantic human expect good performance lexical pattern measure word relational similarity sim aa sim ab measure good approximation hybrid approach combine beneficial stem fact word ignore commonly represent word phrase house house english house house house serve house whereas house purpose house storing estimate english word composition simple become contribution mean sentence hilbert inspire mechanic explore pair pattern raise semantic semantic conjunction yet statement calculus however limitation survey event distributional family distributional usage arguably major progress suggest help person request unfortunately computer force artificial language computer understand human us potential computer enable semantic language semantic researcher semantic conclusion make intuition soon deal organize accord determine processing play important survey show familiar emphasis new believe present matrix semantic human capture kind suggestion journal artificial intelligence research publish ai access national pt pt computer little mean human limit computer computer action computer analyse begin address limit survey semantic currently matrix yield survey broad range three category take detailed project category goal survey semantic provide new perspective less familiar field make full use computer currently understand mean technology language yet impact impact deep semantic space semantic general sense mean phrase language concerned approach semantic survey distributional represent aspect language retrieval system many concept represent space apart distant document document sort order success inspire extend semantic english human school age adequate relation attain multiple analogy question college average work accord context pattern fundamental linguistic three believe possibility introduce type corpus much semantic code basis system measure national resource similarity generally often phrase document lead measure semantic due distributional hypothesis distributional similar meaning often tensor connection space must derive graph represent matrix imply adjacency matrix derive event frequency emphasis frequency bring explicitly connect distributional hypothesis vector cognitive frequency text semantic machine learning classify item vector classify collaborative recommender typical system people correspond item product rating poor fair excellent person many mathematical well term frequency cognitive prototype often prototype membership categorization formalize however usually numerical human frequency extensive measurement usual typically subject item subject item technique analogue relate entirely appear cognitive argue aspect ai plausibility promising area research survey semantic currently comprehensive date survey approach semantic growth ai researcher semantic serve unified encourage area pointing area survey make framework term pattern see importance kind draw matrix address matrix application potential actual exist summary omit matrix nlp cl system cognitive arguably semantic far write survey art semantic introduce perspective familiar area reader basic algebra text book concept perspective semantic good information retrieval recommend reader familiar survey understanding beyond familiar natural reading recommend article accord get corpus text framework involve generating discuss linguistic review process linguistic processing mathematical semantic model plain raw linguistic sense parse section look linguistic semantic document raw frequency operation compare describe optimization concept semantic detail look present retrieval library explore build representative review module build system open application semantic serve short historical view semantic give row section generalize idea phrase book collection discuss context consider occur alternative semantic question limitation discuss state hypothesis usage people mean work define specific hypothesis bag distributional hypothesis distributional notational matrix bold capital letter denote scalar number document convenient discuss document page mathematics set allow bag matter bag bag element element bag row member bag document document bag represent bag word frequency tend relevance word apply capture document document matrix suppose document row unique column let row th frequency th zero since document use whole vocabulary document likely pattern signature tell document seem tell frequently lose phrase document surprisingly seem aspect semantic arguably extract author word section document topic measure document treat engine pseudo document relevance document row matrix document may context sequence character distributional word context hypothesis justification apply measure word may derive occurrence window dependency rich context dependency preference position see various word word reveal thing language primarily interested physical usage context physical building main derive frequency include semantic argue sense co occurrence word say company keep pair row vector pattern cut work purpose pattern similarity pattern find propose distributional co occur tend solve co suggest pair similarity vector relation material material second member tend pattern relation co occur pattern tend semantic word row vector tend relation extend distributional column pair tend pair suit measure similarity similarity suit measure distinction similarity depend correspondence property correspondence degree relation relatively similarity whereas cat relatively relational relational word relation material present semantic cognitive science relate share bank trust company car frequently think kind white kind colour sound prefer relational whereas relational computational share share semantic term relational involve mean semantic occur frequently calculus corpus usually often possibility document triple pattern matrix measure similarity word triple whereas pair pattern triple build grow increasingly rare contain phrase together triple pattern grow increase break tuple triple correspond triple matrix go beyond tensor scalar order tensor high tensor term word tensor preference tensor example correspond word english correspond join row represent slice tensor element english similarity slice question token instance symbol type token consider ever try ever fail matter fail token ever token token type fail line token matrix type token token nine column ever try ever fail try fail try fail token token document ever try fail try try fail matter matrix token binary token document otherwise row integer vector token row ever token represent token vector typical sense deal word token specific context rather mention relationship define characteristic frequency mention five repeat interpret vector work something implicitly assume something like statistical human usage figure people text piece piece frequency intend include word context pair tend tend indicate relevance query pseudo vector term distributional context tend tend similar semantic co similar tend relation similar raw context linguistic raw tf document search bias document weighting tend yet normalize co occur document form method text tf work word variation value replace perform wide measure semantic pair th row th correspond number time row raw frequency value word product expect random semantic relation distributional hypothesis negative give high relation indicate uninformative know problem bias consider I hence increase decrease another event smooth raw replace laplace laplace depend frequency small laplace towards simple way improve information represent occur content however carry little weighting maintain semantic discrimination computation compute intensive share coordinate e vector share frequent precisely little discrimination weighting describe highly occur context keep word conservative show precision matrix compute reverse compare match zero elegant operation term document operation svd thin svd mention truncate question english language indexing apply way svd present way look reduction occurrence three singular form produce corresponding minimize error frobenius discover mean word context svd create capture force context force correspondence improve describe noise think span space span specify reduce rank amount think compose variation high describe truncate svd discover occurrence occurrence appear indirect occurrence similar define recursively low occurrence truncate discover co occurrence sparsity truncate svd k dense sparsity insufficient svd lack svd correct likely another incremental another incremental truncate svd miss treat analogous parallel factor canonical equivalent discover survey empirical tucker decomposition order tensor ram projection subsequent research alternative index iterative scaling allocation discrete four equally well smooth truncate word frequency truncate implicitly frequency explain semantic measure raw cosine angle word two word frequent rare word short irrelevant thing cosine opposite degree cosine raw frequency vector cosine smoothing weight yield converted inversion q ir lexical circle ir measure normalize measure hellinger kullback measure involve word similarity cosine coefficient find similarity focus overlap lee shannon linguistic similarity measure measure group high measure cosine jensen shannon recall frequency sensitive mi lin frequency score similarity tend frequency sensitive prefer word determine measure believe determine appropriate similarity inherently frequency compare smoothing apply problem bad case parallelization multiplication observation scalar vector coordinate cosine overlap decompose nonzero q pairwise row efficient leveraging determine solely share nonzero reduce cost computation determine vector share building indexing change retrieve share nonzero nonzero nonzero coordinate nonzero efficient experiment semantic pair large web average leverage describe coordinate tf coordinate power coordinate interested solely run processing fast open software package implement thousand allow sophisticated parallel execution programming start index streaming part input read dedicated trading increase parallelism increase building index reduce column approximate efficiency project svd perform computationally intensive randomly impact computational little score especially top vector indexing distribute row represent accuracy mostly zero number cosine approximate cosine two compute vector index task locality lsh another approximate projection control tradeoff irreducible polynomial create collection document task remove web lsh general technique map row signature lsh preserve similarity preserve cosine similarity top cosine projection provide indexing lsh similarity task corpus lsh indexing however large corpus outperform lsh efficiency measure cosine near cosine external similarity cosine implicitly internal measuring value linguistic mathematical unsupervised vector general nothing machine aside task literature specific type discuss system source interested reader project text engine library foundation arguably wikipedia offer indexing rank primarily content document image video decompose field store implement content correspond correspond store use field allow string document also text point classified spam retrieve match column document field instance content consist instance index tf store schema identify define function build phrase query date restrict sort update occur search index programming language foundation open searching index present full http web create document index software offer web seed parse web document page seed page index book explain
synchronization process transition generator sense note remove observer directly hide state rather observer internal observing length observer state observer machine length observer know observe simply every finite observer machine word machine abc exactly asymptotically observer exactly time machine since contain almost every infinite word turn asymptotically ref disjoint finally synchronization observer block observer expect observer state machine observer function previously symbol eq observer symbol close close rate synchronization result synchronization consequence lm lm lm w w observer internal state constant know hence q since implicitly conclusion state observer machine rate constant machine alphabet state probability say restrict consist state irrelevant state observer observer currently observe initially possible observer generate similarly observer currently new observer govern machine recurrent strongly connect always original def follow assume order state order case block transition machine row state joint observer symbol joint recurrent observer observer observer observer word k jj w know pair convergent follow k j I n repeat observer state algorithmic theorem topological pair mm box box distinct mark end box already mark mark repeat path initialization replace j pair mark convergent marked box fact prove minimal distinguishing proof omit ref time mm distinct find convergent check either convergent distinct pair exact machine show exact machine observer observer phenomenon map efficient test machine ref similarly synchronization turn qualitatively similar result hold method plan generalize countable fellowship research project physical intelligence project view finding author either express department result alphabet fig name e arbitrary unless follow fact real let radius course radius triangle result linear algebra fact finite version ref linear operator banach define restriction state machine refer know short finally equation take normalize eigenvector maximal never divide eq therefore follow b pt algorithm corollary observer internal use treat exact synchronization observer number treat sequel observer average fast observer well additionally synchronization rate exact machine synchronization state state interest include synchronization machine mean completely symbol observer ever machine internal result consequence enable observer output include stationary particular identify qualitatively distinct synchronization exact infinite case treat sequel alphabet let variable generate future begin observer prediction measure review block uncertainty observer symbol symbol decrease limit observer prediction observer prediction asymptotic interested prediction source convergence restrict stationary simplicity state state label consist correspond node label initially machine pick symbol labeling symbol fashion denote machine visit output symbol generate kernel stochastic sequence state observer internal machine illustrate definition example alphabet transition block
base measurable discrete distribution impulse value independently also probability sample independently impulse stick break r r pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt kind specie sampling attribute view initially call scheme sampling scheme poisson partially admit atomic counter distinction atomic versus important posterior mixed base impulse mixture actual index index convert irrelevant group actual index countable mutually exclusive number assignment primitive tree partition top rgb f rgb rgb rgb rgb rgb bottom display bottom process cm cm cm rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb rgb cm cm cm cm rgb cm cm rgb cm rgb cm rgb list subset order store order need roughly generate generate entry impractical know bin sampling size element order partition occurrence instance order slightly representation set order order bias sequence probability partition occurrence member list follow take probability impulse partition yield form index occurrence sequence index partition represent impulse need probability poisson dirichlet order decrease stick break rgb rgb rgb rgb rgb rgb rgb rgb cycle cycle rgb rgb cycle rgb stick broken stick length break proportion broken stick part remain proportion break stick part suppose random variable poisson sort value call discount literature concentration community inverse parameter use sort act give form break inside mixture indistinguishable ever convenient show log pt plot show hundred effectively say effect discount great show discount roughly like whereas discount geometric roughly common poisson dirichlet extend poisson dirichlet distribution yield finite countable dirichlet formula dirichlet sort definition summation stochastic base index sample distribution long atomic chinese restaurant customer item item enjoy chinese restaurant crp biased index chinese sequence integer index accord crp bias order crp alternative vector get domain property subsequently formula remain index atomic distribution irrelevant bias crp sample table restaurant statistic sample multiple count distinct call size index accord bias denote ordered occurrence size sequence crp sampling gets sort suppose distinct time atomic safe associate sequence biased exist index dirichlet subsequent give dp process random concentration partition parameter dp finite definition categorical say hierarchical broad sample distribution sample length sample independently accord give weakly partition basically true expect slow learn use surely surely use space continuous distribution another distribution atomic posterior sample weakly mass discrete discrete one give model diagnostic version see crp consider finite definition px n hx denote increment lemma term present size partition occurrence list call go chinese restaurant use crp chinese restaurant biased length pi lr occurrence follow definition chinese restaurant distribution note partition represent ccccc partition key definition rgb rgb rgb rgb c rgb f rgb rgb cm k rgb rgb rgb cm cm rgb rgb rgb cm rgb cm rgb rgb cm rgb node second row partition partition complementary bottom rgb cm rgb rgb cm rgb h cm rgb rgb cm cm rgb rgb f rgb rgb rgb rgb cm cm rgb rgb rgb rgb rgb bottom middle top form bin indirect bottom b l order detailed partition represent see complementary change one partition template complementary chinese restaurant functional definition partition let convert conversely convert correspond sequence occur node correspond tree depth partition see partition root node expect size average subsequent parameter discount pt figure instance discount next third label colour child whereas fourth plot label maximally node leave discount schedule split lower also infinite counterpart understand summation start simple draw circumstance follow measurable q inverse explanation start cluster circumstance follow simplifying nest simplify distribution theory improper prior posterior represent improper prior may posterior improper infinite define difficult infinite vector must prior full hilbert extension prior improper good improper improper sub pp measure instance informally believe improper normalise posterior sense property proper plausible standard proof integral exist check improper measure project improper variation define say improper sequence closeness measure variation improper prior sample proper improper detail improper definition use parameter atomic base distribution notation definition eq definition stick ii v n biased order improper prior posterior formulation attribute chinese restaurant process stick breaking stick proper improper sampling crp sort whereas improper correspond sorting distribution discrete identical calculation whenever hierarchical around impulse partition instance cat bias order formula size bias unable version introduce next occur restaurant definition order sequence match latent word column give sequence list cat size instance row word appear map index cat cat cat cat need return corollary prove hierarchical consider finite base sample constraint b one need table indicator contribute compute value start choice discrete indicator corollary indicator indicator appear explicitly forget ratio easier discrete especially moment properly moment base probability vector moment prior lr conclude aa moment apply finite require discount rapidly store log recursion slow requirement discount memory place cache value store save considerable memory repeatedly ratio ratio store recursion table resort storage number ratio present bit calculation proxy linear less accurate ratio recursion ratio mostly proof give augment distribution significantly recommend fitting induce two chinese restaurant scheme generate chinese restaurant dirichlet simplification dirichlet simplification infinite form improper dirichlet finite vector soon proper normalise alternatively probability order dirichlet stick covered dirichlet discount concentration discrete dirichlet discrete base dirichlet present indicator multinomial dirichlet dirichlet remarkably conjugacy apply conjugacy expense derivative unstable accurately compute communication centre mm height dirichlet mm com national
point estimation slight tendency variational place iteration convergence likelihood mcmc method estimate approximate density simulation compare plus minus mu mu mu minus mu tw mu mu mu minus mu minus mu mu minus plus mu minus mu minus mu newton mode update k mu plus minus theorem proposition principle involve focus statistical deal candidate consider computationally approach variational log greedy space step screen large inclusion potentially extend algorithm involve diabetes selection matching pursuit variational mu plus minus mu plus minus mu plus mu mu plus minus mu mu mu vector vector unknown independent mu plus mu minus mu plus mu minus mu mu mu deviation predictor form hierarchical prior refer wide range idea convert bayesian inference fast especially problem approximation independence variational leibler posterior result key contribution proposal role section method selection posterior model probability methodology demand need application greedy pursuit processing lar see excellent family algorithm greedy assumption application serious inefficient parameter chapter number model covariate use involve flexible extension approach demand setting framework develop use aic bic together forward contribution greedy linear currently attractive work efficient section diabetes datum regression consist construct coefficient stop predictor path diabetes stop iteration predictor enter closed maximize novel greedy two benchmark research technical approximation method detailed exposition orient introduction variational refer different idea convert parametric write difficulty application posterior involve integral variational proceed formally attempt kullback leibler mu mu minus mu minus mu plus mu mu mu plus minus mu identity minimize leibl divergence often energy due non negativity leibl low maximize leibl divergence useful bayesian selection bind express mu minus plus plus minus l mu minus mu mu minus mu plus py mu mu plus mu mu expectation put mu mu mu mu mu mu minus minus need iterative fix write design mu minus mu mu minus dy mu mu maximization minus plus mu minus mu plus variational plus mu minus mu mu mu plus minus mu mu mu plus minus mu expectation easy minus mu mu mu nz nz mu mu plus minus apart normalization gamma normal log computation require fitting generalize linear appendix write th diagonal obtain mu mu mu mu minus mu plus minus mu mu plus mu mu optimization retain bind approximate close possibly high matrix avoid follow td mode response x w lower retain specify initialization first perform ordinary fit initial ol use selection ol mention far far helpful drive low approximation apart mu plus minus mu minus py plus mu prior likelihood plus mu mu minus mu mu mu plus mu mu plus plus minus mu mu minus mu mu plus mu mu mu step add previous variance efficient perhaps likelihood condition marginal distribution consideration rule replacement strategy thorough review see bind approximate leibl divergence structure variational distribution product normal minimize kl small good experimental suggest maximize tight present strategy ranking prior give currently active mean variance write predictor independent priori minus plus minus mu plus plus mu mu mu mu set mean model mu mu minus minus mu minus plus mu mu predictor slight abuse plus minus mu mu mu minus mu minus encourage parsimonious small complex one option minus mu mu mu mu plus mu mu minus mu agree extend bic advantage require still encourage add rank possible update part stress factorization purpose inclusion outline correspond row add mu minus minus minus mu q mu mu mu minus fitting extend optimize value maximize respect variational posterior mu plus minus mu plus minus mu mu mu plus minus substitute end plus minus mu minus mu plus minus low predictor write optimize writing minus mu mu plus mu plus mu mu mu minus mu mu minus mu plus minus mu plus plus mu mu plus minus mu minus mu mu mu plus mu mu agree rank residual point detail later value far normal approximation n q l good value variance assume mu minus py mu mu plus mu minus plus minus minus mu plus mu minus mu mu normal mode obtain mu mu mu mu mu plus mu minus mu mu mu vi vi plus mu mu mu series mu plus mu plus mu minus mu mu plus minus available find good newton stop mu minus mu mu minus minus eq q plus mu minus plus minus mu back calculation write mean optimize stop store pc c lc lc j lc describe regard add current hereafter variational short like forward greedy widely scientific field encounter add backward elimination mistake consider burden factorization step bind current model sum base minus mu mu mu minus mu minus plus mu mu minus mu minus replace relevant need current quantity correspondingly plausible current plus minus minus plus minus mu plus mu minus mu minus plus plus mu eq mu mu plausible current plus mu minus minus mu eq mu mu plus mu mu initialize backward elimination stop lc vc lc lc j lc else hereafter might meaningful restrict search inclusion include restrict search candidate algorithm dc v removal elimination j pc j pc j j pc c v j later compare implement model shape allow response flexible bic fisher inclusion rather plausible var burden order var mu plus mu minus mu plus mu ny qx mu mu mu normal plus plus minus mu plus minus v mu mu mu mu plus mu minus minus mu respectively reduce greatly newton get estimate experience rank standardized optimizer rank q I rank absolute standardized residual agree frequentist predictor residual frequentist extra parameter penalization greedy tuning final plus log unlike greedy encourage parsimonious extra tuning penalize bias desire shrinkage well zero confirm stop make valuable high problem var practical benefit processing aim near four water validation serve variable measure spaced nm think range nm second final consist predictor response treat single popular ol fit current first predictor fit assume variable water set obvious see response clear absolute value residual fit response predictor select reflect ols plot nan model analyze assume nan th response predictor predictor visually mean employ result make prediction examine usefulness mean plus mu mu mu yx mu mu mu mu mu minus mu mu mu minus mu minus mu py mu mu mu understand mse sample var summarize estimate var probably reason var water also analyze bayesian transformation transform response wavelet bayesian likely value report comparable wavelet predictor prediction select anonymous separately model rather multivariate compositional justify transformation response mu minus mu minus plus mu mu plus mu mu mu denominator value mu plus minus mu I mu mu mu minus plus
stationary z classifier infinity every depend refer infinite double sided viterbi double side double hmm exist infinite viterbi alignment moreover viterbi satisfied positively nonnegative stationary code operator x rd rd rd rd stationarity fact infinite alignment finite viterbi choose rd rd double alignment process denote restriction integer note side viterbi process variable imply double sided limit smooth j py almost show inequality x r risk let consistently viterbi would hide viterbi follow call viterbi depend function mistake viterbi misclassification viterbi convergence state use surely theorem alignment ergodic viterbi alignment well risk bound go negativity boundedness risk sided infinite alignment stationary ix vx ii element theorem prove rd first term converge surely term surely auxiliary exist proposition moreover hold py almost surely analogously thus cauchy sufficiently q imply stationarity recall unfortunately hold provide surely stationary ensure hence sequence let big cn r remain py v rd constant lemma ready term converge prove let borel shall q imply moment approach section indeed counterpart inequality immediately probabilitie difficulty however viterbi convergence hold n nx nx state introduce sure viterbi borel adjust viterbi conditional integrable prove rhs viterbi alignment integrable integrable theorem state together give likelihood p n v ij v shannon remark entropy implies interpret conditional entropy difference alignment theorem eq every state also notation e let finally let cluster suppose pi py imply state arbitrary exist observe arbitrary loss bellman principle every every q since px sf inequality proof analogous proposition bellman principle cm theorem axiom section conclusion condition conjecture exercise section lemma mail se keyword hide alignment asymptotic segmentation risk goodness double stationary irreducible depend process call non observable emission shall assume emission measurable borel algebra generality hmm literature usually stand regime observation hmms widely recognition bioinformatic processing refer integer drop consist formally look state finding every let problem framework specify shall denote thus viterbi viterbi stand viterbi minimize often preferable pointwise loss classifying count misclassifie sequence pointwise posteriori alignment viterbi shall risk risk minimize clearly necessarily minimization importance apparent minimizer hard enough viterbi obviously integer estimate tuple show viterbi quantity classifier deal risk viterbi value grow viterbi obvious viterbi piecewise shall fairly hmm limit main limit risk constant depend viterbi alignment limit viterbi alignment segmentation sense risk big viterbi alignment classification alignment alignment model know asymptotic risk theorem concern ergodicity sided sometimes call regular ergodic follow state recurrent process couple stationary ergodic stationary version support copy e lattice span since chapter fulfil claim value satisfie theorem hold ergodic since begin limit immediately separately shift trivial consider process together def extended function almost realization statement exist first first element function alignment independently infinite alignment add alignment existence property alignment alignment infinite viterbi restrictive infinite viterbi prove assumption recall cluster intersection support emission cluster consist single state existence
establish show oracle dimension simulate accuracy useful tool polynomial retain advantage suffer curse pattern first incorporate selection paper compare nonparametric moderate include generalise et subject bandwidth fan although lasso recent relate linear include version lin zhang adopt et al aim thereby paper explicitly dependencie pair identically relate predictor smoothness discuss assign increase rectangular form assume bandwidth kernel vary function asymmetric respectively give bandwidth matrix possibly different size unbalanced lead undesirable dimension redundant possibility infinite dimension everywhere fairly exposition generalise find bandwidth become point large value decrease cost variance linear accordingly first traditional cross validation assess need eq lx affect increase variable influence influence remove area reduction dimensional uniformly response significance redundancy eliminate dimension variable significance perfectly dense illustrate adjust dot dash line modify bottom bandwidth entire imply include thus somewhat improve second top consider significant possible selection cross validation select tuning parameter different domain consider hx hx thresholding square redundant extend although unstable approach percentage variable exclude fit explicitly redundant zero lasso problem asymptotically naturally numerical polynomial whether initial bandwidth thus bandwidth bandwidth term function cardinality variance adjustment need expression shrinkage let step bandwidth case define expression measure shrinkage satisfy deal region domain behave somewhat arbitrary result force exclude still final advantageous situation despite ensure estimation increase suggest algorithm intensive work simplify expand equally meet bad grid stop simplification redundant classify everywhere grow easy fix great ability exclude circumstance initial instance proportional use employ offer practical advantage theoretical work addition initial allow variable long constant mention recent attempt assign attractive shrink greedy approach shrink cause change step dimensionality fit local penalty follow reduced penalty suggest comment difference presentation paper performance point ensure active redundant everywhere already strong nonparametric property algorithm technical find restrict attention situation could screening partly also dominate useful depend univariate introduce page uniform consistency estimator h first fairly occur follow purely convenience asymptotic almost everywhere apply cutoff circumstance entire minimum region estimation distinction tail problematic would uniformly good globally global case integrate establish theorem simplify version include hold degree estimate ensures consistently estimate scaling apply proof relevant notice illustrative open intuitively irrelevant thus region exploit small set latter correct form redundant everywhere correctly tend property local adjustment correct separate correct redundant everywhere ensure approach concern trivially correct tend everywhere probability tend result selection result cover decrease remove redundant actually whole fit elsewhere adjust locally follow variable algorithm perform oracle far locally actually square generalise boost approach r package respectively tune choose local include standard obviously fail htbp name description b linear fit relevant local generalise additive spline multivariate adaptive htb example methodology error simulation parameter average poorly redundant incorporate actually well redundant design specific estimate deviation std ols inspection good dependency particular improve performance reasonably nature strict effectively selection represent grid variable irrelevant wrong pattern broadly near boundary slightly large cover remove completely example distribute keep relate cutoff hard redundant dimension cutoff remove result exclude variable threshold need cause compare severe cutoff clear broadly correct improvement htbp use al air plus cube root response york fairly aim predict concentration variable temperature wind smoothed figure dependence part temperature wind flat imply high expand could useful comparative leave one make use observation build suggest model method offer traditional fit variable plot panel non perform curvature ol show use variable dependence fairly complex significant notice label redundant highly wind relatively useful approach std ol engine ratio engine air observation fairly equivalence clear htb std particularly suit produce improve model inferior linear section incorporate bandwidth variable increase understanding dataset potential effect removal partial removal variable redundancy apply property satisfied approach demonstrate classical regression concern predict response predictor dimensional often predictor
compare asymmetric tailed tailed skewness correlation alternative alternative normal beta cauchy laplace c uniform beta laplace beta beta laplace logistic result package package recommend distribution powerful alternative test student nevertheless perhaps cause dominate force close normality sign similarly might alternative recommend test test independence mention independence note complement theorem test powerful version skewness easier common normality widely field overview normality base characterization underlie population independent normal propose coefficient normality asymmetric discuss propose lin estimator skewness sample test simulation indicate notation central replication cube equal coefficient fisher normality sensitive normality skewness tail normality consider moment use manner alternative expression correlation enable consider correlation unbiased formulae somewhat easier standardized excess identically skewness nx standardize calculate moment involve know moment eq expectation expression little standardized tell value allow satisfy statement readily verify equality equality hold ii conjecture never similarly conversely moment obtain replace moment counterpart estimator nothing relate skewness scale test normality furthermore er lemma whenever necessary moment exist show slow prefer carlo underlie close nan hypothesis normality
occur france likely source south france occurrence majority case year triangle generate triangle year method triangle represent dash line period period consecutive week week second correspond link consumption period line stage week week week case age surveillance human surveillance seem valuable tool time cluster could public surveillance develop computer detection still remain necessity ensure public intervention way proceed order quantile promise future also plan investigate extension spatially acknowledgement thank head center provide implementation language none le universit france fr business school paris paris email fr france email system early health surveillance return count particular infection surveillance method surveillance france et theory return surveillance surveillance number event expect enough department death health publish classify three broad control review two step event week day statistical comparison statistical alarm observe significantly expect base count count current count occur g year incorporate regression easily trend model try reduce influence past avoid associate weight take trend past reporting delay incomplete inaccurate report surveillance system delay particularly problematic surveillance reporting concern surveillance encounter report delay surveillance report early surveillance system surveillance multiple statistical analysis daily automate method early cluster essential public health surveillance order generate development surveillance extreme early detection cluster report surveillance system cause human france cause laboratory confirm death occur year surveillance present description observation event develop last contribute surveillance perform surveillance hundred paper mainly illustrative purpose frequent variability event france mention impact comparable year see count within period series illustration give week last occur period five previous year alarm band realization distribute non probability distribution recall represent rewrite return quantile inverse associate able time standard exceed explicitly bound return level return period event purpose extensively assume follow generalized pareto peak threshold context use bound return level develop large small adapt associate upper tu namely confidence delta concern asymptotic confidence v ex impossible whole family function non function choose sub power real number last final follow low necessarily cover enough provide satisfying define alarm consist step explicitly period plot return period return plot read observation return bind theoretical return associate justify lead return period observation well variable interval since new period backward time hence see alarm second notice set one level finish plot associate observation existence least observation pt I illustrate return return period represent upper bind observation correspond respectively deduce return period equal
side sided sided student parameter obvious laplace plot region value small slope student illustration test cc student side solid sided freedom panel panel satisfie recall side satisfy meet student laplace rate another likelihood semi cover student prove proposition entail estimator bandwidth converge summarize introduce mh determine non follow behavior sided value semi differentiable derivative differentiable f side location student side sided student f derivative particular student corollary g laplace rate obtain begin latter require imply table side meet tend tend estimator location connection symmetry property likelihood ratio model side complementary contrary side always consistency sided location model estimator satisfied estimate sided laplace imply consistently estimate estimator derivative order study bandwidth converge rate summarize row satisfy lemma appendix hold bandwidth table gaussian regularity h differentiable convergence table side side sided sided sided laplace highlight test location side make possible conversely side slow property procedure procedure result procedure asymptotically tight fdr critical asymptotic power improvement rate parametric connected test require derivative conclude class light rate plug side laplace slow arbitrarily centrality distribution plug study plug asymptotically powerful procedure procedure far specific application exceed require extend interesting research realistic typical aim two value alternative proposition h h derivative negative x prove concavity x ratio student freedom centrality converge absolutely absolutely dominate j follow lemma entail regularity ratio freedom centrality proposition series entail derivative product series dominate tend concavity student student model proposition likelihood f decrease j conclude proof g therefore go central theorem fulfil yield k h conclude equivalent proportional choice choice prove converge proposition converge obtained hold unconditional asymptotic fluctuation start state procedure consequence converge surely surely go integer enough mm entail number procedure less great write empirical observe consequence fact alternative hypothesis unconditional theorem mt mp respectively consider establish translate unconditional setting replace therefore procedure unconditional adapt theorems unconditional threshold simply unconditional model threshold critical recover proportion plug among lemma converge almost far mh proof almost sure sketch inspire converge surely nan converge item concave fluctuation thus fluctuation negligible therefore taylor converge g dominate finally conclude proof theorem consequence lemma let b recall fact delta g proof x conclude lemma side make assumption differentiable tend simple f onto satisfie differentiable prove extra entail write bt b statistic test statistic false plug procedure control estimator simultaneous hypothesis test parametric wavelet method functional image dna microarray high throughput sequence set goal infer true definition discovery one widely risk multiple testing problem expect false positive among procedure test independent test independent procedure fdr typical go infinity plug construction powerful fdr target study correspond fdr influence plug kernel density statistic testing statistic identity denote respectively symmetric typically fulfil laplace exponential satisfie side function sided well non concavity hold consider sequence test integer side sided test characterize I terminology deterministic setting originally introduce set specifically random independently identically conditional setting satisfy literature independently identically distribution vanish restriction natural unconditional paper situation identity entirely characterize characterized concave generally regularity regularity assumption assumption express density alternative take return index hypothesis reject call testing reject correspond negative ii measure procedure I error measure discovery widely denote among test discovery proportion define risk type error rejection high type may error power testing positive hypothesis quantity depend r v whenever thresholde fdr prescribe order hypothesis rejection fdr previously use another context fdr regardless specific simulate less mt information interpretation mt set henceforth short entail decrease remark procedure sense maximum nan large threshold still maintain natural apply level estimator stage geometric interpretation rejection converge powerful adapt estimator may write value true nan hypothese one converge stochastically dominate uniform include set plug add numerator control estimator view order density integrable identically mx call asymmetric rectangular kernel characterize introduce formally connect depend test test specific testing testing prove multiple almost intrinsic sense regardless multiple obviously limitation g statistic satisfy ratio near critical result asymptotic let proposition asymptotically nan power generalize critical critical multiple critical illustrate sure result extend soon prove theorem cover procedure procedure value rate soon connect fdr control light connection interpret oracle level powerful thresholding fdr value procedure problem thresholde fdr contain procedure include organization extend include modification tend parametric rate section fdr plug achieve plug section test location moreover regular q bias positive estimator decrease plug consistent meet meet shape asymptotic bias variance characterize variance positive h differentiable h bias regularity near regardless trade natural resolve trade optimal choice asymptotically correspond prove convergence parametric estimator density derive g require match regularity proposition kernel estimator optimal bandwidth mse proposition convergence regularity proposition assumption kernel importantly convergence well asymptotic convergence recover envelope assume estimator converge strong regularity minimum away converge additionally proposition lipschitz derivative unnecessary increase estimation regularity rather aim rate plug limit associate false proportion fdr controlling procedure plug depend set unconditional relatively technique adapt completeness procedure bandwidth consistent define procedure asymptotic converge rate dominate result plug procedure proposition differentiable case bandwidth critical procedure asymptotic procedure unlike nan former summarize meet threshold asymptotic procedure particular whereas asymptotic value imply target fdr positive fdr asymptotically plug controlling decrease threshold remark name oracle g g power
kernel parameter select mean law free consequence simulate trajectory limit reach simulate rate result allow monitoring procedure behavior result nan follow walk state discuss behavior series behave change condition shall discuss e omit notation satisfie satisfy ensure sequel notation satisfy let process sequentially update ts define assume satisfied hold sequentially update l process asymptotic kernel ratio limit c brownian central corollary stopping monitoring procedure residual q innovation choose sequence walk correlate change walk stop concern design monitoring yield linear slope detection residual limit thank careful computing simulation final appendix proof result eq functional pz p p ax ax nx ax linear matrix recall ts yield since ts ts ts ts ts j j may eq lemma tr ts uniformly functional easy application formulate change modification exposition map show assume number bind stochastic product topology consequence process weakly thus argument weakly continuity virtue previous position result process associate stop sequentially thus modification assumption theorem may weak combine functional vi linearity remainder induce observe proof pt remark update residual stationary error institute statistic abstract behave stationary important particularly arise financial statistic engineering study detect walk theory monitor control chart test residual central corresponding limit chart finite keyword autoregressive chart sequential classification address institute mail de walk production degradation fail reach threshold financial walk appear price asset random hypothesis finance address increment random walk important economic product sequentially compatible control series answer lead completely wrong conclusion elementary change limit implication rich nonparametric control classic chart discuss detail random brownian detect know mind particular monitoring surveillance detect walk hypothesis soon article investigate monitor relate root detail unit hypothesis stationarity use statistic dispersion sum dispersion statistic et unit lee schmidt stationary study refer al test powerful robust rate walk stationarity versa sequential monitoring stop establish promise preliminary study involve error walk allow nonlinear observe sequentially time observation satisfy motivate polynomial assume detect control favor cover motion drift brownian motion substantially polynomial trend stationary chart stop evidence compatible hypothesis behave walk polynomial establish stationary provide may surveillance particularly signal classic confirm study square residual apply residual often classic computing introduce horizon monitoring stop application monitoring conduct however modification infinite straightforward discuss error sequentially residual constructive representation asymptotic brownian motion asymptotic change fraction random finite property proof result remain form unknown integrate set clear stationary change state walk integrate order specify satisfie stationary ar equation note coefficient lag exactly sec hypothesis assume mild making weak strictly eq brownian equivalent brownian combine assumption yield nonparametric time popular class series recall satisfie put classic time define version tt calculate appropriately statistic formula call nonnegative decreasing increase intuitive recent work condition kernel define instance back increase chart convention horizon stop asymptotic moderate limit monitoring stop may early stop favor alternative control rate favor stationarity indicate control characteristic exposition shall monitor start eq inference begin evidence get walk compatible simply appropriate scale replace place mt vi loss rarely brief exposition weakly yield pf p subtle value define right open singleton closure function appropriate define exist f continuous separable weak study term regression behave chart inf associate turn negligible central theorem process theorem notation intercept base x process former sequel stand ts polynomial fix limit play crucial role main right ts increment scale statistic degenerate distributional process summarize correlation inverse need
preference rate construct subject interest recommender extensively adopt mechanism history transaction algorithm advantage traditional filter music filter hard represent content three primary hybrid show deduce primarily algorithm introduce describe filter base item continue algorithm use recommender amazon com security recommender result filtering email bad associate annotation message annotation rating share annotation message although good drawback query recommendation recommend article find user algorithm similarity similarity category understand algorithms understand similarity closeness attribute tuple help compute tuple attribute categorical vs base pearson correlation item dimension attribute rate know calculus formula another rating show interest good give pearson user rate show pearson cosine et al similarity recommender user item base belong user item user user step first one user user associate weight importance third recommend high create user performance algorithm find user similar algorithm user datum one user pearson pearson correlation cosine similarity approach require item rate user item simply directly captured repeatedly come look define pearson item capture look rating give category database rate similarity cluster find recommend suffer drawback user item website user rate portion rate user item lot insufficient form greedy generation near require computation grow number change database since recommendation conclusion item user recommender recommender recommender make past analyze idea future historical look user make item implicitly rating category item algorithm step step scan past rating give similarity item item algorithm compute item large sure recommend collaborative computed implicit specifically opinion item user rate item formula explicit item average rating implicit implicitly compute item correlation boolean item equal user base major disadvantage build item construct similarity pair rapidly scale accurate recommendation scalability amazon com website book million customer million amazon com recommendation collaborative work rate item item element item similar item record similarity improve performance amazon com build recommender offline create expensive costly item offline item recommendation online rate recommender system reasonable recommend course user properly recommender domain recommender system metric evaluate finally recommender recommender differ poorly word recommender thousand thousand opposite movie recommender vary accord value similarity way extract get system thousand million item time system cope growth suggestion close recommendation user preference recommender system understanding suggest look gain trust evaluate recommender compare recommendation metric mae user item receiver operate characteristic roc metric prediction interesting regard recommender example visit user recommendation useful visit mean recommendation know recommendation metric sometimes recommendation order recommender system reliable accurate reasonable request per evaluate efficiency set compute item big grow quick look problem item trade quality efficiently give rating item knowledge customer memory computation recommender system calculation two perform time run hour speed calculation technique hierarchical search recommendation thousand request efficiency vast item trade prediction scalability consider maintain hashing obtain reasonable recommender rate lot rate call percentage item recommender agent prediction face employ recommender challenge issue rate establish recommender item redundancy user rating propose implicitly behavior approach rely filter call dynamic automatically recommender give recommendation preference accurate become moreover since recommender recommender trust need name birth code email http preference express recommender system user express identify company netflix website every recommender exposition user recommender system also security recommend recommender often target book recommendation book recommend amazon recommendation item may type attack item either attack attack rating rating item affect recommender call recommendation another item recommender way attack know stop study collaborative find propose hybrid combine try efficient recommendation categorization item appear extract cluster algorithm base preference similarity list merge depend item recommendation top recommender system cope item make suggestion approach massive growth ensure term employ user result extract suggest accurate propose deal whole portion due implement apply amount memory much less calculation recommendation term approach item issue apply system able user item profile program name algorithm comprehensive algorithms evaluation cluster association rule feature www os index open http wikipedia http en wikipedia loading file format part file gender student child movie category table table movie movie child provide software sample criterion program modify manually make dataset logical website movie title view rate user generate dataset entry order user category belong pre step file select class appropriate recommender objective recommend title category predict specific would also child movie hand child action movie recommend profile base accuracy experiment classify recommend movie title belong select
relative single process cubic computational preferable gps data gps input collaborative filtering notably winner netflix include drift rating svd predictive use factorization entry pmf possible closed markov mcmc posterior distribution various variable construct carlo typically straightforwardly predictive quantity interest integrate evaluated hyperparameter typically capture relevance covariance control likelihood marginalization process necessary function carlo posterior invariant chain state evolve close e hasting hamiltonian base require treatment detailed subsection slice mix impose several collection typical recently method transition gp number function covariance correlation process play critical effect affect hyperparameter mix due strong constraint despite therefore application marginally distribution hyperparameter specify result conditional mix chain cholesky hyperparameter hyperparameter covariance new unlikely noisy update work well hyperparameter change advanced develop method utility score national appeal approximately censor evaluation player side team play home relevant outcome reason new model game game team team pmf type see score perfect match score negative team tend ten dyadic e different feature contribute contribute enable side use provide relative pmf censor data eight four week ask prediction game past prediction month could entire model metric mean rao winner rmse winner accuracies rmse assign number spread spread yield point prevent tie single unit score take single cause point view evaluation spread expert identity report market force refine line spread imply mention previously conditional entry typical coefficient censor rao store state compute covariance component mixture weight good incorporate temporal fluctuation aspect vary appropriate end handle include slice sampling infer gap mass true different standard bayesian pmf method temporal home number ten chain censor interval year state chain warm start year run start iteration score keep chain component prevent pmf heavily influence previous prevent advantage relatively decomposition improve extensive sampling sampling remainder iterate approximately minute modern hyperparameter span game prediction mild believe effect covariance notable baseline pmf model improve home away alone consistent evaluation interval effect complex clear game home away prediction illustrate model information home paper induce pmf mcmc carry make prediction prediction notable process specify allow variation slow variation star reflect something g address issue nonparametric insight kind latent true indicate winner home temporal home away home away probability winner leave right expert bottom r r r pt r cm cm acknowledgement wish thank valuable placing institute research name probabilistic pairwise relationship collaborative analysis among area actor difficult pmf model side coupling pmf gp function vary smoothly successfully use date model interaction interaction item link salient feature task observation netflix user movie associate rating predict unseen pair sort relational pathway document data treat structure word occurrence pmf model powerful dyadic current pmf pmf often available identity collaborative example rating incorporate low feature however simple generalization probabilistic replace feature place prior introduce pmf relate probabilistic incorporate feature pmf typical probabilistic collaborative set explore later rating observation game use interaction whether user filter movie idea information apply pmf uninformative reduce pmf pmf marginal pmf share one correspond player allow hierarchical hyperparameter strong prediction capture length scale
notation real l undirecte give membership assignment node cluster simplify take connect connect assignment assignment node generally social network definition external node outside relative find characterize commonly definition set property graph give community consider clique community search define property every value sometimes entire pointed often definition community decade see lot topic community good concentrate internal community perspective community belong may describe allow community evaluate expand community maximize whether accepted spectrum clique influential essentially clique recent release synthetic benchmark graph become possible study benchmark generally number world social community overlap like overlap benchmark benchmark place overlap exactly community base explain community avoiding develop scalable highly community structure graph realization model unweighted loop represent connect network review interest work notation describe assume community assignment describe assign graph independent node bernoulli probability community assignment end inter community model draw assignment refer quantity proportion refer difficult em algorithm among thing conditional heuristic maximization search graph cluster use value greedy strategy assume community integrate likelihood integrate create mass mean selection bic technique overlap similar treat parameter integrate allow expand new name overlap blockmodel full column assume draw bernoulli community replace connection sigmoid natural extension estimate use allow restrict consider restrict form clique sbm share integrate cluster allow another heuristic community assignment place allow integrate connect connect independent connection tendency tendency node connect simplification become imagine every treat integrate possible permutation community partition equivalence matrix differ belong use community contain zero q community assign allocate community choosing summing write speak consider bayesian however refer attempt maximum computationally cluster estimating maximize complete likelihood show result clustering gaussian remainder emphasize primary integrate analytically give eq consider nuisance totally yet appear choose triple sample modularity small community calculate million edge maximization add community increase community decompose update remove community order consider allow convenience change eq pair depend whether change simplify fix unknown constant ignore decrease little moreover whether update unique column find introduce multinomial would expect find community output attempt node think community community graph initially community consist node directly add maximize expansion high far small contribution negative clique small community dominate whereby continue decrease unless consecutive expansion improvement subject expand community count community expansion claim expansion informally densely connect removal positive occur edge expand assignment community take place end inspire remove depend current estimate change seed impact fewer share old pair node one extra one share associated calculate node g g make change old ignore search expand factor maximize maintain ratio community update ratio search diagram skip fix investigate logarithm running time fast capable fast get result scalable trivially objective variant modularity fast maximize partitioning find hope progress restrict algorithm facebook five distribution often assume empirical graph strict law average five assume node community facebook user facebook divide six community nod
notational simplicity indicate belong assume throughout design deterministic allow effect large complete deduce q q due restrict likelihood regression add regression fix vector component minimization section likelihood mix section posteriori principle density correspond variable I marginal vector I regularization bayesian bic motivate degree freedom selection example cross criteria bic selection base simulation due lasso penalize may penalize loss study deal knowledge high smooth build upon prove response assume k estimator lie respect since use negative log risk coincide kullback sequel drop require condition eigenvalue bound triangular allow study notation consider support optimality form adaptive also different b automatically order set support appendix could involve adaptive penalize give oracle penalize estimator estimation imply error absolute bind distinguish argument corollary estimator meet restrictive similar result adaptive model version restrict eigenvalue current full recall eigenvalue hand parameter take motivated q penalize assume corollary set appendix main keep calculate employ inexact focus mean consequently omit ordinary cycle involve kind first notation prove achieve defer support information p section p jj value appendix support implementation website every cluster give appendix convexity achieve optimum performance kind mixed effect remark provide support hereafter classical linear mixed package furthermore standard choose overview conclusion effect aspect appear covariate incorporate effect covariate coefficient penalization thereby get large relate covariate aforementione cover parameter simulation remarkable incorporate effect estimator large effect focus effect use diagonal covariate elaborate paper suggest subsequent restrict group penalize intercept assign slightly grouping increase square use suggest approach production covariate measure expression issue covariate specify matrix deal determine apply validation effect coefficient bic lasso include seem reasonable fit wherein additional cardinality fix order fix effect cccc table considerably small indicate might dominate whereas slightly penalize maximum likelihood difficulty thereby deal substantially challenge gradient descent algorithm provable numerical study real datum remarkably incorporate support proof mail consist ensure secondly completion refer parametrization exist inequality norm assumption present result increment low increment constant term constant discussion assume throughout see eq omit deduce restrict notational degree n nj st survival variable eigenvalue centrality claim set eq technique nk formula define proof theorem comprise main present fulfil verification check drop slightly parametrization one proof vector eq make proof appendix matrix strictly positive definite exist correspond excess get eq first term w constant q right q apply restrict restrict give arrive corollary consider appendix solution cross grouping structure ordinary lasso calculation gauss element positive fisher information constant min r p suggestion take fact respect stepsize eq truncate simplification reduce especially setup set package employ cholesky triangular penalize reach choose eq reduce use active specifically cycle reduce stationary check assumption fulfil precisely proper convex continuously simulation study elimination example intercept deviation run tp positive emphasize coefficient indicate h ccccc tp coefficient subject penalization c ccccc penalization active slightly cardinality set select model coefficient active variability penalize bias effect problem z penalize estimation grouping structure provable coordinate descent achieve decade statistical allow large idea pose may speak inference lead asymptotic behave lot dimensional powerful name method reason couple feasibility optimization exhaustive selection square estimation statistical lasso work numerous respect establish equal optimal would would emphasize obtain design get small something aspect involved show lasso consistent selection non value neighborhood stability condition neighborhood rather weak restrictive sufficiently say huge reduction stage method restrictive assumption l non requirement eigenvalue requirement sufficiently eigenvalue stability lasso everything generalize penalization finally prove selector lasso positive besides include subsampling assigning regard homotopy lar coordinate gradient descent efficient datum incorporate grouping group effect besides effect extension concern longitudinal
business parent remain limited conventional analysis would overall behavior opinion conduct medium utilize customer business count sale figure average type role tail apply copy book instance place great importance public transmission sort viewpoint book analysis contrary sort conventional normal sampling internet also transformation past distribution generally inferential benefit technology grow momentum distribution video product video almost service activity sphere call population make age article analytical central distribution sample asymptotically normal useful sample distribution directly precisely calculate convolution considerable technology statistical analysis use distribution benefit individual content service content determine meanwhile software also benefit service enable perspective view relatively unknown become contribute service value introduce statistical software analysis discuss software base research internet population sum population principle calculate elementary size practically impossible size prominent limit kind extremely characterize much concentrated asymptotic probably also great unified research investigate treat distribution population calculate ad hoc recently size tail valuable present specific method tool tool microsoft calculate mathematical explain formation population worth look differently relevant release public drive valuable assess site analyze develop google statistics site department statistical service concept share private share public set amazon service public become next several representative service article organize relationship user service service separately single service amazon com site stock case site often able comment sometimes future service collect general public internet site product service user content video sharing service share site numerous allow comment total user become value piece feature service tag various tag content summarize manually content search separate tag system service social service various improve similar user knowledge resolve computer system sometimes collective intelligence collective intelligence environment normally since certain service measure tag user experience google search perform search service tool audio file allow speech mistake game google element contribution ignore thought drive behind example site comparable user user extension characteristic collective knowledge modern internet service form model analyze growth tool analyze example follow evaluate use previously way video video site team evaluate tag basis importance site proposal tag text site universal various service rather compete section analysis software package essentially convolution arrange file format value store hash hash convolution read convolution arithmetic operation long theorem useful time spaced example convolution memory consumption thought calculation feasibility accurate value calculation include convolution complete sufficiently equip convolution width determination skewness degree central meet derivation derivation probability transformation however explanatory information probability win person change place significance impossible apply hoc convolution calculation hand win first adjacent chapter discuss software practical social service site store involve various tag recommend similar tag service boost site important thing common wide internet service service share site evaluated comment share comment tag site indicator content site website tag comment determined sum model sale business management product customer provide sale strategy sufficiently theorem law meet sale internet rare average customer attempt significant continue stress set comprise user include user analysis aim identify tag would include illustrate tag activity basically expect law number customer decay tag increase tag allow maximum tag tag develop among move illustrate main frequency site distribution treat clearly user large central relative tag histogram long histogram choose tag observe obtain easy rare term value tag assign tag tag parametric test multinomial distribution another readily calculate modification trying determine tag appropriate problematic allow compare assign assign skewness skewness relationship customer total tag number index site tag high relative customer correlation read basically zero variation indicate search site customer site quantitative extracting define site rare figure illustrate rare customer positive increment axis axis recall rare account site rare value account customer determine site rare probability volume customer customer often rank upper percentile lower probably difficult determine threshold line fail student pass fail similar event ordinal tag probability via customer next calculate meet eliminate operation plot common axis sample central limit dominant grow account total majority small substantial axis limit avoid effective analysis item consideration generalizing examine application follow one content receive induce description tag make easy speech recognition correction
action relative game inequality nash payoff maximizer subject instance dynamically maximizer cycle maximizer period period bind crucial maximizer action action paper game player obvious sum sufficient maximizer cycle submatrix probably paper call contain game subject violate directly relative payoff maximizer never suppose maximizer choose action period improve relative payoff period symmetric payoff action non path cycle along cycle relative player action finite game subject positive payoff improvement cycle cycle contradiction g assume maximizer maximizer could payoff already must rule prove cycle converse game payoff generalize cycle cycle play cycle column exist generalize generalize payoff strictly payoff contain maximizer select row relative next period play maximizer action e period another maximizer cycle argue necessary saddle saddle generalize saddle subject paper submatrix profile rise submatrix subject game game symmetric sum game follow finite sum game example game matrix row diagonal payoff essentially game maximizer period maximizer period period maximizer must contradiction maximizer payoff matching counter consider payoff game additive restrictive say function separable let separable payoff symmetric induction suppose step maximizer maximizer merging apply yield maximizer directly maximizer ever relative payoff separable indeed exist corollary proposition application compact function continuous function e pure equilibria convergence totally resp xx difference consequence sum game difference separable payoff totally decrease theorem potential game symmetric potential py py symmetric payoff game essentially proposition demonstrate payoff write apply essentially cost price write ax ac cx benefit assume function study decrease essentially pool resource common resource game two outside activity common pool resource denote maximizer pool resource likewise return resource effort effort game van among individual relationship opponent symmetric payoff corollary essentially arm arm totally concave function player search player search trade another payoff game separable separability payoff counter game game player relative separable yet maximizer zero maximal useful obtaining equilibrium explore implication notion potential besides notion potential potential game symmetric game game potential px px py py ordinal potential game py py ordinal generalized px py x py ordinal potential ordinal potential paragraph ordinal possess nash finite ordinal potential possesse sequence profile exactly player sequential path strict player action strictly payoff strict contain focus relationship payoff game possess ordinal potential cycle cycle cycle cycle construct action action sequentially turn maximizer strict note e maximizer maximizer cycle I converse counter follow possess construct strict game ordinal game subject cycle finite game improvement cycle imply converse generalize payoff game converse true paper strict possess ordinal payoff potential exact weight potential ordinal symmetric game weighted essentially game definition quasi payoff space symmetric exist peak exist space abuse relative payoff game symmetry payoff x x xx x game subject reach finitely step order always large strictly negative likewise upper analogously maximizer maximizer never maximizer maximizer claim exclude exclude maximizer choose choose claim hence simply maximizer element far step period repeat reach reach stationary leave relative euclidean f fx finite dimensional imply model nash game nash demand game nash player demand amount within receive relative player symmetric payoff finite payoff subject payoff remark improvement ordinal game neither paper many game economic possess natural aggregate player game game resource view aggregate study os absolute payoff second totally monotone argument ax ax ax yx sometimes shannon aggregate show action inverse cost require forward game converse single inequality q since relative payoff improve reach improvement possibility maximizer claim action lemma maximizer payoff else new would maximizer choose new step new start repeat stop complete contrary eq contradict prove present extend payoff symmetric demand cost player win proportional bid player bid bid symmetric payoff game al os game game corner finite strictly contrary q contradiction ax ax contradiction ax xx strictly show step maximizer payoff nontrivial maximizer repeat relative nontrivial maximizer take corner show either contrary strict contradiction payoff player payoff throughout nontrivial maximizer corner finitely sequence reach finitely corner reach summarize symmetric game function symmetric complement importantly relevant game complement fair seem game prop payoff heterogeneous public pool resource effort relationship arm game potential payoff prop ordinal nash demand games modular game prop modular generalize thm game need aware limitation analysis primarily game scope paper show game inverse demand write quantity profile cycle induce yield clearly require among thus recall truly sophisticated whether sophisticated experiment payoff go wrong consider suppose start equal maximizer choose price maximizer could marginal cycle example show crucially heuristic class leave conjecture chen interesting discussion national international conference helpful comment symmetric player game decision necessary variety example game competition pool resource game effort arm opponent payoff game ess games games game behavioral stress rule human make capability heuristic decision plausible adopt bad situation long heuristic heuristic look rational currently mutation pool rational exploitation opponent evolutionary follow heuristic even rational look maximizer class relevant play computer human subject easily exception action
say scale tree say law corollary part straightforward convenience define posteriori decode optimality span edge apply coarse exactly cardinality forest formula tight give converge complete natural law different manner regard attribute describe abstract forest oracle knowledge need fraction conclusion
pa array dim array dim pa pa pa nb pa col col pa density conditional e deduce code exp cx col py density normal variance yx yy sx sx cx true col true x independent distribution large conditional distribution large obviously truncate poisson conditional distribution gamma distribution sample gibbs sampler code grey main col rao checking execution program ten correct initialize initialize st col grey st st per se integrable infinity constrain conditional give well simulate joint initialize I exp st exp col add type invert cdf represent gx yy yy predefined sigma mu inf inf sigma precision col col x col col add alpha alpha alpha sigma alpha sigma simulated histogram integrate histogram satisfie detail simulate markov alpha else nice fit concentration stationarity pass fail st fraction nd window var give moving modify alpha x else stationarity var pass fail window fraction window posterior sigma r beta sd sigma mod sigma sd mod sigma sigma beta else beta prop sigma prop sigma prop sigma jacobian probability additional running diag est mix chain completion factor seq seq ks nan le ks converge converge pointwise value connect exercise get example sequence beta check via program ess ess subsampling justify mcmc figure compare evaluation chapter none strongly manual book monte publish solution paris student master whenever appropriate code name grateful student solution manual incorporate manual manual demand strategy find book come ignore book universit paris www books complete reproduce manual stress r come solution solution behind argument effort put manual study introduce algebra conditional notion cover reader lose obviously suggestion manual independently correct almost code page prefer code code student fast code along page automatically substitute code request two odd one access version since become book possess manual please explanatory self explanatory execute seq variation array dim replace interval quantile histogram quantile get region array dim array dim replace confidence interval datum book type environment base information sd function false sd na else na else sd na na else sd look description database modify format recommend assign rather assignment create sep mostly explanatory recommend function save write need transpose internal fairly see try allocation fast inf inf numerical lead fail high enough lattice library produce representation tree index tree estimate std value pr intercept tree plain non false array consider self explanatory instance loop whether integer exist entry box entry remain plain grid sum pool wrong break stop outcome solve cdf u fy fy easy calculation histogram tail pi array dim tail histogram numerical box compare exact range acceptance accept acceptance bind therefore attain easy contour maximum value truly take exactly try code cp array histogram array cs use create alpha cs alpha alpha see histogram use see pareto uniform variate function spread sum sum lambda lambda user lambda user lambda lambda exponential good appropriate return reject obtain maximum since require modification derivation e lead optimal mean ratio maximal lead exercise density square hold derive also chi rotation vector lead variable central chi include centrality nc chi square scenarios ii df df fast since likelihood density improper prior mean mean proposal inefficient credible repeat experiment lead col ty l legend numerator cauchy approximate directly x type col col gold evaluation ratio estimator separately variance get digit normal replace comparison clear normal simulation compare identity see expectation distribution thus approximation pi hx gold col estimate probability compare inclusion exercise book exercise I iid q fy c la la est type px sep col convergence plot clear like appear picking seem produce tail tail straight integral exist sampling eq integral integrable go acceptable quantity x according accept reject empirical mean deduce pi c code exp gold col gold inf pi gamma inf exp col gold col jumps jump differ evaluation absolute pt low e variable accuracy use q approximation simple accounting lead exp exp evaluate w effective preferable choice w efficiency simulation estimate unknown self simulation hence bias harmonic let tell n marginal try marginalization rate ok check look plot seq estimator produce decompose term q result likely correspond likelihood q quite similarly consequence jensen slight read establish since exercise read complete square exponent opposite monotone another exercise exercise exercise convergent estimator denominator exercise indicate exercise programs rao col grey b col grey col sensible b sd type col grey col grey add b col miss distribution beta x grey n col grey gold clear notation replace accept acceptance time associate numerator among probability reject subset uniformly accept choose w j hx hx hx hx hx hx hx hx us density optimal accept reject second moment know approximated instrumental density simple mu add da mu surface sample impact surface binomial constraint find obviously hence therefore subsample constraint simulate simulate circle pi col performance simulation domain via acceptance rate mixture minus mu package reproduce program like log factor calibrate convergence optimization exhibit mode circular program path mode mode mode matrix prox mode c prox schedule sa sa illustrate experiment four outcome mode indicate principle involve power logarithmic involve thus impact remove start point stop base binomial x col grey col gold range true program obviously likelihood apply surface five mixture question capital namely datum relie problem c em produce component notation question density miss normal upon know therefore normal exercise exercise simulate chain sd add hasting accept property transform decomposition derivation detailed exercise integrate x metropolis simulation alpha alpha col lines alpha drop alpha b alpha function c b alpha acceptance change respectively reject candidate max col reject col add simulation give hasting gamma implement col ga efficiency implement g metropolis hastings ga col add quite use early miss correct txt header final add save simulation graph assess col main col strong across rate unstable converge interval question must exponential available run glm binomial pr intercept dispersion freedom freedom aic beta result check col red failure exp add col gold temperature metropolis x sd beta proposal else b b else acceptance acceptable explore c col col l col failure seq exp col grey add seq
absolute distinguish coefficient context test invariance proof hardness inspire game geometric result list concept class learnable noisy consistently explain common address agnostic agnostic class hypothesis require nearly say hypothesis perceptron boost study widely system label linearly determine agnostic agnostic agnostic class constant correctly label exist agree note hardness agreement always example draw essentially subset decision imply hardness proper decision list describe detail hardness learning exception hardness learn rich broad hardness proof invariance hardness invariance principle np hardness cover strong condition unique learn decision list study agnostic equivalent ability come function refer class equivalent long know complete later improve david et result hardness approximate agreement every coordinate whereas work hypercube maximum li hardness approximate agreement david et subsequently finally tight hard distinguish consistent prove hardness allow clause decision hardness hardness problem exception number hardness complementary minimize disagreement evidence hardness agnostic even proper agnostic major al distribution decision list learnable presence random perceptron separate margin robust mild noise hold adversarial et give non agnostic learning give agnostic hypercube analogous algorithm analytical purpose illustration test suffice unique game encode boolean conjunction refer theorem integer connect hypergraph vertex vertex label vertex say strongly vertex sake clarity sketch special games conjecture constant integer positive strongly label weakly fraction strongly agree agree clearly statement convenience hypergraph e coordinate coordinate vertex coordinate every labeling thus label correspond idea fraction weakly fix identity permutation every complicated ex coordinate write procedure output negligible conceptual clarity label f r rf v intend problem linear function satisfy completeness close nature refer hardness approximation notion close influential coordinate coordinate little resolve use critical recursively influential none critical denote say close index influential coordinate influential appropriately counter employ recent independence outside agreement negligible operation query pr distribution carefully infer structural independently bit sample construction pass probability write expand I note large fraction substitute two distribution invariance principle unable c condition value conditional column still invariance principle coordinate j enforce hardness hardness game reduce label cover complicated consistency check encounter hardness commonly map label vertex several label mostly identical correspond edge natural extension unique game hardness game fact execute reduction precisely cover vertex principle certain fourth moment smoothness term reduce np great uniqueness property mention convert game hardness unconditional hardness inspire avoid geometric tight factor relate fundamental space among semidefinite unique game hardness np tool notion coordinate w k w critical contain geometrically subsequence weighted fall interval proof regular define pr distribution fall probability bit bit fall set coordinate regular define pass statement pass least list intersect lemma pass intersect immediately less approximated critical h matching introduce whether suffice show moment match degree influential fix rewrite c possible randomly ensemble c ensemble coordinate one lemma moment moment also unbiased ensemble spread principle average get two influential unlike traditional unclear whether number influential define generality geometrically imply know eq h ok discuss case overall index show conjecture although hardness assume describe purpose proof e r define produce label conjecture integer labeling fraction consider v v r agree agree probability ok labeling vertex randomly reduction good rt additional smoothness problem agnostic give hardness property exist constant integer hard distinguish e un strongly every weakly weakly least fraction addition fix vertex pick contain map find start un e l produce example refer projection follow v output example completeness smooth weakly parameter agree probability combine main remain correctness claim completeness I ei agree label strongly obtain agree complicated section almost edge group group copy almost lemma formal nice appear appear generalization section let agree v vertex nice respect I projection vertex nice vertex nice vertex follow property vertex average argument ok property therefore overall denote generate agree denote projection brevity shall v k v c j notation fix ensemble r invariance ensemble claim degree ensemble spread invariance principle imply eq claim hold average setting ensemble moment degree moment moment agree condition condition c n therefore v I substitute give I tt ok e matching show identical condition without w define I geometrically notation similarly h claim pr g correspond lemma every chebyshev inequality eq recall ok case h q lemma replacement change overall combining agree first statement weakly define suppose agree fraction agree nice good nice exist intersect labeling overall strategy since discussion hoeffding let real random unbiased chebyshev real generate copy generate write claim pr I vector pr claim claim verify apply inequality z independent suffice pr fall invariance principle r ensemble random satisfy ex random ensemble ensemble k c
move proposal random metropolis simplify refine random metropolis proposal add third purpose tail third explore make easy leave mode walk sampler normal component vector identity walk value show random walk walk local mode proposal adaptive hasting density stage estimate target adapt heavy tailed stage density construct iterate adaptive component ten time normal normal obtain stage begin iterate maximum schedule describe tailed density include strategy mode effectively computationally article normal third identify estimate normal harmonic stochastic normal set tune approach report tuning see sensitive particularly naturally distribution flexible density transform proposal multivariate low degree freedom whose help explore copula dt cumulative sequence copula marginal normal freedom copula estimate marginal distribution degree freedom profile copula value small iterate mixture weight component draw location finding copula schedule proposal often chain symmetric generalize allow implement sample multivariate generate respectively accept reject accept third would fix throughout stage component unnecessary third performance without discuss rwm walk rwm metropolis normal cluster mn cl describe independent metropolis hasting distribution copula normal cl sampler proposal term compare take sampler accept metropolis proposal sample divide generate factor autocorrelation lag otherwise lag low sample iterate rate account define interpret sampler attain attain draw sampler sampler take extent language affect sample logistic regression different prior coefficient second intercept double laplace lasso double exponential spike tail normal distribution third prior component normal assume suggest variance example work unconstrained covariate list discuss l year year hour work marginal market experience hour run three target double normal obtain fitting normal prior ht exponential intercept acceptance rate size algorithm min max median max rwm rwm mn cl cl exponential rwm rwm cl cl cl normal rwm rwm mn cl cl rate adaptive walk result compare adaptive metropolis hasting proposal version well home home relate otherwise list ratio american history slow pay pay slow account insufficient history payment record otherwise self otherwise otherwise year otherwise history equal history start initial distribution adaptive generalize matlab normal summarize marginal normal cc normal exponential mean intercept table equivalent sample computing walk high metropolis especially normal prior copula high rate normal multimodal ht min max min median min max prior rwm rwm c cl cl cl rwm rwm mn cl cl normal rwm rwm cl cl cl proposal freedom give copula coefficient reference therein prior parameter take mixture normal distribution worker survey logarithm list carry bayesian independent run double exponential case table marginal posterior estimate age worker year square age education high school post secondary complete college omit cover collective omit otherwise trade service otherwise omit mid west south west east never omit category less year otherwise visible otherwise cc normal normal mean intercept acceptance equivalent size computing scheme distribution independent adaptive proposal acceptance rate low l min max min median rwm rwm mn cl rwm rwm mn cl cl rwm rwm c mn cl adaptive integrate effect response probit cumulative parameter ij covariate q proposal include adaptation iterate mean similarly accept choice design may whether choose cancer presented ask would binary covariate l know otherwise patient due patient cost effect double mixture normal scheme unconstrained hasting algorithm random algorithm standard error identically random proposal importance initially table summarize cc cc parameter normal double normal mean table show acceptance computing time high exponential acceptance least min min rwm rwm mn cl cl rwm rwm mn cl cl normal prior rwm rwm cl cl propose copula scheme generalization design proposal hasting normal scheme reliably study much acceptance rate walk copula copula normal complicate multimodal arc grant dp thank computation intel ghz ram platform compute function inverse cumulative mixture normal file matlab speed normality reject fit density
interest approach tend g composite block exponent agreement reliability bound constant curve achievable rate block signal also compare rate poor curve demonstrate channel complementary function superposition axis give stay block close curve codebook whereas considerably small say square decoder unknown whether decode near capacity decoder convex discuss decoder superposition break achieve correspond residual specify residual difference contribution column attain exponentially exponent slightly optimal moreover least square achieve exponent superposition code level rate capacity least successive aspect power rate capacity need achieve allocation decode reliable decoding rate capacity express square make challenging convex constraint nevertheless constraint decode convex move decode unclear reliability power communication language dictionary linearly combine issue field number reliably zero partition complement recent recovery case channel highlight coefficient value control coefficient minimum conclusion reciprocal well allow converse constant capacity projection constant achieve square analysis least fix capacity capacity capacity input white band specify spectrum decomposition code suffice near need concern communication near channel decode empirically moderately decode rate mathematically prove case channel code base low check code aspect art reliable decode restriction alphabet distribution exponentially small fix contrast code decoder decoder moreover beyond uniform distribution investigate extent regime one alphabet require pack power implication marginally jointly empirical notice superposition idea superposition adapt quantization applicability quantization pack development benefit code pair shannon binary code target available challenge concern code exponentially large step toward practical rate inner drop capacity inner must order least consequence outer noise difficulty superposition comparable outer fraction achieve remain comparison false discovery significance development number arise incorrectly provide additional possible subset selection within section address discovery superposition gaussian user channel send sum put shannon purpose feasibility identification achievable another channel channel power arrange successive decoding relate successive decode superposition code apply individual however part reliability high rate design attractive single amenable channel access brief provide core superposition code reliability discuss composition outer code correction mistake matter henceforth logarithm suitable base calculus log conclusion state base use derivation moment generate function deviation exponent construct normal variance correlation coefficient otherwise understand minus infinity bivariate joint mean let increase maximize near optimize expression simplify composition evaluate ratio expression match occurs include say decode superposition least reliability superposition code allow term first subset correspond zero specify choice subset solution achieve superposition code number section incorrectly channel capacity shannon capacity statistic approximate interval lemma refine mistake send value achieve proceed bound probability appropriately design union give interest subset intersection difference mean give conditional normal govern send respect test term whereas depend express adjustment normalize equivalent density construct hypothesis give helpful rule provide ratio reverse event contribute term side outer among prescribed bind side bring outer iterate involve simplification jensen expectation inside yield recall independent accordance denominator denominator numerator entail choice normal correlation claim exponent mistake occur behavior similar order theory difficulty correspondingly combinatorial analysis denominator pay term interpretation logarithm likewise event interval test split event event event dependence average difference much decompose form standardized normal multiplying maximize consequently reduce quantify function find generate ts give rise appear expression complete proof dotted explain mistake consideration correspond proceed exponent exponent superposition code incorrect expression replace exponential partition superposition bit member bit equivalently note control polynomial size want size order inverse gap capacity instead decompose r coefficient combinatorial plus exponentially v ask exponent difference shape comparable c ce v two likewise v enjoy bound consequently find sufficiently nonnegative small choice end accordingly exclude nevertheless ratio end value insensitive maximum whole limit strict positivity zero within set interior determined ratio end right replace bound accordance ratio derivative respectively accordingly derivative development determine suffice determine whether ratio less certainly ratio case recall associate derivative second third eq evaluate evaluate first give right magnitude indeed take root claim v claim contrast small magnitude produce claim form expression agree argument extend continuous moderately equal near thin error exponentially bottom minimal bind specify high fraction minimal moderately observe reduction require extra via exponentially characteristic gr gr lr ar bound derivative rate equal quantity critical interval sensible establish exponentially reveal exponent stay gap exponent positivity produce match regime prefer conclusion least mistake recall string coefficient previously vector zero freedom magnitude draw send superposition term receive string receiver section implication dictionary least mistake section rt rate capacity bound integer lemma follow constant fraction mistake probability exponentially exponent precede choice fraction rt make requirement c r ref consequently lie tangent derivative see otherwise latter include situation bind line since non decrease development tend tend consequently multiple inferior exponent follow bind control case half less fraction arrange accordingly quantity two exponent ii bind way first choose previous match addition say exponent exceed bound minimum likewise minimum exceed accordingly sufficiently small become least remark tail distribution geometric sum arrange yet mistake small introduction inequality proceed proceed lemma substantial mistake aside exponent proportional mistake small device considerably indeed mistake mistake suitable code thereby small probability compute random sequence individual well ensemble implication review code discuss role rs code outer code mistake rs code element message rs symbol q add find convenient symbol give code outer superposition block first equal view represent symbol outer composition bit symbol outer superposition receive square decoder think symbol rs mistake property code correct since code albeit rs code minimum distance symbol string identical code symbol composition obtain partition superposition positive partition superposition concatenation code obtain composite less equal implication implication section dictionary receive sequence send exchangeability column average write appropriate average conditional expectation likely behave similarly indeed markov p verification simulation repeat geometric implie small codebook leave line communication average maximal say armed empirically dictionary satisfie requirement proof facilitate provide make joint permit simple power condition accordance dictionary match expectation enjoy markov inequality except exponentially control power case formulate dictionary examine role decode power square terminology arise setting transmission wireless equal sign code first sign superposition code section nonzero sequence distribution index choice likewise power across section simplify average deviation chi accordingly chance exceed via chernoff dd normal approximation average outside deviation instance capacity high hold precisely carry mistake long average code likewise subset superposition without sign distribution input independent zero likewise overall need square expectation plus norm input random variance l subset independence draw chi chi equal lb yield average matter power among normal coordinate chi probability nr e nd rr near rely decode sign provide power sign interference produce section lead equal choice sign property conditional subset contribution section mean uncorrelated sign conditioning mean shall deviation equal presence sign lead variance concern square uniformly exponentially union column except conditional allow size small bad show capture typical conditional inner value normal moment equal moment difference independent half x e gr j nd union dictionary conditional equal give concentrate
ray coordinate regard space joint live f response separate overlap split signal especially different log covariance know extract signal inference optimal usually derive principle except rigorous gibbs technique joint space easily spectra reconstruction datum gaussian noise principle help uncertainty minimal approach e power basis disjoint support band band cover space band inverse assume independent inverse exponential cutoff determine jeffreys end calculation effective hamiltonian accord band internal gaussian average expand introduce two free energy ignore correction mean wiener filter spectral assume invertible filter provide unable unobserved space widely field estimator perform irrelevant energy respect template respect calculate integral energy systematic taylor fr expand around energy second alternatively average minimize ks calculate surrogate energy scheme might argue build could markov carlo right synthesis build direct mcmc analytical analytical combine taylor expansion gaussians mixture moment express leave verification synthesis future minimal principle theory maximal relatively straightforward suitably parametrize degree internal entropy hamiltonian surrogate pdf variance free principle tackle normal poisson spread early calculation understand surrogate gaussian correct previously well propose complicated combined verification future easily maximal allow construction reconstruction inference spatially help concept permit tackle perturbation discussion manuscript lin tackle technique within field minimal free understand gaussian full cross optimize three normal background count ray galaxy iii unknown construct free measurement signal encode information retrieve strategy construct minimize prior ground state would function literature entropy result reverse origin certainly sense often numerous possibility much entropy since lack irrespective force inference ideal provide combine posteriori principle energy regard functional pdf minimization entropy latter logarithm signal pdf usage concept freedom approximate gaussian connect theory degree former complicated sect entropy motivate principle derive sect maximal application principle optimize approximation concrete provide sect log sect reconstruct sect posterior obtain sect conclude sect possibility plausibility sure impossible uncertainty obviously theory generalize logic different possibility introduce contain aspect reality aspect retrieve possibility vs datum sd denominator fraction highlight statistical mechanic hamiltonian usual mechanic signal lead ad hoc temperature permit narrow phase space pdf sect field correctness sampling guess suitable energy might hamiltonian minimize hamiltonian signal principle denoting mean thus solution field augment information hessian uncertainty introduce often ie image I ie image equality field ie usually argue ad hoc assumption entropy package plane knowledge lack reveal prior method ideally likelihood ensure maximized constraint temperature parameter weight closely map want enter formalism prior identify assume irrelevant additive constant pixel etc generic either peak several prefer reconstruct commonly signal physical measure spread pdf boltzmann posterior signal introduce internal energy free fully dependent free provide field value entropy entropy functional restrict space obtain entropy state complete lack uniform signal possibility return alone expect analogy use quantity minimize free calculate pdf necessary go sure understand construct free energy suggest partition function temperature narrow posterior importance pdf delta peak locate entropy permit mean temperature low center pdf tail guarantee reconstruction slightly since reveal aspect pdf central differently accurate sect partition read far motivate temperature free calculate connect directly take moment function g restrict free energy explicitly energy internal energy entropy internal posterior average mean dispersion fs fs pdf modify posterior q md solving yield energy gibbs internal therefore gibbs variation eq energy respect optimal map hamiltonian gaussian location set evaluate gibbs hamiltonian q equation seem temperature denominator opposite hamiltonian gibbs free another measure leibl kullback divergence characterize role equivalence cross inference step since last leibler divergence kullback maximal cross however minimize use posterior relation regard covariance minimize principle also hold gaussian later sect relation differ calculate need specify taylor fr expand coordinate integrate energy point calculate analytically field index since result internal odd product ns symmetric internal entropy construct accord optimal internal solve derive depend order hamiltonian match free hamiltonian indeed coefficient case interact minimal use inference noise ray ray count galaxy proportional spatial position permit linearity description support theoretically start count likelihood separable internal calculate analytically put calculation response exactly ray exhibit spread single detector come indistinguishable direction galaxy distance generalize response treatment galaxy case see galaxy problem via map principle spread model
htbp structural extend eliminate column constraint margin corner entry require ultimately stem equation submatrix kl namely nj odd expression hybrid within gibbs know central metropolis eq carry sampling four assume take posterior figure htbp posterior report list credible table credible simple configuration configuration level posterior propagate cost set restriction sample gibbs sampler realization estimate cost credible cover great pattern previously would analyse system cost th estimate posterior aggregated proportion credible proportion evident due section principled narrow gap htbp see belief study relate scale estimation proportion uncertain due natural viewpoint explicitly proportion yield update proportion conditional nevertheless integrate nuisance derivation numerator resort method complexity main uncertainty specify behavior study region variability incorporate pattern instance available region obtain extra defining alternative traditionally refer albeit fully reflect hierarchical candidate multinomial dirichlet mass q informative attain distinction preliminary approach commonly estimate remain count furthermore bayes posterior offer principled incorporate seed distribution count follow multinomial flat adopt distribution adopt gibbs new level iteratively previous section become list sample conditional know walk central step normalize z ij kp kt simplify realization chain acceptance update initial configuration ij ij candidate otherwise reject set random initially take low solely credible arise use observe square htbp informative wider credible length bar square posterior dot see proportion parameter make variability wide credible htbp suppose preliminary keep flat posterior count affect cost summarize show credible marginal compare proportion solution probability solution c conditional display randomness attribute inferential posterior bar square posterior list influential two list c static estimation traditionally regard optimization contingency formal pattern make classical function artificial configuration incorporate principle entropy maximize principle able classical map history behind traditional yet benefit solution solution number alternative preliminary purpose insight region however dramatically traditionally nonetheless fix acknowledge uncertainty make random hyper build contrast informative seed configuration accurately carry besides also able hypothesis explore flexibility bayesian really explore configuration pay need closely generate need assess propose try comprise future direction include efficient scheme improve proposal fast implementation version serve traffic step jointly assignment modeling propagate step aspect refined camera variation dynamic acknowledgement motivation nsf grant static viewpoint novel cast study factor maximum identify solution usually next propose devise obtain several approach highlight source incorporate consider region divide include usually restriction row margin eq constrain immediate static provide broad treatment study many decade contribution generalize decrease cost dc ij incorporate observed regard possible previous study balance factor know define iteratively balance convergence heuristic formulation maximization micro state associate configuration equivalently entropy maximize q would coincide make regard form mathematical program certain constraint second many configuration implicit formulation propose show even instead optimization classical model solution maximum estimate besides consequence generally quantify propagate framework regard usual fully approach consistency satisfy randomness come initially belief observe margin arise next incorporate small study multinomial proportion nonnegative hyper improper informative micro important role area behavioral perspective correspond random multinomial logit model set covariate pair cost incorporate another inference update observe prior self maximum note balance entropy formulation proportion recover define entropy maximize principle maximize uniquely measure amount entropy justify partial consistent familiar subtle difference formulation constraint effectively configuration proportion feasible proportion guide soft argue formulation knowledge seem posterior
operator mm condition asymptotically semidefinite via recover follow one hold psd find psd unique minimizer conversely none psd psd similar threshold psd set hermitian analysis find empty suppose otherwise maximize increasingly tight simulation curve via optimization theory fit almost perfectly suggests quickly limit minimum study need recovery constant suggest trace actually uniqueness suggest psd sure positive interesting need mesh way important recovery threshold weak weak nuclear minimization recently gain problem simulation far sensing threshold suggest grow linearly size need three recovery weak threshold analysis special semidefinite matrix discuss simulation address measurement project matrix dimensional gain attention practical measurement recover fact low program turn replace nuclear heuristic close relaxation nuclear refer program sdp study recover isometry success rip result minimal sampling recovery nan gaussian success threshold establish explicit oppose wise new recovered minimum novel space minimization find threshold basically give separate vector compress positive semidefinite analyze xu basically strength match exact compressed sensing simulation indicate threshold tight shall nan space compress sense call column I e unitary unitary basis matrix decomposition unitary positive increasingly denote call nuclear frobenius act iff ensemble variance circle histogram square converge nothing normalize value similarly normalize note limit support word mi mn degree freedom normalize order set particular hermitian several lemma make later proof large ix particular find obvious norm norm mesh subset unit sphere uniformly haar function introduce analyze strong modification rectangular probability follow fact iw iw rx unique nan regime determine compressed space haar view establish matrix sure nan intersection careful actually equality get optimization clearly hand sort increasingly directly addition one choose nc z c else reasonable lead reason worst basically contribution region expectation separately follow eq gaussian vector singular z fx p value constant iid ph h ns na use exact schwarz large combine upper thereby measurement substitute substitute use otherwise sufficient operator numerical strong plot support recover measurement iff q note suitable write hold minimizer tight find start gaussian analyze nan subspace haar establish necessary intersection bind discuss unitary uniformly I unitary transformation unitary show successful depend assume sec w increasingly similarly let denote hand w w need problem lemma increasingly result let program q combine mesh technique sec sec h circle iid analyze let sec nr f q combine h show sec use combine asymptotically upper use otherwise gaussian n q numerical calculation threshold section weak threshold recover weak simulation strong threshold recovery sense prevent repetition derivation repetition threshold unitary tw matrix left analyze upper diagonal unitary transformation basis transformation assume block firstly sort absolute similarly notation note essentially basically correspond repeat base arbitrarily iid need use let therefore exponentially combine prior yield threshold least mesh freedom nuclear simulation measurement region region include notation symmetric denote semidefinite psd stand semidefinite hermitian denote unitary hermitian entry diagonal variance order create one let unitary order singular define triple limit cumulative definition x ix follow counterpart semidefinite root since psd give program psd matrix matrix x psd matrix want positive semidefinite solution analyze call psd nn via unique unitary hermitian psd eigenvalue submatrix submatrix immediately psd transform long minimizer psd recovery without tw however last small perturbation change hence last psd matrix psd whenever tw tw v tw tw analyze separately nan argue nan restrict iid also imply unitary identical
perhaps I finish stanford would ask know old von day history von generation monte von page publish stanford von expand brief exponential distribution yield intend report joint u discover stanford stanford report mathematics publish issue last publish exception lee publish publish follow von day still publish von illustrate normal von differential von obvious sample evaluation number generator normal von generate enough algorithm von minor interval convenience correct convenient generate sign historical take trial trick correctness experiment well store power reduce reduce call uniform generator von uniform normal von expense describe interval example eq principle necessarily development people give sampling book improve box fair say none well tradeoff depend machine generator pool maintain transformation number function spherical symmetry normally suffer paper corollary power series von elegant compute polynomial refinement
monotonically exist distinguish go section primary weak synchronization synchronization exponentially fast machine ref essentially say observer observe exponentially variable belief states generate state combine synchronization theorem strategy exponential theorem subsequence argument markov chain finite irreducible chain equilibrium fr class irreducible chain state irreducible chain extend periodic block initial linearity lemma strictly joint eq machine constant claim claim edge exist b exist constant large taylor l exist hence q consequence thm exponential exact existence everywhere synchronization establish machine pointwise sense def exist constant thm constant prop lemma machine rhs eqs give word define observer prediction fast observer borel thm asymptotic synchronization treatment involve result exact ref observer uncertainty vanish observer prediction exponentially class countable machine hmms acknowledgment support advanced research project physical intelligence via finding interpret view imply department synchronization synchronization observer source vanish observer average predict ref synchronization machine observer internal state number measurement observer analysis differ qualitatively sense observer machine constant entropy follow provide essential definition result picture synchronization use state section average section synchronization use theorem machine entropy vanish exponentially approximation exponentially entropy sec background result thorough reader ref presentation mm xx jx state depict direct state edge symbol require strongly generate start choose stationary machine pick symbol edge consequently fashion machine visit symbol chain normally observer even extensively derive even consecutive transition even transition probability generate mm markov symbol l machine machine originally probability history equivalent finite immediately apparent synchronization establish ref extension machine edge follow indicator state iw markov whose probability visit move generate symbol strongly well edge relatively output transition distinct relatively fig bottom h length pair distinguishing must finite distinguish observer internal state observer internal study procedure observer know observer able machine time state observer observer define observer generate word observer know infinite set weakly sequence observer observer uncertainty sequence exact machine almost machine necessarily exact observer question thus exact always finite exact word finite quantity observer average uncertainty length output observer randomness turn determine average observer symbol observe stationary monotonically observer asymptotically observer optimal observer symbol induce observer know current machine close machine relate average rate primary study synchronization intuitive picture formula basic observer machine five symbol merge symbol symbol state path observer w p kp p observer exactly merge impossible path observer asymptotically relative path however synchronization concern absolute normalize synchronization quantity probability machine probability path eventually merge state asymptotic synchronization average time likely path synchronization formula analogous previously
overcomplete frame theoretically experimentally overcomplete dictionary ensure optimal orthonormal basis endow property frame propose generate dictionary operator optimization adopt keep terminology duality devote dictionary although overcomplete dictionary pca derivative reconstruct seminal field learn overcomplete probabilistic lead cost perform alternate difference advance compress lead pair decode transformation efficient natural dense introduce encoding module build define penalize augment code product input term time code new vector require minimization rely proximal year empirical evidence proximal underlie algorithm considerable amount devoted topic reference several use sparsity particular introduce tree structure attempt cast experimentally recover dictionary code dual classification intensive hand implementation training goal encoding operator optimality filter learn dual induce sparsity respect reconstruction since know gauss sparse dual dictionary successful towards respect force rely consist convex constraint presence make minimization proceed contribution differentiable part forward backward split convex solution optimization literature dictionary extraction smooth proximity projection descent inner coefficient adaptive discuss section minimizer minimize gradient proximity correspond plug obtain quadratic equivalent denote constraint formalize indicator set resp belong proximity update choice achieve minimize evaluate explicitly choice towards rate similar eigenvalue convergence fista modify fista fista sum two iterate define step replace modification allow quadratic convergence achievable theoretically experiment confirm replace sec break complete outline optimize carry adapt equation popular warm atom reconstruction atom mean element replace achieve iterative stop upon reach iteration find hundred reach case require optimize code confident accelerate easily experiment context discriminative image order dictionary vector obtain learn aim dual control cc converge dot decade synthetic bottom recover necessarily atom first corrupted gaussian algorithm coefficient stability principal converge transpose pca assess angle span decrease image reconstruction superposition element corrupt frame minimum frame experiment frame recover dictionary dictionary apply set image benchmark berkeley quantitative assessment berkeley experiment patch berkeley segmentation intensity center relative tolerance stop set level sparsity low dictionary interesting report code coefficient visual pattern look like tend poorly specific atom seem encode dataset sample atom column least dictionary pair mnist mnist binary handwritten train dictionary atom likely overcomplete pre process literature comprise representative digit figure bottom digit atom extremely middle first report dictionary respect relevance aspect empirical dictionary digits change reach substantial iteration correspond identical iteration iteration group
since orthogonal lead note u theorem establish nuclear norm know use solver become prohibitive hundred variable optimization propose outli pursuit special rate interior paradigm outlier diagonal otherwise column zero quite minimize promising property outli pursuit randomly fix generate entry generate copy random outli htb ccc outlier b identical outli noisy gray denote outlier pursuit succeed adversarial succeed fail pursuit outli sample noiseless matrix result separate subgradient variational result note rx lemma last column lead orthogonal pursuit succeed pair correct column must output pursuit assumption strict analysis solution outli subgradient evaluate part condition strict unique w fu eq strict next imply strict objective proof equation complete exist show two assumption equality hold satisfie satisfie hold strict hand satisfie together establish equal otherwise u orthonormal hence follow remark component successful computable nevertheless well outlier recent consider arbitrarily corrupt pca collaborative bioinformatic agent corrupt contaminated yield completely corrupt outli pursuit mild assumption g exact identify corrupted identification corrupt application beyond nuclear line correct seek rely optimality fail present treatment give zero column aside broad restriction arbitrary know non recover identity zero column exactly efficiently motivated component analysis arguably reduction seek point decomposition find approximate form form standard pca arbitrarily quality persistent corruption failure source mean hence column ask exactly exactly identity establish natural convex column identity outlier non outli note paper alternative entire apply rotation change performance noisy additionally work long find project recover exist identify outli identification outside pca application finance exist robust pca suffer degradation dimension outli iterative inverse regime non point non even combinatorial intractable scale problem size particular seminal recent overall paper spirit difference thing fail handle corrupt technique believe broadly interest significant extension precisely principal investigate exact intend ahead need success use oracle seek convex recover correct noise identity outlier analysis broadly corruption corruption hope capture proof work thus result technique contribution outli consider ambient like b identity outlier column corresponding outlier identically singular orthonormal basis arbitrary recover clearly always impose section interested subspace incoherence column recover matrix corrupt always meaningful impose definition column say axis perfectly incoherent high incoherence column support natural side big essentially recover corrupt meaningful subspace column lie incoherence require incoherence svd incoherence condition require extra recover capital letter represent vector letter etc column column onto column denote row column finally projection span v complementary usual matrix norm nuclear depend context unit identity svd outlier recover matrix goal attain corruption constant fraction point corrupted show mild exactly low lie identity natural exactly via pose significant challenge give pursuit generate row find optimum output noiseless extension adapt noisy outli pursuit surrogate combinatorial stand weak pursuit matrix identity column outli statement noiseless observe incoherence output pursuit recover column space identify index outlier lie corrupt pursuit provide note bind outlier success outli pursuit space convert replace corrupted noisy pursuit essentially tight follow universal constant structure impose state noiseless outli outlier arbitrary rank identifiable separate corrupt guarantee possible prove paper exact successful technique condition subgradient desire solution assumption optimum column column outli pursuit recover outlier intuitively nothing leave column dual pair column correct know follow standard main introduction proof oracle constraint enforce column property dual must satisfy optimality solution obtain condition technical detail sequel norm column v tw ab definition discuss outli pursuit possible construct goal pair pair impose precisely support recall projection onto column result truth optimum nothing outlier arises impose problem solution pair construct appropriate subgradient key optimality svd column thing small must cl eq far correct let strict satisfies convexity argument pursuit imply lemma therefore establish strict w equation complete thus oracle determine condition must satisfy seek paper svd let let u complete construct oracle satisfy consider corrupted indeed set straightforward satisfied order recover I outlier immediately hard require include general condition long orthogonality modify eq definite cone away hold follow establishe reveal correction satisfie require strictly strictly satisfie imply simultaneously five show specify
plus pt pseudo random generator describe g unlikely method processor paper computer exponential normal generating normally random involve use rejection often preferable section efficient vector method serial unlikely competitive processor generator available wish normal mean normally number follow old minor uniform generator exactly exactly never return exactly similar comment satisfactory tr use correct correctness logarithm distribute distribute root implementation discuss variation interval compute go reject else go return circle distribute replace independent normally distribute random number execute root expense implement rejection implementation box processor return less number detect random permit subroutine number library approximation use identity four multiplication computation cycle normally distribute describe generalise generator use length uniformly distribute cycle add cycle peak give main component actually actually time computation generation computation reduce computation way reduce fast method b difficulty function taylor behave near suitably make change degree away experiment easily obtain chebyshev appear requires really want allow computation avoid expense uniform arithmetic generate independent use avoid computation normal gain overhead cause method straightforward implementation result table cycle avoid evaluation increase cycle cycle thus avoid box would work fact similar function replace uniformly evaluate exactly method uniform step reject else section deviation triple advance many number handle level version user routine call overhead cycle p method fast box cost rejection generate million rejection random p simple ratio generate else normally uniform number correctness logarithm proof lie outside occur much way section step go else else go lie inside region almost area execute lie region expense scatter thus logarithm partly r logarithm dominate library routine random scatter although ratio method half many number produce expect fast serial fact fast von sample density generate number first else go hard
observe time trajectory randomly possibly depend depend study randomize alternative code subject collect first treatment treatment code treatment outcome summary code cumulative discrete collect month course eight month study protocol large state art low summary clinical exploratory analysis convenience important development value decision maker summary contain indicator exposure indicator treatment month stage decision h restriction broadly classify indirect direct indirect approximate dynamic programming nonparametric learning indirect search method cumulative outcome pre class maximization outcome indirect generalized series check goodness particularly opinion form rather directly contrast utilize misspecification estimator variance indirect recognize effort variance inference e confidence confidence inferential challenge construct learn attractive practitioner extension assign treatment patient stage present baseline th nonsmooth functional nonsmooth learning plug dynamic h nonlinear expansion open problem principle highlight crucial usual goal note linear make treatment vary focus discover distribution addition b asymptotically nonsmooth implication asymptotically mean signal compose variable effect near poorly fix bias confidence move section asymptotic bias asymptotic bias incorrect level coverage confidence great estimator shrink bad converge asymptotic throughout outcome require design matrix covariance plug let asymptotic second reduce predict soft estimator penalize illustrative bias eq nonnegative positive soft nonsmooth predict first appendix assume c approximate estimator plug fold reduction asymptotic learning result large value prefer indeed soft estimator space fix converge bias drive square infinity chen consider uniform situation look fact viewpoint see actually bias especially large viewpoint provide toy highlight asymptotic play important study nonsmooth asymptotic study nonsmooth manner across use generative close problematic nonsmooth asymptotic alternative converge measurable ny ny n q prove lead large toy bias randomize study denote code high value example problematic see zero variable equality approach reduce thresholding nonsmooth covariate indicator precede small asymptotic lead behavior use simulate perfectly monte carlo replication set plot treatment confidence width roughly problematic large asymptotic cause away zero stay interval estimate treatment difference middle display axis red section decrease plot rescale axis reflect range correspond plot rescale insight asymptotic notion closeness sample plot rescale clinical research essential measure construct confidence difficult impossible ii propose additional extension provide inferential occur probability similar confidence idea regular zero covariance method construct set taylor valid say union appeal simplicity especially fail less group conservative regular locally confidence combination coefficient interval standard method n construct convergent estimator limit nonsmooth subject stage treatment effect subject subject distinguish subject make insensitive bound construct c nonparametric percentile denote percentile become potentially tune demonstrate effective drive describe limit behavior limit distribution sequence infinity validity h np equal limit asymptotically mean probability one treatment almost nan subset upper stochastically experience hold choose equal choose bootstrap follow percentile percentile bootstrap thresholding st driven test wide interval cover nominal frequently cover time wide thresholding penalize approach soft thresholding experiment evaluation model stage follow x feature h h h correctly match avoid misspecification comparison generative influence produce occur occur control patient measure al near example strongly treatment moderate however equally optimal moderate near give extension three stage example regular b confidence interval treatment truth show coverage monte replication bootstrap nine st examples st setting stage st nominal coverage conservative average expect whose double among level dataset estimate mark nr two st estimate confidence treatment nominal level construct draw correspond coverage significantly nominal mark st nr st nominal construct dataset mark nr width interval intercept nominal construct two nominal mark nr r perhaps perspective however view intercept poor attain nominal nine example st nominal fall contrast nominal coverage example average cover adaptive behavioral child trial al subset never randomize treatment subject massive subject provide school year precede odd diagnosis diagnosis baseline diagnosis exposure indicator receive code st stage code correspond behavioral modification patient treatment basis two scale al list behavior month stage outcome initial take prescribe month month school year response randomization randomize month nd code initially correspond increase initial treatment teacher teacher rating score code clinical randomize month month first step month prior study vector interaction action main effect stage least percentile residual show sign misspecification short seem reasonably upper intercept baseline odd diagnosis exposure non st txt stage txt txt dependent assignment treatment available diagnosis subject exposure h contain term action main covariate stage form plot residual obvious sign misspecification provide reasonable estimate intercept baseline odd diagnosis exposure txt exposure estimate function reveal treatment first stage decision exposure behavioral modification subject prior exposure recommend regardless recommend optimal insufficient recommend treatment patient availability individual clinical recommend unique optimal confidence interval response case binary fix patient history would th insufficient nominal interact treatment categorical consequently history recommend treatment history strong leave solely conversely confidence interval recommend evidence clinical making conclusion insufficient insufficient insufficient evidence insufficient evidence discuss asymptotic bias shrinkage stage use form b bc mobile device collect patient thereby research match technical point grow function stationary markov mdp setup algorithmic largely characterize rate limit theory management across location treatment location affect outcome neighbor across spatial often resource location rather need sequence function date spatial treatment recommendation spatial learn suppose treatment option feature treatment interaction feature contain even possibility computationally moderate expect outcome code relaxation replace nonsmooth concave surrogate rule h h h h constant version algorithm h h n mild nonsmooth learn occur outcome standard density proof contribute second display q contribute bias variable expectation give part equal take schwarz triangle intercept first identity behave eq supremum must terminal cover assume reward observe stage seek maximize trajectory restriction line spectral denote member value center uniformly real equip np bootstrap empirical measure respectively characterize derive limit convenience dim eq estimate notation without pf pf f pf bootstrap theorem together ad must guarantee continuity finally continuity next step characterize limit lemma accomplish theorem claim follow weak b last old precede display weak law large number vector na tight covariance f pf pf pf n l vc envelope square integrable follow van close subset limit element f boundedness note f f f van uniformly immediately section theorem expansion coefficient useful expansion individually equip norm b b k alternative expression upper similarly analog replace fashion use proof technique lemma present imply conclusion assume verify h k hold loss h h l n identity hand surely let result argument k class see sequence almost almost sequence surely theorem proof convergent weak law bootstrap w establish strong measure spectral matrix first continuity continuity omit three possible model pa pa pa pa use h c indexing example analogous effect furthermore happen standardized treatment versus treatment treatment versus treatment equal example table interval diameter across nine two stage stage stable quite conservative allow grow
pca perhaps dimensionality reduction pca back early become technique extraction datum theory pattern recognition process construct square eigenvector pc matrix broad attribute primarily classical regime recover subspace existence nevertheless solve well precisely suffer non corruption sensor failure grow slowly quadratic consider principal existence map produce low way point corrupt non emphasize rather g distribution considerably dimensionality signal snr go norm scale dimensional generally recover without main surprisingly contrary nature regime fraction arbitrarily accomplish pca propose tractable provably asymptotically optimal corrupted scale tend algorithm kind moreover pca removal subspace removal probability solution lead good make argument rigorous organization reason classical robust setup pca devote guarantee experiment technical derivation performance capital denote permutation robust dimensional regime pca focus robustness corrupt make fraction outlier regime good addition corruption significantly robust algorithms class maximize variance obtain project serious covariance treat volume cover pose knowledge dimensionality seem crucial subsample spirit approach correct corrupt nothing principal pca maximization become extremely dimensionality local maxima grow fast effectively random exist output dependence dimensionality regime category outli bound inverse interest consider standard setup sample fact magnitude dominate counter intuitive hold principal magnitude principal mahalanobis distance corrupted fraction consider mahalanobis small align corrupted follow roughly nearly large extensively exist robust outli iterative various alternative use mahalanobis outlier depth algorithm propose dimensional neither mahalanobis bad pca sample algorithm mahalanobis namely minimum projection pursuit find fraction determinant pursuit maximize robust univariate combinatorial scale difficult yet volume cover low post algorithm project fine produce estimator pursuit non intractable author avoid examine direction direction nearly direction adapt low rank low entry arbitrarily close spirit motivation completion herein every element hence pca setup pca detail signal realization unknown absolutely respect borel constant easily relaxed expense significant notational outlier denote corrupt point contaminate equivalently top seek collection orthogonal achieve vector another angle express represent portion expect express fraction generate long tail become distinguish quantify effect tail weight margin contribute variance theorem scale paper focus scaling scale happen asymptotic together follow etc slowly zero stands perform h pca contaminate md aa infinity immediately sequence bind explain due removal corrupt may remove account outlier magnitude remove principal component go proving imply corrupted scale corrupt side continuous I contrast existence corrupt optimum enough side strictly maximal never outlier asymptotic gaussian cc gaussian uniform perform dimensionality space know form origin generality mapping pca involve pc note former apply pca pc pca since pca contaminate sample concentration must satisfied decay thank lemma exist least dimensional empirical truncated mean convergence since abuse sample heavy obtaining rely class aa f e vc inequalities one next en notation non variable relate step leave exist constant apply note exist q know ib h q prove second similarly let exist prove show either removal good corrupt precise set remain ss project mainly point denote next true removal corrupt point least trivially focus en true remove corrupted high let event q exponentially high would hence good martingale argument random q detail defer appendix proof due event occur increase factor exponentially sample theorems theorems main term small guarantee estimator intermediate lemma value give leave detail bound h theorem state theorem immediately asymptotic performance pca numerical pca algorithms multi variate project pp dimensional regime robust pca magnitude uniform magnitude report result ill condition report test pca pca htbp pp strongly corrupt recovery satisfactory sometimes bad performance pca increase essentially consider versus dimensionality performance sharp become htbp property make dimensional perform generate choose line trend observe although gap decrease since cc investigate perhaps principal
around ball completely contain ball result boundary valid region valid r n k probability least minimal maximal graph differ additionally mutual degree radius kx kx concentration union minimal n additionally need contain affect value thus continue direct vertex exactly maximum minimum ball ball radius maximal graph proof fix connect large become around near probability inside among inside among neighbor graph high connect additionally connect ball drive example remark conjecture max institute popular walk mild distance namely sense graph plant undirecte expect walk electrical define background read science proximity collaborative supervise image computer science various bioinformatic nice practical function compute close short path shortest connect satisfie highly cluster whereas consequently encode distance behave time simple formula accuracy denote distance vertex representation well expression vertex time ij j relate distance electrical represent edge weight vertex know coincide distance see focus deterministic geometric point vertex neighboring point type geometric two whenever euclidean nearest versa mutual graph connect vice base similarity weight application similarity parameter definition random underlie vertex vertex graph construct order able minimal space connect subset bottleneck regular sense constant b lebesgue essentially spike consider valid couple region arbitrarily narrow constant hx cube vanishing generating volume ball depend geometry constant explicitly comprise two technique restrictive apply kind treat well come price assumption region bottleneck unweighted fix distance depend n c n satisfie case px bottleneck unweighted fix eq statement ij ij bind bind denominator converge various main bound independent gap spectral gap suppose general exist probability surprising underlie enter bound term constant bottleneck intuitively diameter contain walk plug unweighted c di ij px unweighted build knn continuous ij analogous hold unweighted additionally field common approximation fully upper low uniformly result apply treat graph adapt bandwidth principle fully beyond remove bound truncate technique treat general adapt compact build similarity hx pn h c ij px px scale density limit rescale enyi graph vertex certain allow enyi partition degree random random put consider laplacian matrix ed exist enyi graph allow self enyi rescale times enyi uv application plant enyi plant plant split equal put edge vertex cluster simplicity loop plant partition plant arbitrarily slow ij result structure graph time grow slow distance converge result grow might degeneracy visible degree place vertex cf read distance carry everywhere easy prove inverse degree lower weight remove w st simplify monotonicity principle increase graph build adjacent set infinity vertex merge parallel parallel exploit law electrical edge add detailed example lead show computed flow corollary flow weight st evaluate formula tight widely geometric space construct let regular width neighbor least edge grid valid grid point neighboring cell u fit bottleneck give graph geometric bottleneck straight stay inside denote let grid minimal point apart grey cell bound st st n general underlie devoted overview straight edge flow edge bring flow see flow hypercube side center hypercube locate path figure hypercube flow neighborhood reverse flow discuss computation contribution step consider current step unit flow adjacent flow contribute overall neighbor neighbor carry flow hamming illustration compute exploit neighbor let flow neighboring size receive flow sum flow edge might step cube flow hypercube dimension line connect hypercube figure layer cube cell layer final geometric graph geometric bind well graph appendix state degree order simplicity lead final statement near note mutual proposition formula approximate th calculation eq times cm proposition first convenience go eigenvalue pseudo inverse satisfy matrix k bipartite compute thus p p n prove little j j time j u j formula eigenvector ingredient distance gap low geometric geometric general special graph unweighte undirected follow strategy spectral general graph connect vertex make bottleneck maximum average undirected unweighted connect respective e maximum path number spectral bound eq gd b selection see tight proposition sure graph uniform argument absence boundary effect set graph unit cube grid cube grid cell geometric cell contain exact later construction path assume b cb cb deterministic cell path simply hamming path cell g cb ca reach adjacent path cell neighbor interior cell path connect force add point path assume connect path path load cube geometric minimal number construct exist cell cell maximal load upper bound cell grid cell center corner pass c size center per center satisfied possibility part neighbor lie neighboring cell interior cell two cell cell edge cell cell pass one cell start vertex intermediate select point treat pass load cube hx path average graph assume small grid denote ball center point path case let ball clearly boundary clear neighboring cell x l definition part use appendix load cube mapping exist n load construction distance always ensure grid always l mass ball eq cell ball proposition deduce analogously p obtain eq c plug ready apply maximal cf proposition path per cube probability c quantity lead replace deterministic radius radius symmetric mutual graph load load mutual bound converge take minimal maximal length maximal collect corollary gap plug vertex similarly apply minimal weighted spectral stochastic q unweighted replace second eq discussion tight eigenvalue laplacian ij walk example proposition directly follow px px nr ij I right treat bound w weight edge ij treat define truncate gauss ij w gauss converge note exploit obtain expectation implicitly weight minimal also unweighted graph note coincide eigenvalue proposition corollary h
mutual information fitness forest span might may take forest tree liu base minus forest obtain forest paper liu random clearly liu capture section deal generalization consider present summarize state work say undirected respectively respectively connect exist particular undirected forest loop connect pair finite say direct information probability nk although express nx nx wish minimize approximate depend equivalent forest maximize along mutual liu else terminate component find relative yx ix x minimize dp minimizing replace ni np np jx structure could require liu variant easy realistic case px x qx qx n find maximize minimize quantity description notice inequality connect description length become ei ni constant rather maximize loop else terminate connect terminate candidate make loop notice training look simplicity fitness stop rest either connect order structure result forest complete n criterion xx probability section random set set obtain countable operation event field generate borel mapping say satisfy say continuous measure kullback available say absolutely hereafter ni n leibler define minimize ii generalizing dx x dx proof express ji jj ii liu sequence I rx specify add terminate becomes suppose respectively gauss unknown eq connect need number may j j ni nx x ni ni j ni liu algorithm general avoid consider avoid
misclassification correctly blockmodel choose equally meet parameter class class identifiability thus meet misclassifie decay illustrate network datum employ publicly network facebook california technology ac indicate student friend yield gender community output partition covariate identify conclude student algorithmic grouping obtain formation reflect show order student covariate account facebook fit residual beyond choice blockmodel include logit blockmodel incorporation covariate parameterize odd edge occurrence vector covariate categorical indicate plus covariate indicate range analogous blockmodel blockmodel intractable employ approach hold constant holding test initialization consistently fit fitting estimating subset top adjacency estimate leibler bound row student logit blockmodel ranging describe paragraph bayesian sample five fold respectively plot suggest low divergence theorem example leibler divergence ab ab square error magnitude estimate reveal exhibit correspondingly evaluation adjacency assignment show row along correspond covariate student divide naturally interact frequently subject quantify increase concentrated connection evenly distribute membership group fit employ student group constant bottom identify category visible grouping right hand student support national institute health office office additional proof bernoulli chernoff respectively ab ab kullback leibler fix diagonal entry distinct see quantity ab ab ab ab q obtain union claim ab ab ab ex term ij x ex ij jx ij apply independent observe event ab ab side yield pz claim uniformly imply blockmodel condition I realization imply blockmodel formally subset upper triangular blockmodel membership generally p ij coincide blockmodel assignment step refinement induce blockmodel assignment blockmodel membership induce assignment respective assignment let admissible blockmodel assignment realization blockmodel refinement observe definition several membership assignment continue nonempty remain terminate index repeat denote number pair form follow hold index whenever distinct triple manner ready follow assignment refinement node one nonempty half iii condition ii theorem thus eventually triple cardinality cardinality nonnegative divergence refinement indexing let kullback leibler divergence equality refinement stochastic network show misclassifie allow grow size finite blockmodel estimate comprise bernoulli hold assignment fitting logit blockmodel network comprise facebook profile residual social blockmodel biological sometimes propagate yet understand appropriate remain statistical blockmodel salient partition distinct interact network concept connection adapt give rise bernoulli blockmodel misclassifie zero even class advantageous expect size relatively graph misclassifie stochastic blockmodel restrictive requirement closely suggest alternative exchangeability generative blockmodel analogue exchangeable sequence infinite exchangeable whose observed conceptual population fit blockmodel describe blockmodel adjacency edge blockmodel probability far symmetric membership vector blockmodel assignment number maximize asymptotically behave grows fast fit fast apply bernoulli assume amongst success assume class blockmodel fitting blockmodel hold blockmodel blockmodel incorrect assignment furthermore identifiability hold blockmodel class pair conclusion suitable misclassifie yield convergence grow class bound constant condition two eventually single
family include logistic regression p regularization term lead order ingredient establish line al covariate mean tail glm type rsc long scale sparsity allow bound glm address decomposable regularizer decomposable regularizer regularizers impose inactive simultaneously predict vector reader paper structure collection recall section vector simplicity use denote natural consider estimator user define estimator cover commonly often refer ordinary regression provide restrict sparse extension sparse set regularizer discuss block block positive constant consider sparse maxima say mp group choice ask matrix answer design matrix constant depend condition hold great supplement rsc sparse sparse rsc condition rip g close apart rsc impose let matrix associate generalization group assume without rescale noise constant group novel satisfy normalization solve q apply choice ambient refer exactly group derive et zhang arise particular general weak block analog ball optimize eq result generalization corollary reduce corollary consist term ball provide rsc derive class regularize explicit high sample size structural parameter play role isolate notion mean difference nominal parameter fact notion convexity high scale interaction decomposable bind explicit convergence estimator include derive model establish novel approximation describe bound completion framework rate combination nuclear apply leverage convexity state variety work simplicity exposition optimal instance fraction nonzero regularization parameter similarly last example type hierarchical regularizer regularizer combination decomposable regularizer fuse author partially nsf dms dms yu nsf n acknowledge nsf grant support nsf grant thank van de zhang regularizer lemma usa number comparable size consistent unless recent work sparse general regularize optimization combine datum encourage structure unify establish consistency regularize dimensional scaling consistency also identify refer strong convexity many case statistic concern ambient dimension large root old back theory g decade research rapid major force allow measure large throughout science project large survey www sample financial dimensional hundred thousand financial trading advance allow measurement thousand protein lead statistical reference image among image imaging lead estimator constraint impose within impose type study example include matrix component matrix decomposition many base form weight application pursuit square solve quadratic apply program constraint type include regularization regularizer value within statistic obtain high control ambient sparsity degree matrix typically bounds decrease metric place type perhaps consideration recovery noiseless g g parameter selection theoretic method model include consistency sparsity also among random field inverse frobenius selection application among group base noiseless arise nuclear finally although primary emphasis nonparametric almost regularize assumption change unified insight answer highlight key function regularizers estimator specialized different corollary author also optimal noisy noisy en establish corollary convexity group organize define notion main discussion consequence devote corollary model group regularizer supplementary file denote distribute cost quantity user setup derive convex unknown analogous frobenius apply certain type provide measure regularizer classical ambient contrast analysis vector contrast statement obtain finite ingredient regularizer capture might vector particular complement space definition large treat nuclear follow give norm regularizer ideal away subspace deviation always triangle tight deviation away subspace property regularizer decomposable respect complement main pair subspace formalize intuition let projection shorthand interest action desirable small theorem fast concrete begin regularization norm usual class cardinality reflect support complement inner give correspond perturbation subspace capture norm decomposable pair subspace form partition put show claim worth strictly sparsity structure norm group likely simultaneously model partition g n correspond regularization past interestingly lead superior performance subspace define usual precede exploit modification order overlap regularizer correspond identify inner product include collaborative expensive enforce manner study precisely nuclear decomposable rank subspace give notation finally nuclear decomposable problem sum low regularizer form weight nuclear norm regularizer work illustrative consequence specifically implication pm procedure often sample pi need curvature certain definition restrict convexity ultimately interest guarantee vector function formalize rsc particular statistical rsc term arise function high series low constant function ambient instance case covariate control tail follow type provide square norm choice group formalize quantity relate error regularizer compatibility respect reflect regularizer restrict dimensional constant establishing restrict convexity concrete membership definition therefore consequently restrict strong convexity hold paper ready although may abstract result concrete consequence specific immediate corollary design result rate matrix report elsewhere establish model rank sparse nonparametric structure curvature tolerance definition subspace compatibility paper strictly constant consider statement program although convex need strictly convex optimum attain state optima regularizer rsc condition challenge since unknown right note actually subspace decomposable ignore term correspond respectively decrease since estimation usual choosing choice sequel component degree work unknown recall subspace contain strong rsc tolerance reason corollary stated theorem belong rsc hold program satisfie rsc curvature grow compatibility compatibility increase third scale must strictly except compatibility dependence arise regularizer subspace obtain concrete rate use require condition three quantity illustrate follow model py nx pi link via vector focus setup describe give certain sense rectangular consequently identifiable impose section generally might weakly sparse devoted discussion solve quadratic choice loss order expansion case precisely establish appropriately previously discuss decomposable subspace natural equal figure strong convexity satisfy eq pursuit could relate condition less van discussion compatibility bernoulli special imply restrict isometry turn substantial dependency al design form refer strictly depend great
therefore family subset mass set result mean correlation obviously combinatorial greedy poisson combination family necessarily parsimonious determine fitting fast greedy inclusion feasible reproduce certain correlation family limit certain purpose adaptive useful discuss potential copula well bivariate discrete early binary vector copula frank g binary generating copula applicable interesting exercise without much practical application false true true corollary correlated context adaptive generate close target sampling proxy easily quickly exhaustive enumeration parametric carlo applications construction reproduce concept fail suitable approach impractical thorough scalar type bold main matrix determinant denote write closure indicator write ii notation triangular binary produce necessarily weight moment first frame parametric family suitable adaptive monte reason give q probability weight respectively analogously normal binary reproduce throughout rest generic since iff full write call notation moment da immediate dirac delta denote monotonicity respect correspond hoeffding apply factorization permit component conditioning already r I trivial certainly marginal logit family check requirement list parsimonious easily reproduce observe simple impractical monte product target distribution rest deal vector generalize permit vector multivariate auxiliary copula underlie build family work explicit necessary generalized review mapping obtain interpretation representation write eq convenience provide discover allow high try parsimonious interaction term however negative normalizing solution np unconstraine little practical virtue explicit moment explicit mean basic rather give formula yield include normalize write dominate definite value explicit recursive marginal expression sample fit note moment sample symmetric matrix fit fast requirement parsimonious via moment usually rule via factorization family matrix ever definite use carlo proposition derive probability assume contingency refer log contingency well study modeling approximation previous section derive marginal fit univariate probability target function probabilitie logit triangular logistic conditional eq special conditional exponential note possible family component practice parametrization seem impact carlo multiplicative close procedure construct sparse work parsimonious l I variable simple accord unit logistic draw conditional predictor firstly interaction secondly cause even conditional remain component parsimonious identify association intercept conditional family log weight family solve condition reweighte newton w I converge monotone thus maximizer complete separation way algorithm ignore lack proceeding problem jeffreys variance grow beyond threshold logistic family combine approach elaborate cause extra decomposable parsimonious likelihood computationally intensive chain factorization exactly although marginal section turn family copula copula family draw auxiliary distribution dependence rather gaussian natural option gaussian repeatedly cumulative speed parsimonious already sparse identify unit interval firstly correlation really matter adjust bivariate correlation extreme chance definite parsimonious copula identify component accelerate sample easily set task standard denote bivariate newton evaluate
sign negative objective loop correlation zero coefficient sign one eq analyse performance compare six representative dimensionality algorithm principal component discriminant discriminative locality zhang locality preserve database et pca reduction project linearization laplacian linear embed combine pca produce face discriminative dimensionality reduction face variation gender pose appearance consist vary size gender pose illumination expression age randomly select image contain individual gender expression configuration database normalize gray equivalent recognition firstly separate projection testing project matrix finally neighbor test subspace supervise retain number project sample subspace remain image remain test five recognition calculate dimensionality setting fig seven unsupervised pca design less retain neighborhood lda ignore margin level elastic fig outperform consistently keep rate select robustness computational cost proportional select produce computational outperform smooth coefficient path mark circle active mark feature feature head contour feature corner face sequentially feature feature select significant small one powerful obtain matrix term net incorporate dimensionality obtain impose elastic net patch transform penalize square thus angle patch alignment local patch coordinate secondly minimize directly weight group elastic net add rewrite lasso lar solve lar loop sequentially element active direction keep active lar conduct basis advantage aspect margin discriminative improves interpret result perform well algorithm component discriminant locality locality projection supervised preserving still error different retain compressed tool concern valuable norm far improve accurate penalty penalty could lasso smoothly scad fan li reweighte elastic zhang advantage adopt variable still long support grant project university wu science usa sparse reduction penalize directly lar popular learn therefore indirect manifold elastic net transformation lar adopt sample well datum representation margin maximization projection elastic reduce fitting projection various dataset reduction primary focus representation al li et li al al space subspace meanwhile particular original preserve remove decade reduction fisher discriminant regularize gaussians project label find scatter scatter draw gaussian harmonic assume draw gaussians merge basically manifold dimensionality locally et al laplacian le li al hessian generative et tangent alignment zhang linear measurement seek dimensional global geodesic distance pair measurement preserve proximity undirecte weighted pairwise exploit tangent tangent information align provide coordinate obtain eigen analysis build estimate hessian locality embed al neighbourhood preserve systematic zhang al zhang difference reveal ii embed framework g locality alignment liu manifold et wang smoothly scad fan screening fan selector selector dimensionality basis variable reduction develop achieve use coefficient problematic could linear combination reasonable feature selection coefficient property lead response lasso randomly therefore elastic net achieve grouping become dimensional subsequent regression especially important factor original decrease variance fit increment bias generalize interpretation relationship variable important practical minimize subject sequentially utilize acceptable lar propose subsequent principal nonnegative sparse discriminant projection al et definite programming use programming label discriminative class utilize particular manifold reduction g locality sparse absolutely sparse impose penalty loss lar objective direct therefore currently indirect manifold net obtain subsequent impose penalty combination criterion base dimensionality equivalent objective lar apply patch encode alignment alignment unify coordinate patch low directly minimization put new elastic adopt construct lar write correlation lar add set change distance control lar basis basis basis thorough face face dataset dimensionality manifold elastic lar effectiveness face conclude discriminative let objective dx different error minimize base aim low dimensional representation euclidean preserve actually popular criterion maximization combine interpret advantage improve computation reduce model interpret algorithm intrinsic reduction elastic framework obtain manifold discriminative sparse discriminative directly use least angle lar reduction find manifold discriminative reduction via alignment encodes geometry align utilize minimization criterion incorporate geometry reconstruct sample patch framework geometry patch k pf ix df I maximize local geometry k relate group patch representation expect sample possible relate sample class coefficient dimensional consistent alignment coordinate select global r ns rewrite sum alignment alignment worth space implicit adopted minimize therefore function q error model enhance represent error adopt classification however challenging apply classification formulation discriminant multi mild indicator l lda reduce fine draw bayes dimension sample usually flexible l indicator flexible representation near neighbor nn nn reduction point dimensional meanwhile consequence center belong center jx jj pi therefore dimensional representation indicator trade parameter expect projection matrix projection characterize fan projection lasso relaxation penalty although lasso regularize angle lar lar towards zero accuracy lasso follow penalize select care select impose projection elastic overcome retain favorable detail norm increase data response combination grouping effect property projection grouping demonstrate lar efficient lasso penalize penalize penalize least lar lar design particular I simplify instead eliminate objective respect e eliminate objective lar obtain optimal net rewrite q constant accord lar algorithm apply solve penalize linear lar lar entry coefficient increase another direction lar variable nonzero entry loop optimization size lar determine lasso eq simply ignore generality large sequentially coefficient important determine inactive different inactive coefficient sparse conduct follow inactive add current direction coefficient correlation equally preferred direction vector correlation variable loop direction assign sign simplex project correlation active decrease prefer origin space et distance normalize correlation identical correlation variable complement lar mathematically accord I new add direction coefficient accord loop elastic double shrinkage correct lar time huge dimension loop van inverse gram particularly loop accord rule calculate accelerate computation lar structure calculate cost well discriminative six subsection h training tw dimension
divergence gaussian map solve arise year independently show optimal control bellman become admit see relation leave case consideration us briefly recall formulation policy conclude trajectory specifically model dynamic implicitly q make discrete fully solution drawback importantly state latter cf claim show dimensional let q make corresponding stochastic optimal particular solution trajectory hold small though question principle note eq however case divergence zero problem write see condition suggestion back matching maximization consider close relation energy minimize free energy matter correspond partial furthermore choice generalize although interpret policy guarantee log likelihood despite control task equality likelihood jensen inequality observe necessarily tight coincide fundamental finding explicitly present apply give optimal provide interpretation stochastic optimal solution base propose control kl theoretical intended present propose optimal control divergence kullback leibler divergence distribution introduce discuss duality result demonstrate relaxation arise iteration infinite horizon finally relation previous field term multiple slice distinguish control wish infer aim inference horizon take form relate classical choose restriction previously posterior use note case additional strong present subset etc variable extra therefore could avoid introduce formalism helpful develop may state relate discuss arbitrary uniform stochastic one require merely little practical consequence control remain formulation corollary bellman minimize restrict delta argue present policy certainly answer specific suggest update subsequently extend horizon discount derivation horizon recursive pc give boltzmann similarly question control tx preliminary aim near assumption asynchronous converge pointwise constant informally aim finite horizon sufficient policy time easy analog yield therefore analogy old schedule asynchronous schedule give step policy fall immediately guarantee furthermore schedule
measurement applicable many popularity reference therein stage initially subset stage subject idea extend stage approach examine broadly experimental design computer machine type impact microarray contribution gain attain behind ds portion eliminate fraction appear promise consideration component retain previous propose however good quantification initial work manuscript extend previous strong entirely characterization follow review fundamental limit localization localization detection dramatically give measurement auxiliary review detection threshold summary support proof facilitate next baseline noise budget localization false support false discovery proportion component elaborate estimator estimator threshold implicitly establishe limit localization use measurement procedure recent metric utilize noisy sharp asymptotic procedure fundamental limit detection amplitude amplitude zero detection observation hypothesis alternative signal non zero recall exist sum tend conversely test sum false alarm tend possible consider describe process retain zero eliminate step precision allocate precede measurement make budget budget step measure component therefore factor ds localization word amplitude depend level ds capable signal purpose investigation assume zero component refinement rp x w j j amplitude measurement sense kp precision ds algorithm ds I follow ds empty successfully identify slowly reliable require amplitude exceed novel improve initial arbitrary amplitude adaptive gain section main characterize sense ds lemma quantify finite ds I gaussian tail tail mp component begin quantify j z lemma proceed refinement z z assumption refinement ensure event occur union bound obtain z jj require examine situation allocate allocate prove throughout fp fp gp k explicitly ease prove ii detect absence identify location zero follow slight analysis develop analyze index retain ds lemma characterize k ds consist p abuse ds k tail together immediately ds follow apply construct event occur tend event verify quite useful recall consider recall large enough j clearly negligible sufficiently show k know one output ds tend ds z event conclude k bind zero immediately ds conclude recall simultaneously tend conditionally definition converge conclude present ds demonstrate predict quite implement ds procedure simulation theorem lemma allocate precision allocate step provide snr result detection square root first last step equal first beneficial crucial step control r choose adaptive ratio location ds operate output trial generate square ds measurement highly successful happen ds ds successful finally ds average detect demonstrate extension provide ds ds gap arise snr much less arbitrarily course pattern observe upper figure visually arise rational discover sense indicate different accurate occur ds outperform sense examine non sense fdr different signal solid dot number signal allocation case snr sampling threshold result snr exhibit ds threshold fdr ds adaptive snr less dependence interest problem develop theory practice design improvement achieve analyze sense ds ds detecting weak localization ratio snr roughly low adaptive problem interested practical employ future investigation characterize sense devise experimental notable improvement claim sense ds even follow small time indeed choice ensure proof ds alternate measurement combination direct adaptive compressed literature sequentially combination regression show theorem proceed consider separately phase scope begin minimax false discovery coordinate formal optimality hypothesis case retain signal distribute standard fashion ready zero p p q p clearly converge proportion reasoning take p converge show thresholding discovery proportion easier upper necessary albeit insufficient easy control discovery spirit bind false p trivially mean
contamination extreme instrumental density poor rejection sampling use rejection procedure make target distribution optimal yield plausible need acceptance adequate acceptance greater extreme return iteration sampler solve call monte compute trace penalize carlo expensive problem proceed approximate density however approximate form individual multivariate member multivariate distribution procedure choose leibl writing approximation fully q quantity conclusion value pt deviation give deviation mean variance consideration simulation step alternative simulate dispersion triangular element probability define dominant thus partial factor fix minimum eigenvalue work run parameter roc curve run entire computation draw distribution simulate extreme break gene microarray contaminate extreme outlier simulate contaminate equal diagonal contaminate finally compare develop simple many covariance usually robust marginal procedure definite minimum matrix refer algorithm run norm warm start initial iteration lead guarantee unimodal place observe drastically start place performance relative lose simulation good job recover true perform surprisingly moderate discovery perform datum freedom provide rate interest extreme therefore estimate graph dot exclude solid observation extract part observation increase infer level pathway expression measure normality deviation stand exploratory plot gene inverse column include potential outlier obtain find test pair gene draw conditional approach biological interpretation choose run edge low conditional available estimate one remain estimate coordinate manner variational step also leave lead conclusion give take find run procedure use keep believe procedure compare biological find group despite correlation three pathway figure identify achieve plausible pathway original figure perform far relationship behave identify algorithm robust graphical model expensive great gain generally prefer alternative sophisticated monte approximation degree run suggest prior freedom line put increase quickly classical alternative model small line employ degree freedom conditional purpose lose degree freedom run last tune information validation often tend desirable goal rule compare throughout alternative remove penalty small decrease finally remark particularly rather misspecification involve degree maintain gaussian graphical model conditionally give conditionally uncorrelated distribution nsf dms support graphical useful explore structure multivariate interest result include penalization penalize provide use approximation carlo gaussian attract lot interest undirecte conditionally conditional infer nonzero element classical solution ba search optimize penalize likelihood concentration regression subsequent optimization arise maximize penalize algorithmic development deviation contamination drastically graph apply experiment screen become remove entire outlier discard variable fairly concern constraint case outlier contaminate transformation provide selection highly datum build upon use em cope contaminate maximization em penalize likelihood multivariate done analyze expression compete finding independent normal likelihood perform irrelevant lead function cone absolute entry large validation tune operate partial maximization row hold maximization descent briefly review show hold update cycle row diagonal freedom expectation notational convenience vector distribute chapter sampling distribution lead observation arise namely constraint impose correspond factor variance even illustration inequality pt take expectation gamma recall despite still prove pair correspond follow conditionally indicate latent interpret conditional correlation prediction neighbor appeal lack property distribution way issue equip gamma treat use line liu hide complete abuse indicate constant omit complete sufficient statistic follow simple pt update estimate constraint search undirecte however would section penalize avoid put maximize penalize account miss datum e calculate take step step eq yield maximize quantity exactly iterate describe call force convergence chapter concave usual able guarantee obtain well contaminate observation reduce contaminate loss away good contamination deviation normality part long bottom handle situation alternative
applicable scheduling result armed bandit problem rich context medium access simple player play maximize cumulative performance gap know reward reward policy characterize respect player average inherently tradeoff armed bandit sample ensure arm policy well often reward formulate generalization multi armed allow markovian user resource associate evolve irreducible transition allocate resource receive allocate resource classic arm potentially reward resource user resource arm challenge traditional armed bandit resource apply diverse switching input match output schedule wireless need allocate assignment problem learn maximize usage enough apply cognitive come markovian design policy markovian matching resource super polynomial policy static restriction logarithmic polynomial resource organize section present per polynomial user certain describe state arm show still example idea treat across state evolve markovian reward markovian arm assume arm value provide logarithmic extend simultaneous base asymptotically logarithmic arm reward finite achieve logarithmic utilize hoeffding regret recent policy provide asymptotically cognitive bandit markovian reward paper slot reward generate chain transition identical space also bind user independent arm recent liu result single state markovian prove upper element proof however markovian user arm arm cluster provide reward regret arm model linear static number time arm close build bandit formulation storage matching yield polynomial resource time reward reward case work state distribution allow support consider bipartite predefine decision slot resource policy resource evolve irreducible chain unknown assign resource receive depend state allocate user resource evolve resource allocate resource mutually allocate resource denote steady resource interference user zero interference cover scenario setting policy observation select resource user history resource assign throughput policy resource permutation formulate combinatorial multi armed bandit matching resource kn armed bandit share component express weight weight resource design armed bandit reward respect like averaged tend combinatorial armed bandit apply ucb across arm arm store slot decision alone several ucb ucb storage reward arm play analytical motivated policy correlation well use store resource slot slot slot play time clarity necessary step reward play permutation accordingly solve maximum bipartite j nn j accordingly summarize notation htbp index use resource indicate refer resource slot slot slot resource resource steady access resource z k kn k armed traditionally bound expect play take time arm although notice consequently quite loose linear arm novel uniformly theorem irreducible algebra increase reward markov chain denote satisfy lemma matrix j z j ng f resource pair mutually note evolve assign resource play resource index irreducible visit main regret state maximum policy expect policy denote c ic explanation counter slot arm play play case pick arbitrarily say increment pick arm summation equal define false pick arm arm could kt kt kt kt mean pr bind l kt upper upper could extend
moreover address synchronization ref roughly set synchronization ref topic address nonlinear continuous dynamical system stochastic effort close connection work cite symbolic dynamical perhaps intuitive one ref serve synchronization question come state internal organization get randomness issue synchronization review ref highlight synchronization review synchronization ref sign measure diagram ref use joint information x random measurement outcome value throughout limit sequence l outcomes length unlike field course mind symbolic discrete dynamical invariant notion apply spin deterministic probabilistic length eq set sequence bit kind reduce process infinite ref property process solely one property irreducible output one extract alphabet raw compress precisely shannon bit shannon theorem operational meaning channel transmission note dynamical spin mention future generate something one optimize shannon think channel actually channel determine closely random tell share much noisy block specifically sublinear effectively length speak various reference entropy excess merely hierarchy derivative integral discrete length converge excess control speed step see discrete integral determine clear information quickly block intercept ref diagram appear compactly introduce operate slight additionally improve go operator process capture reflected give one attempt calculate hierarchy analytically addition hierarchy distinguish hierarchy trivial example turn conversely process excess entropy reference introduce line prediction communication wish probabilistic minimum good capture information future building good behavior channel mechanic equivalence purpose forecasting process state state fig ref denote matrix consist denote causal state future past moreover optimally predictive know causal word share past structural start symbol sequence state importance reflect representation important component know measurement uncertainty vanish capture minimal store order entropy shannon causal complexity bound excess view memory store basis capture asymptotic causal state normalize transition complexity also directly directly rate produce historical show excess entropy excellent capture approximate job compare description arguably however mind constraint preferred benefit elaborate description model state transition describe address alphabet variable denote machine presentation organization space relax make distinguish several kind presentation notion reverse third predict discuss two discuss recall result theory stochastic complementary entropy typically type output markov use stationary il need entropy convergence stationary il deterministic ref result note serve motivation later generalization derivative integral block entropy derivative ensure consider integral fact block entropy sum state thus sufficient necessary sum state integral sufficient theorem limit curve constant entropy fig take block entropy contrast entropy drop formulation measurement formal entropy entropy offset excess fact derivative recall presentation observer hide internal introduce ref causal observer uncertainty series vanish observer observer answer observer answer though presentation state observer formal synchronization synchronization presentation word word markov admit weakly observer turn weakly state vanish fast l implementation particular state impose sequence due duality synchronization return briefly interpretation give excess internal information sequence synchronization recurrent know transition accord consider useful consider state current symbol observe next converge exponentially add q importantly synchronization starting identify conclude rr lk state entropy define ref converge surprisingly give give far distant ref finite symbol know odd precede making conclude contribute piece comprise distinct contribution piece record estimation length short correlation view amount infinite role contribution synchronization clearly range spin claim spin chain rather introduce ref spin slope l first monotonically monotonically compute entropy estimate entropy analogous technique hold ref focused uniqueness define property process represent entropy excess remain solely lead several capture perhaps quantify characteristic meaning must behavior future roughly think irreducible arise information past future causal reduce presentation curve presentation sublinear block state entropy start work hide future stationarity entropy eq use stationarity ref entropy information share past quantity nonzero complexity intercept sublinear part block proof proceed identically namely take prof prediction must extract perform away representation state justify bi draw degree system presentation state past future past future presentation state uncertainty determine rather presentation intuitively excess later diagram tool point visually sum excess work simple verification approximation limit desire recurrent causal process synchronization happen window tend infinity either always generalize presentation irreducible symbol kind past alone synchronization define ref reference property describe understand block reach concept generalize understand h might particular turn one presentation term term presentation briefly exposition elsewhere observer state presentation regular observation presentation state observer absolutely uncertainty state reach vanish visualize difference asymptotic eq soon observer markov order exactly equal unlike indicate opposite language physics call freedom mention sec observer I look motivation order think length state uncertainty synchronization presentation markov presentation synchronization maximum entropy observer extract bit leave irreducible observer learn observer synchronization presentation helpful presentation synchronization order synchronization slightly let permutation periodic cyclic true equally informative observer observer contrast observer observe instead synchronization cyclic q periodic process sum sum minimal extend recurrent state synchronization give long arbitrarily word despite synchronization repeatedly observe symbol observer consider always order states motivate synchronization intuitive leave detailed discussion properly sequel preferred presentation process interested understanding particular would analogous classification presentation result presentation possibility quantity zero redundant state look exactly ref finally purpose presentation property fact diagram happen define property presentation past state move must addition gray generalized excess dark diagram potentially depict finite random theoretic diagram however component sigma algebra vanish simplify atom dramatically make atom vanish past excess similar calculation correspondingly state induce infinite state presentation induce past define difference excess diagram particularly simplicity derive causal play map state state requirement entirely diagram presentation relationship complex converge linear attention exactly lastly overhead require generally associate presentation uniqueness irreducible process must track bit information correlate future also simplify causal newly integral simple alternate explanation intuition relax constraint leave weakly state infinite minimal causal partition effect fig demand determine diagram indicate consequence nontrivial refinement examine growth reflect curve additionally happens reach since expand way combine variable two expand refinement entropy positive state presentation entropy care complicated follow three presentation fig original causal history straightforward unchanged presentation weakly asymptotically presentation interesting depend integral presentation domain weakly asymptotically long operate recurrent correspond past covering causal every induce weakly induce presentation induce history also finite recurrent product cover capture causal prediction unnecessary unlike entirely call new represent become make synchronization follow weakly l l large synchronization infinite necessary synchronization denote sum piece interesting impose maintain synchronization one mean feature describe straightforward presentation asymptotically history degeneracy break due derive induce rely contribution since finally remove requirement present change whole presentation break much large examine diagram one future beyond think feature maximally utilize additional remark complexity well small sometimes cite one sequel diagram plain make make oracle like without without presentation mixed ref evolve distribution state presentation history transmission complexity discussing indicate one long dynamically lose allow net repeatedly lose converge linear large growth acknowledge entropy block state diagram leave less presentation bit value information copy history cover twice visit coin flip analog reverse symmetry mean development started discuss synchronization block entropy quickly compete reflect give presentation summarize entropy immediately addition track comment development block entropy note analyze directly entropy convergence another important establish final information speak principal subject presentation diagram sigma atom turn measure process c c concrete operating synchronization information contribution information reflect occur length reflect information connection synchronization previously point turn appropriate process beyond describe presentation generic amount finally component implicitly presentation justified concept familiar physics immediate hierarchy go predictor show outside trust presentation make role uniqueness corresponding alternative second wide may calculate information production example convert either ref circumstance resource ready randomness observer prefer recall synchronization control note essentially achieve reflect analogue block entropy counterpart hierarchy ref title usually depend symbol asymptotic property logarithm entropy modify drop boundary give two relate relationship redundancy thm entropy difference l definition
posterior setting strict result additionally cl j code take sample ghz ram burn shape average obtain post mcmc abc cumulative factor specify abc bayesian mmse cl present marginal I turn smoothed great abc high reflect notice aim demonstrate reach reduce distribution change material justify effort require ultimately ensure chain mcmc chain alternatively smc sampler finding paper var predict easily use claim addition predict sum claim year uncertainty conclusion demonstrate frequentist relative demonstrate frequentist bound frequentist lowest close unconditional frequentist residual frequentist free claim novel advanced abc intractable posterior chain chain assess metric abc predict claim via algorithm estimate cumulative claim accurate empirical approximation entire claim valuable reason centrality measure tail var risk complete thank mathematics university award partially dms north finding recommendation express view national cm assumption notation remark bayesian methodology demonstrate classical numerical abc distribution abc parametric simulate directly sample without requirement crucial claim capital value error chain computation chain monte mathematics department email science bag mathematic bootstrapping justification deterministic chain cl model uncertainty cl reproduce cl bayesian abc standard markov carlo mcmc set parametric effectively methodology allow abc methodology evaluation likelihood parametric assumption make overcome present numerical novel abc embed procedure able model us methodology model demonstrate obtain distribution claim point obtain bayesian benefit uncertainty analysis analyse cl unconditional frequentist estimator mmse frequentist estimate addition set obtain predictive classical bootstrap procedure rao integrate parameter analyse achievable bayesian cl summarize contribution cl et distribution intractable result intractable methodology inference potentially poor outline begin follow parameter model construct bayesian present directly illustrate develop synthetic datum via outline development structure triangle give c year claim simplify year period claim time j I underlie rather recursive claim give good involve follow unobserved quantification associate predict analyse uncertainty reproduce cl free bayesian chain bootstrap additive cl j independence density density claim different cumulative q conditionally residual satisfy q p residual slightly involved claim assumption conditionally posterior implication develop make prior distribution prior give gamma cl priors cl likelihood analytically model distributional cumulative claim distributional prior free cumulative claim standard analytic perform long free another use w set abc allow make variance highlight distributional mutually exclusive idea select maintain particular important prior enforce strict develop fail prior satisfie bayesian factor cl cl tail enter discussion simply choose parameter inference context distribution denote posterior allow bayesian cl mmse additionally find approximation line equality exactly cl justify predictor via mcmc bootstrap obtain claim full claim empirically take obtain claim mmse alternatively predictive numerically integrate uncertainty approach result risk security calculate calculate predictor cl around analytical chain carlo mcmc abc free bootstrap previously long moment estimator frequentist approach bootstrap table numerically predict claim associate uncertainty bayesian approach distribute monte carlo require markov sampling distributional regard involve evident skewness instead truly abc facilitate intractable additional encountered working methodology typically methodology see concept embed abc novel back transformation abc intractable abc novel mcmc present numerical procedure numerical abc carlo simulation al description methodology specifically work consider wise intractable abc replace set summary statistic I augment tolerance statistic abc intractable marginal free joint integration intractable statistic converge see reference therein discussion accordingly low possible base sampling algorithms abc informative justification theoretically achieve rejection methodology model paper mcmc abc observation want summary datum assume procedure hard g equal step trivial analytically mcmc method accept denominator posterior computation evaluate extend class analyse measure city block efficient standard especially scale additionally assess serial markov chain sampler autocorrelation chain diagnostic conclude intractable reason normalizing target appear numerator result detail mcmc choice metric study metric comparative distribution unchanged denote generate conditional appropriate choice analysis impact mix joint chain tune additionally round produce acceptance typically design constant datum run version markov show markov chain random analyze behavior serial chain tail due scale euclidean covariance diagonal recommend euclidean trade simplicity diagnostic variable use
velocity via adopt triangle red circle symbol correspond green symbol simulation solution line weak problematic configuration consist wave tx contour lb simulation fourth third version successfully recover observe computation instability application instability engine role natural phenomenon formation domain practical recognize offer physics instability challenge illustrate fig interface perturbation separate wave interface immediately height divide mesh initial interface initial amplitude indicate domain boundary periodic condition bottom boundary boundary velocity boundary reverse one pass interface right reflect wave transmission wave perturbation amplitude interface peak heavy light gradually go light heavy wave reach solid interface transmission wave wave light second wave reach interface continue growth interface instability interface continue grow satisfactory numerical wave air reverse interface observe mechanism mention see medium wave show letter propose flow heat appropriate application heat thereby validate benchmark always show three appear conceptually future xu zhang acknowledge science national foundation china li acknowledge national research program china cb transformation possibility give b way transformation compose iy iy iy ix iy iy ix iy ix iy iy ix iy iy iy ix iy iy iy I equilibrium j j xu li china university mining technology laboratory institute mathematics box china decade lattice boltzmann prominent flow application range pre flow dynamic boltzmann equilibria lb model flow importance wave dynamic lb speed lb lb version flow follow constraint low heat broadly lb flow local calculated expansion velocity line roughly lb lb refer chen al euler well leading refer xu also specific heat degree energy original lb lb lb moment space projection subsequently stream discrete model degree rate moment adjust obvious cause instability effort past speed flow lb enhance stability nearly lb recently construct group theory equilibria expansion equilibrium function regardless heat description formulate flow specific heat express relaxation various moment physical quantity density tensor letter seven g associate equilibrium reference incorporation permit seven equation seven v choice simple detail eight correspond momentum mode stress energy transformation upon gram lead energy consistent divide moment energy two mention group one moment functional one equilibrium basic principle momentum energy correspond equilibrium distribution accord seven relation lb energy
gaussian type handle hyperparameter tractable exception assign gamma augmentation update conclusion product flexibility decomposable encourage induce product graphical practitioner belief problem edge practitioner cluster emphasis place derive model prior examine demonstrate graphical field properly interaction yield lastly american voting amongst concerned problem sometimes learn decomposable popularity mainly tractable graph set use find model accommodate posteriori rely decomposable update give posteriori current accommodate form effort encourage interpretable together exhibit block moment prior handle class product flexibility specification particularly spatial valuable complexity straightforward interpretation examine decision management addition sample important overview exposition let subset say subgraph clique order clique undirecte empty might undirecte decomposable associate decomposable imply decomposable graphical clique graphical model traditionally graphical structure represent conditionally independent give graphical factorize complete cardinality perspective posterior dedicate specify proper decomposable space crucial graph obtain estimate limited distribution bring decomposable often assume prior e prior mass intermediate size binomial number edge maximal suggest motivated result peak beta give marginal default imply interestingly medium sized number decomposable considerably decomposable list decomposable straightforward mcmc scheme prohibitive dimension edge often undesirable string show sample node uniform make interpretation long string tree reality class amongst separation clique move beyond prior model call clique alternatively clique factorial term clique trait interpretability result graph even completely graph preferred independence respectively clique clique value bottom relation clearly demonstrate relative binomial highly separate euler absolute first kind limit exchangeable consider four allow control clique reduce limit partition multinomial use able number clique insight determine policy variability spatially enable management handling thus require accurate effort statistically understand connection yield valuable reason firstly plant management benefit early additionally extreme event rate lastly prefer yield practice portfolio look undirected graph production thousand california department considerable wishart yield give covariance inverse wishart ratio wishart likelihood induce inverse wishart focus specification look expect yield characteristic instance namely prior control contrast put penalization separation clique encourage clique pursuit mcmc length million binomial graphical prior save move clique determine figure high graph mass evenly relative product evident figure form specifically string together plot reach conclusion prior string prior suggest correlation majority plant fall early grow contrast graphical plant decision graph bayesian decision gain understand graphical prediction performance year year simulate evaluated test result evaluate graphical prior sparsity responsible avg turn american
vc smoothness depend one smooth function non opposite interval define obviously former easy describe understand admit part half smooth function inherently complex describe right specify evolution give reconstruct exactly vc dimension smoothness therefore specific visualization different cluster write useful peak description peak position case consumption frame power hardware time switch implement body literature context take dependent piecewise text convenient affine approach inspire technique use instance indexing idea build piecewise level series translate symbol general assume approximate quadrature scheme setting provide straightforward therefore give value approximation segment parametric derive piecewise linear reasonable median provide robust take segmentation resolution computationally intensive bellman showed program different know need segmentation find accord criterion replace accordingly form summation operator aggregation operator crucial minimizing idea quality additive partition correspond partition winner fs fs j kl fs fs fs final backtrack phase index partition output provide backtracking loop figure leave dataset near spectrum feed spectrum segment segment figure segment htbp far naive approach evaluate partition efficient link naive implementation dominate fortunately recursive cost st segment general flexible numerous peak huber etc error piecewise representation former latter indeed easy together piecewise value independence need prevent request drawing approximation belong bellman boundarie interval connect segment piecewise suffer jump could specify segment simple description continuous variation list derive continuity let define make interpolation strictly interpolation belong segment optimize former approximate summary function issue rule address build piecewise segmentation noisy framework relate sample build summary homogeneous tackle case functional error use segmentation set diameter quadratic request quadratic set function homogeneous seem simple measure functional aggregate individual first measure quadrature consist build readily induce mean curve linear bit several curve curve htbp contiguous model approximation partition use measure straightforwardly difficulty efficient curve rule problem simply consider maximal distance curve evaluation fast might compute simplify problem one value observation simplification choose apply optimal segmentation accord nm summary natural cluster previous section application suboptimal optimize derive previous section q scheme independent segment amount global segmentation way tackle optimize sum term programming summary optimize partition segmentation feasibility depend aggregate operator function summary therefore cluster summary distance give partition stable availability programming computing quantity median mean segmentation rule solution tie algorithm cluster distance tie break might never break tie begin imply optimal segmentation unique alternative give obtain piecewise per cluster full spectra frequently piecewise via constant function define rewrite optimize allow section assess distance formulation functional way variation prototype modify mean candidate version version cluster segment regardless cluster fortunately dynamic remove resource cluster segment respect resource partition minimize minimize assignment firstly mention segment segment calculation variant efficient request secondly measure optimize programming indeed segmentation cost calculation segment minimal give computational assignment simple description seem resource assignment segment cost summary uniform reduce introduce arbitrary e time optimal assignment regardless optimisation case piecewise already mention summary compute cost cost scale criterion induce computational motivation introduction avoid rely suboptimal phase alternative representation homogeneous prototype segmentation optimal determination alternate optimum mean partition distinct therefore predict addition algorithm initial configuration three possibility experiment dataset solution start partition piecewise summary initial configuration uniform resource optimal allocation resource assignment picture phase give case result reach resource assignment improve resource identical k give remain favor nevertheless winner help practice favorable case purpose experiment time slow quickly generally assignment dataset study start follow mean k latter iteration acceptable compare alone initial configuration analyst neural base self give exploratory analysis world user build batch help induce acquisition display curve dataset htbp htbp conduct cluster criterion rough dendrogram figure small visual som arrange som analyst decide rectangular superiority som prototype arrange som contain som organize grid horizontal axis prototype however infer grey som summary average segment section iteration display som exception immediate starting summary slow phase cluster follow original map consist consumption record home give consumption minute displays htbp analyse similar clustering som rectangular axis encode global consumption horizontal shape shape interpret noisy nature htbp htbp htbp obtain result som algorithm stable summary segment cluster piecewise constant summary adapt load electrical power consumption summarize noisy counterpart even map power consumption start follow consumption consumption pm day home empty cluster consumption day week end resource emphasize difference resource expect piecewise constant summary figure grey prototype configuration represent load grey curve curve assign rough strong constraint consumption peak outline accurately approximated series segment expert easy addition nature increase segment improve marginally error notation compare error total internal variability dataset relative error quite acceptable strong summary type provide analyst selection guide exploratory functional piecewise linear homogeneous compute optimally program optimally number summary
coordinate coordinate remain list di jk u u jk jk derivative I say dimensional frame define function lie correspondence subset correspondence hausdorff take take prove part curvature identify second lie value frame tell angle tangent v contradict prove claim correspondence let small distinct correspondence bound derivative curvature condition jk jk disjoint counting jk jj follow function claim cover cover restrict domain note distant distant hausdorff vice versa hausdorff minimal ball specifically center curvature cube globally way hausdorff around manifold cover manifold choose hausdorff prove almost entropy need depend uniformly vs vs hellinger hausdorff c b dm dy yu bc hellinger hellinger rate c relate hellinger hausdorff cb bx normal claim x imply two distinct whose contradict claim tangent radius bs b either segment intersect happen c contradict conclusion hausdorff pilot dd manifold cover pilot overlap exist b dx let mx b b nm ij prove hellinger conditional mle manifold n half least observation j relate hausdorff hellinger claim exist v write shift orthogonal equal integral volume radius shift c ab hausdorff estimator choose nn c constant h practical generalization estimator large follow partly r partly partly distance boundary dy dy b dy dy dy dy partly ball outside complement radius tangent virtue partly boundary almost surely lemma imply c dy dy dy establish hausdorff conclude open question assume noise derive minimax rate substantially involve rate near boundary support report important achieve modification estimator hope homology infer reason difficult regression fast highlight distinction topological versus hausdorff exist adaptive acknowledgment comment paper thank comment eq manifolds volume support write define disk q follow u u manifold second coincide b portion radius center sphere summarize manifold see b u upper satisfie v prove ii also since inequality either imply inequality consequence pt minimax hausdorff manifold noisy depend dimension manifold unobserved variable riemannian precise give manifold open center radius risk infimum measurable function value manifold low expect hausdorff bind though exist estimator rate logarithmic theoretical establish tight construct estimator require smoothness vast manifold dimension therein interested estimate literature field computational geometry noise draw manifold specific notion different exception estimator show properly closeness hausdorff distance precisely dimensional ball center euclidean measure lebesgue sometimes volume integral integral measure density hellinger q affinity measure section generic may expression shall concern compact without embed informally mean look like contain denote tangent regard hyperplane regard size large unique onto embedded constant quantity condition reach large intersect sphere space disjoint thus let angle tangent define eq product geodesic suppose geodesic connect unit parameterization curvature mp mp mp v dp n manifold assume draw manifold distributional critical lead simple derive proof report let number lebesgue uniform density recall infimum recall lebesgue mb dy vb dy exist projection want upper line upper minimize take upper simple geometry dy dc locally parameterize surface dy dy ml mb du b du u dy mb upper follow change projection onto latter u du consequence lemma every support projection tangent version le space metric let draw correspond appropriate topic pair manifold le roughly speak subject hausdorff dimensional hyperplane sphere origin create new let constant infimum theorem section lebesgue le set upper achieve appropriate intend establish simple contain maximum distribution lebesgue hellinger distance hellinger logarithm h sequence call actually construct triangle u p result simplicity ready converge optimal maximum half manifold pilot sub hellinger lemma distance
term train competitive well svm contribution order super object overcome yield lie rigorous extend early significance tie application cluster review background widely statistical science comprehensive permutation multinomial computationally feasible avoid rank perhaps exponentially depend case tie incomplete rank largely theory e assume basis probability object preference worth logistic style comparison extend proceed select object stagewise verify sum interpretable incorporate tie rank finally completeness treat symmetric technique learn active topic retrieval g see basic ranking science system often object query pair describe parameter machine divide ordinal assign ordinal label simplify pair drawback order model wherein readily boost simultaneous permutation goal permutation phase output probable permutation appear rank problem object rank object ranking x might document return search response document document ideally contain order return indicate relevance create document rating view assign indicate sort essential contribution perform know object rate consider split partition wherein th rate model among union permutation partition empty subset object rating disjoint singleton idea hard partition super growth behaviour unknown challenge efficient generic tackle order partitioning order among give way partition generic proceed generative subset element remainder select large process continue model wherein reduce singleton advantage partition need specify advance truly brevity clear th remain subset furthermore subset contain possible non partition summation distribution interpret standard significantly small order exponential general computing term alone th eq monotonically increase represent denominator efficiently object towards admit decomposition value essential maximum parameter reader refer substitute w use potential affect local function discard write specific log likelihood carry base take reduce dropping dependency subscript auxiliary array ng k compute imply compute linearly log linearly log mention eq summation rhs summation involve possibly cost often query return object exactly query object relevance query highly issue begin occur rate object enable unseen scoring specify exponential local subsection function case interpret probability possible centre ps resort mcmc start chain subset run centre sample typically choose local distribution idea proportional potential tie worth unique fact choice backward manner shorthand interestingly reverse eliminate bad reasoning limited free kp backward admit place graphical model g probabilistic receive value concept object role membership state space rank rank state subset assignment mutually exclusive probabilistic network markov direct scope handling pairwise basic assign instance preference rao tie tie handle tie create object ordering emphasize advance group super yahoo challenge currently large query relevance irrelevant perfectly relevant yahoo unique contain query two normalise cumulative gain ndcg reciprocal ndcg metric position constant sure gain I put emphasis rank implement several ranking pairwise differ hinge essentially implement variant tie handle implement first implementation result see gibbs hasting sampling handling ignore tie simply document relevance tie order accord sort except bfgs bfgs stop less gibbs mh stop representation normalise roughly mean product order yahoo experience correlate thought correlation pearson whose find threshold ccc order ng ng conclusion first second order feature performance yield either outperform baseline method scope dataset yahoo pairwise train fast slow mh address rank tie generative probabilistic suitable inference yahoo demonstrate rewrite eq empty subset last use non collection object since configuration object appear account weighted contribution towards subset kn kn n kn pairwise ar overall respect w x derivative pairwise describe detail tie rao follow x tie parameter want w e x p w derivative give x w w recall probability px px w w w simplicity optimisation w p j w w theorem corollary address probabilistic super combinatorial unknown order stagewise space
continuously differentiable appendix carlo ideally summary close affect approximate average justification informally globally case sufficiently small argument make try statistic impossible statistic take different requirement solely like abc consider calibrate estimate posterior inferential monte carlo formally calibrate modification call noisy abc calibrate produce recommendation accuracy summary statistic unknown result give previous end firstly calibrate show calibration accuracy abc posterior guide statistic consider abc ignore randomness statement state repeat event probability abc posterior appropriately credible calibration posterior abc posterior idea calibrate noisy involve change algorithm account extra replace abc produce abc calibrate derive variable model link consider use individual inference special calibration inference behave get center furthermore noisy make h regularity abc part show abc uniform suggest difficult calculate result detail accuracy monte abc monte carlo variance monte quadratic want decay trade calibrate give single number summary small recommend combine inference analysis light find approach work abc average noise add guarantee result loss appear approach abc independently estimate theory statistic calculate pilot run abc non use aim appropriate parameter informative avoid value ii uninformative prior implement assume hypercube range parameter range observe pilot run truncate region choice time various take appropriate simple work use neither wish explanatory consider introduce value possibly simplest different beneficial f power produce well response simulate explanatory square I abc use th summary within region adapt run iv weakly pilot composite method linear assume appropriate difference approach approach use abc advantages abc uncertainty get follow abc tends result statistic indirect abc summary refer fair automatic acceptance simulation analysis calculate individual loss shape attractive simulation straightforward inversion calculate likelihood numerically eq skewness typically assume throughout leave restriction study implement automatic possible explanatory explanatory automatic compare compare regression indirect abc also fourth appropriate informally link skewness linear evenly space power choose model range range use parameter appropriate linear final abc first keep subset advantage efficiently uniform exponential perform pilot statistic overall dominate simulation stage automatic procedure choose statistic square implementation automatic abc table abc accuracy correction use abc run stable regression correction great despite poorly pilot abc correction accuracy parameter extent semi comparison predictor except average third investigate pilot automatic implement pilot improve semi regression correction datum set numerically quadratic semi automatic datum set remain unstable true indirect show asymptotically estimator indirect see semi automatic auxiliary initial depend true illustrate produce value simulate set indirect inference many indirect inference loss detail estimate interest parameter identify abc indirect substantially variable ht pilot indirect automatic abc pilot indirect semi pilot indirect ht state number evolve stochastically reference model density intractable simulation reasonable abc base rather give many improve detail set time bandwidth prior state value liu west sample show may inconsistent estimate ignore simple implement algorithm add observed value algorithm abc uniform sequential algorithm implemented choose show bias observation vary three sequential abc noisy abc overall abc appear make observe picture abc accurate abc difference model independent interest initial observation metropolis likelihood summary statistic value affect method distribution mention improper prior place assume negativity automatic simulate dataset lag coefficient square zero semi automatic abc pilot run pilot acceptance automatic dataset comprise zero dataset zero discard training automatically subsequent summary statistic nest explanatory small additionally variance large add observation instead suggest thus summary base synthetic abc regression adjustment produce raw credible synthetic synthetic coverage regression semi automatic model easy often suggest analyse simulation g queue service uniformly arrival queue initially time simulated dataset true draw uniformly service time magnitude avoid situation evenly spaced quantile automatic pilot analysis replicate quantile explanatory construct power minor split form queue calculation indirect result automatic abc comparison latter semi automatic directly abc accurate accurate advantage indirect simulation estimate accurately accommodate summary expensive ht indirect analyse genetic marker thus htp cluster type mutation event appropriate birth death mutation mutation simple sample exist parameter likelihood depend reflect restriction avoid need unlikely restriction positivity posterior reduce distinct number observe retain semi automatic pilot comparison parameter pilot rotation line abc interest explanatory comprise cluster size large cluster semi automatic use abc posterior indicate place less high marginal automatic htp simulate abc abc summary statistics abc average automatic abc reduce automatic summary statistic give bad argue justified parameter approximation abc accurate result abc combine set abc empirical evidence sequential lead biased could genetic combined genomic implement attempt add abc implement rao manner semi automatic implement main summary mean optimal quadratic evaluated method implement comparison semi abc example two give similarly estimate semi automatic abc less accurate alternative motivate approximate incorrect statistic idea liu accuracy sampling normalise mean square estimator size case immediate equality potential gain importance occur vary monte consider transition primarily
little addition many individual circle never customer circle otherwise compare geodesic distance many customer geodesic distance interaction geodesic average expect eight customer include friend friend reach customer customer expect reach interact identify customer span customer c seven customer substantially find customer customer well customer space present refine space computationally feasible extract use rich period might lead really underlying generating depend similarity small cell phone interaction occur among customer heterogeneous practical occur customer among customer treat straightforward update even refer evolve may censoring similarity censor concern like phone call record universe interaction customer reveal people ever may email face face contact connection alternatively phone call cell phone structure another observe useful diffusion innovation occur fashion people one interact regard heart model importance review collect present interest structure geodesic measure latent relation likely area practice identify influential customer opinion frequently diffusion research influence e innovation suggest understand influence among customer focus seem service selective opinion would network latent yield interaction propose bayesian latent dirichlet process well network analysis obstacle involve estimate concept yet fully researcher possess customer maintain heterogeneity interpretability classical machine readily admit latent believe interaction one try recommendation rather observe geodesic distance assume correlated preference behavior test practitioner allow would think might output space configuration latent physical distance sense spatial among people profile characteristic model observe propose correlation function interaction one geodesic distance order conduct infer explain business conduct mobile communication device become available latent behavior select choose specification incorporate nonparametric area abstract nature correct choose offer improve lead fitting examine likelihood use full test characteristic approximation adjust prior normalize rr rr likelihood estimate prefer preferred find difference posterior derive specification open exponentially bernoulli closed remain specification denote duration observation period likelihood way could sufficiently observation period wise unobserved heterogeneity gamma specific shape parameter integrate distribution formulation section definition radius word herein subscript probability let simulate multivariate ability trade draw origin draw clustered origin begin laplace center fold constrained become uniform infer tradeoff placing bound half add add special power balance appear combine transform center covariance matrix multivariate gamma effect sampling general current variable bernoulli trial r simplify combine depend care marginal integrating multiply assumption independence prior normal normalizing simulate walk place likelihood univariate sir might choose base slice draw adaptation direct reader summary terminology distribution take z point number point include person differ singleton person locate people contribution else person singleton call z union vector already set vector set normalizing need thus exist depend yield people mass likely prior singleton probability people one likely interact interaction customer make similarity interaction model scalability infeasible nonparametric moderate scalability find insight latent customer activity interested customer affect relate market among customer diffusion leverage efficacy incorporating model show improve forecast new customer likely interact efficient leverage connect link interaction many accommodate customer simply customer interact share information customer literature thing record observable correlate determine individual interact illustrative parsimonious theory similarity develop output understand customer interact behave model fundamental behind individual space working determine incidence difference clean observational among customer phone call record transaction observe individual two people generate latent interaction similarity identify similarity service paragraph activitie link know customer direct relevance practitioner heterogeneous population segment characteristic attention might contain company segmentation company phone reason model challenge key modeling offer valuable obstacle observational binomial use want break ignore unobserved heterogeneity restrictive interaction intractable challenge similarity characteristic interpretable bayesian nonparametric process essentially scalar purpose salient characteristic realize cluster location mass individual smaller fewer distinct likelihood segment bayesian nonparametric algorithm infer interaction probabilistic allow interact circle customer possibility customer may interact power interaction datum dataset specification incidence interaction distance validate showing improve respect metric literature interaction calibration predict interact unobserve heterogeneous among simply represent scalability nonparametric insight get segment offer finding literature similarity mix effort able distinguish customer computational dp inference wide variety include indicator transaction among interaction treat process govern network differ across call call call call occur phone call might ultimately conditional rate incidence upon similarity likely unobserved individual individual measure space people distance distance way pattern purpose article treat coordinate persistent though interaction phenomenon abstraction reality dimension relative coordinate vector datum observe heterogeneous across distribute mixing would imply sense heterogeneity similarity heterogeneity heterogeneity shift focus individual mostly unobserved trait characteristic unobserved coordinate lie space function distance induce dependency contact positively mean determine evaluating go people function among issue mean iteration value formulation moderately become number discrete latent mass distinct large lead value avoid integrate analytically certain want know dirichlet nonparametric prior distribution although dirichlet back new analyzing across estimate home probability oppose interest context finite paragraph dp think course model parsimonious require mass dp around realization lot dp lot less dp generate discrete important determining cluster purely depend either know though draw directly trick integrate analytically treat process mdp see distinct likely mass illustrate work univariate cdf color realization fewer high cdf histogram draw draw mdp deal dimension rich specification basic idea prior detail introduce restriction concern simultaneously uniquely latent space determining constrain prior distance origin introduce much incorrect prior origin translation define mode origin prior generate specification empirical censor second statistic exploit exponential computational fundamental much dataset never duration choose exponential pair gamma common degenerate hierarchy homogeneous heterogeneity later integrate three latent worth add interaction common maintain heterogeneity appear datum contact homogeneous latent full latent nonnegative latent contact contact select geodesic along short another candidate weight dimension distance space differentially also distance parametric specification equation distance subject empirical context assume customer claim show level parameter ran dimensionality log marginal decide contribution forecast empty allow assess goodness distance histogram call network article week axis log proportion dark dot log actual dataset represent cluster vertical replicate rather value infer close whether empty lot baseline actual dot outside many statistic existence interaction whether contact assess fit perform calibration period nonempty empty interaction duration want ability model lift lift likely contact period completely model percentage top empty great chance lift metric always lift metric variant result empty non remain geodesic distance break tie way lift metric exactly would expect assume empty likely contact sort distance long empty improve dramatically full contact nevertheless assume independence across ability latent behavioral interpretability link formation accounting calibration condition geodesic geodesic contribution model scalability amount dp group every generate outcome dense continuous non empty china mobile distinct likelihood much aggregation homogeneous likelihood available form dp group likelihood pattern dirichlet keep zero happen likelihood zero plus observed latent dp social least large much likelihood must individually latent likelihood likelihood compute requirement extent dataset change ultimately front influence grow small test work practice successively original value weakly compute tp size mass grow incremental change decrease though continue low rapidly though number number nonempty fast increase dp feasible effort likely come aggregate
proof concentration realize latent dimensional condition observe realize volatility take variation process brownian w w theorem law iterate dt repeat process x conditional time frequency ambiguity base interested compute moment term use realize satisfy obtained ignore fast auxiliary moderate presentation fourth two decompose generate simplify without ingredient calculate easily use fact even presentation generate bound I x x q iterate derivation normal second old work term follow condition eq valid condition theorem observation go choice moment function observation final satisfied observation diffusion process bound let brownian motion see theorem write note utilize result conclusion involve q normal presentation grow moderately bound moment suitably bound real replace need iterate expectation normality similarly bound matter need moment appropriately inequality proof omit satisfied proposition fan nj department business management technology mail yu department financial nj portfolio exposure effective increase stability vast require high matrix volatility volatility high portfolio propose pairwise propose vast compare portfolio concentration inequality estimate desirable volatility asset exposure constraint study carefully compare daily trend portfolio time period advantage frequency empirical consist stock portfolio portfolio assessment frequency mean portfolio finance yet practical portfolio selection number know depend volatility lead short portfolio portfolio market introduction exposure non exposure parameter portfolio theoretically optimal portfolio empirically portfolio exposure little accumulation effect empirical portfolio practice challenge portfolio statistic challenge medium period week say realize still portfolio selection portfolio allocation exposure provide reduce volatility capture dynamic volatility adapt volatility expense size wide availability estimation volatility year frequency volatility volatility presence market asynchronous price brownian satisfy process grid become stochastic calculus realized realize quadratic co see example high covariance tend bias toward zero signature affect way bias estimation integrate estimate extend improve zhang separate jump presence wavelet issue address realize reduce achieve integrate non absence first zhang study integrate integrated factor extend study perspective financial engineering topic datum asset handle trade former definite whereas far estimation error control paragraph base pairwise accuracy result pairwise though former implement outperform method adapt comparative advantage rapid demonstrate portfolio govern maximum thank vast overview portfolio datum perspective asset well simulation section technical process denote log price dimensional brownian instantaneous stochastic independent portfolio hold return history short simplify problem literature consider risk practice return constraint avoid negligible exposure exposure problem put equivalently proportion expect unless path current rely approximation even ideal observe continuously window historical reasonably rely continuity vary volatility reasonable rely stationary small hold short recent datum problem usually volatility nonparametric usually prefer million natural question whether result stable exposure constraint problem exposure estimate eq risk portfolio portfolio exposure respectively maximum accumulation exposure drop confusion show risk indeed oracle want portfolio quantity usually grow slowly number reveal exposure approximation semi estimation allow maximum advantageous method high trading former matrix trading several synchronization scheme propose zhang efficiently available datum idea say price asset price obtain previous yield vector available trading clearly stock synchronization integrate synchronization advantage method scheme trading call point estimate definite thank exposure portfolio selection far rich volatility help efficiency portfolio univariate volatility two volatility realize volatility wavelet realize price process pairwise integrate particular element estimate scale realize pairwise price assume actual observe observe transaction price logarithmic stochastic assume assumption mainly replace time process really study asynchronous discuss optimal realize covariance asymptotic normality zhang reveals simultaneously remove due due adequate vast depend replace subsample condition differently integrate volatility case facilitate reading condition integrate integrated process interval condition process candidate parameter condition inequality readily bind log observe market frequency clearly large accurately observation pairwise theorem proof observation time hence clearly see somewhat hold semi optimization yet even symmetric semi definite decomposition p minimum eigenvector remain positive semi transformation diagonal satisfie projection result keep asset study turn decide keep pairwise call covariance covariance pairwise call distortion effect serve matrix risk portfolio compute range profile numerical method number time distribution summarize clear pairwise scheme yet minimum pairwise insight risk approximation p w daily risk compute compute latent range characteristic frequency trading day stock various characteristic portfolio risk portfolio absolute difference pairwise portfolio result pairwise tendency tendency absolute pairwise turn absolute covariance method expectation portfolio exposure allocation risk compute risk exposure exposure agree exposure tight obvious method former latter bind exposure secondly flat due intra day trading condition increase increasingly unstable portfolio become basically actual risk become flat comparative especially portfolio strategy include asset price conduct simulate price trading day day record trading time trading asset mean asset trading day asset capital pool low strategy day use day daily matrix portfolio allocation exposure frequency trading trade use make portfolio allocation frequency integrate projection transform definite optimization portfolio trading day adjust portfolio portfolio hold trading day portfolio portfolio characteristic trading day calculate risk portfolio whole exposure stand portfolio strategy usually relevant exposure strategy present omit standard actual risk optimize profile significantly accurately short horizon provide graphical detail simulate trial intra trading portfolio trading day daily max median number position exceed whose exceed std min frequency covariance estimator frequency covariance short frequency frequency short c simulate intra portfolio trading day daily weight min whose absolute exceed std max short estimator short covariance estimator short c length low range exposure theoretical shorter consistently low portfolio figure slightly surprising low outperform instability realize curve attain figure fall specification couple study exposure outperform secondly risk approach mild exposure explanation accumulation dominate give day stationary portfolio low frequency assign weight around one asset portfolio exposure risk minimization asset stock stock call stock average stock index create co company publicly united market make individual change market condition asset frequency stock highly intensity trading trade median stock day cover birth financial hold period sep stock accord day daily low trading day estimator trading bandwidth since pairwise trading day estimate use exposure risk price arrival news characteristic portfolio characteristic portfolio stock daily risk risk minute return graphical characteristic std max min short short frequency covariance estimator frequency c index std frequency short high short high covariance short frequency estimator weight reveal term portfolio pairwise base exposure low frequency approach necessary short vary local one support strategy outperform
list concentrate many display membership recover matrix actual focus scenario diagonal present group volume bic group membership membership close bic simulation accord probable grouping node show network message along accord membership determine row matrix row order figure indicate inference capable membership prediction lie line inference algorithm email focus send message distinct message send month cc field language transaction result website post link comment chain focus hundred assign comment information include close community interest website resource social comment link comment link comment comment interpretation email represent group link top select discard comment network node per remove transaction transaction send one category comment user activity normalize measure bic suggest mixed membership quite focused element thus deal assign probable ht grouping probable identify appear send htbp matrix figure characteristic cell dark plot band correspond send member group member appear frequency observe frequency suggest remain behavior r probable summary nine calculate probability belong would vary considerably range suggest member group weight likely response member version transaction measure indicate identify identify message message pick predict overall performance message comparison link immediately transaction version send combination fig low htp green hierarchical htp red cluster algorithm compare mention availability observe measure hierarchical transaction use convert transaction count hierarchical apply receive frequency label mixed method hard classification membership ability co hierarchical use frequency multiple membership include communication pattern development variational accurately indicate interesting competitive membership transaction novel performance compare cluster issue study scale transaction cluster network transaction explore relate multiplicative respect varied minute computation cluster fit linear way bernoulli permit transaction impossible email transaction extension exclude nan might transaction information incorporate vary version activity could membership change vary fit partition interval interval extend structure behavior govern distribution membership node version transaction label transaction label label transaction additional message label group label allow topic membership message group category unclear practical support natural science engineering mathematic technology list communication email social network convert node capable model rich node transaction general flexible notion enable accomplished algorithm indicate email extract website clustering superior discover predict email text become consist actor relation relation canonical people relation friend relation always communication individual occur time call e call email involve one additional transaction message content g header network transaction email potentially email obvious develop shall assume transaction transaction thus observable transaction transaction discover structure future combine transaction group role receiver nod social node play role interact assume two role social easily research office propose hierarchical inspire membership detailed network structure develop review section novel performance soft cluster simulation present conclude summary scalability future seek toy adopt convention represent column fourth message transaction transaction transaction message receiver pair convert threshold figure small moderate toy summary count co message thresholded thing directional relation frequency seek edge decrease extension transaction section seek network inherently transaction occur transaction probabilistic efficient uncertainty develop develop send within list represent message node unobserve membership collect node model incorporate additional node belong membership allow transaction membership potential list conditional group membership node membership membership mix draw email select multinomial value select equivalent draw employ elaborate membership nonzero element email random membership potential membership email member tb draw membership draw pick e among node dirichlet interaction interpret differently depend email receive send member member restriction collection allow capture possibility large group high intensity intensity element pattern member notion section illustrate distribution specify variable assignment every row transaction row encode correspond email estimating infer posterior integral pick distribution posterior kullback use factorize dirichlet multinomial variational variational empirical multinomial omit fix simplicity initialize initialize normalize inference order number develop bic compose term term membership node receive email node trial decide receive exclude membership average membership compute probability write transaction likelihood bic eq number propose inspire relation receiver pair seek pair relation observe conditionally outcome membership behaviour incorporate allow relation similar represent edge relation direct require simplification
side et classify spam work lexical work lexical high lexical accuracy build base online use lexical al author outperform focus lexical lexical show algorithm batch work lexical dataset context non draw sum lexical classify al et al chen safe microsoft mention introduction aim web rely base verify manually member site verify month may carry collect appearance characteristic spread yahoo yahoo yahoo generator generate generator collect collect maintain collect mid collect dataset recent different query server responsible level domain implement engine adopt module feature answer could play site site team team server network country complementary former external collection collect collect depend load team extract lexical external feature l com http www com http cn com uk http www de http www http r www form auto www com form file domain ip max file name max yahoo yahoo lexical external feature lexical extract complement break token constitute feature token token domain name file extension token appear argument token part constitute binary bag et address iii name iv follow lexical detect feature classify five dot word similar type name ip number domain name name number use iii token dot token address name case category instance file page file dot file address ii name put name serve write server often include part include value lexical real team date site piece prior learn yahoo yahoo yahoo yahoo bad bad combine yahoo dataset dataset get classification online op weight machine knowledge turn become introduce online indicate otherwise receive train datum give predict op addition sign vs online algorithm train batch train meanwhile label use batch batch achieve perform classify svm construct large hyperplane involve determine lie svms investigate svms discuss operate online receive predict use receive update op update continuously update predict ty update suffer drawback account make adapt enough make binary confident notion address confident ip indicator update sure even domain mistake update avoid lot change formally maintain represent capture new weight pick vector prediction continuously label make close old distribution kl correct datum big require memory category examine considered presence feed slot likely drastically weight domain formally regularizer ty tries preserve valuable old much running update feature memory reader well pt p cm compare lexical svm lexical yahoo lexical op lexical evaluate lexical lexical lexical yahoo noise lexical ii lexical full evaluate effectiveness lexical working summarize pt cumulative svm svm daily compare lexical kernel svms matlab box svm validation yahoo similar examine svm svm svm batch svm svm instead set size svm svm initialize batch cumulative misclassification classified observation update essential svm performance fundamental svm observe third outperform retain high experiment illustrate model continuously lexical feature light weight light memory algorithm cumulative op error rate cumulative lexical gain lexical gain lexical yahoo yahoo bad pt conduct evaluate lexical feature full examine op dataset matlab yahoo good yahoo initialization pair cumulative yahoo due report cumulative rate pair involve note bad pair regardless suggest classifier agree discussion purpose op regardless use lexical full feature lexical slightly yahoo slightly lexical lexical alone achieve summary lexical feature lexical feature alone c c cumulative mis classify fp mis gain gain yahoo yahoo effectiveness lexical report feature boost pair range yahoo look mis classify negative number mis mainly come mis reduce range increase range false arguably spam summary lexical set examine yahoo give number label vice cumulative yahoo various achieve dataset maintain accuracy accuracy summary achieve pose advanced online outperform op pose seek affect performance examine importance long memory dataset notation introduce similarity denote lexical threshold name token file index file interpretability subsequently number similar dataset tail complementary cumulative significant distance size limited accuracy need produce explain batch result classification accuracy batch algorithm batch memory online limitation see effectively explain batch pt head function cdf cdf observe significant mean batch base mean classify essential maximum distance similarity unless update prohibitive similar classify memory behind rapidly op op reflect update section svm op dataset two retain history rapidly implement lexical classify implement requirement loading avoid accuracy thorough comparison classification feature section option divide component core component classify newly model component maintain stand one maintain run service maintain model label yahoo detection add classify split another option core server maintain run internet detection advantage server several drawback amount traffic yahoo scale mobile device life keep service mobile device persistent connection date practical bandwidth mobile device need various feature publicly add implement scheme internet attack detect lexical accurate avoid noisy uci california sophisticated carefully select lexical lexical vs lexical purpose
suboptimal filter inaccurate parameter posterior negligible hasting restrictive tune firstly negligible correlation parameter state secondly surface growth irrespective error modal strong flat mode average novel development use particle mcmc joint process non metropolis objective examine growth complex surface examine via markov capable sampling non careful surface multimodal sampler failure mix concern typical sampler additionally use structure fit precisely address fit first observation modelling abundance source weather detect failure device error bias argue detect model simplification difficult linear allow work presence introduce estimation date reason first non compose posterior design process structure four include strong population section estimation surface important discussion part synthetic newly methodology conclude recommendation development gaussian realization bold bold scalar discrete methodology filter sequential denote th chain particle involve iteration model consider discussion modelling regard density strong effect generic typically population rather mechanism reflect variability assumption transform importantly flexible literature five stochastic dynamic describe individual growth population reflect source variability growth behave exponential growth log rate per birth death discrete familiar logistic growth model density growth strength population exhibit dependence denote growth rate positive equilibrium latent exponential population unknown determine must effect transformation birth rate limit thereby size describe effect mechanism limitation population effect unstable equilibrium occur stable equilibria effect equilibrium equilibrium discrete study dynamic discussion note nest model set density bayesian unobserved time interested opposed estimate state jointly posterior give prior observation process nonnegative give prior c parameter prior observation generic eq latent specify inference model minimum error mmse mean evaluate evidence context estimate follow quantity quantity present considerable challenge dim variance require present recent space sample innovation combine mcmc advantage update entire particle chain thereby dimensional even strong model consider seven space additional hundred state particle proceed hasting static component construct via adaptive metropolis scheme mcmc chain marginal static model improve enable much rapid mixing number second component construct estimate allow sir note sir filter posterior proposal approximate p smc sequence recent review smc methodology acceptance produce generic j mcmc proposal appendix probability design proposal low dimensional monte marginal acceptance empirical law converge filtering particle path adaptive parameter distinguish kernel combination sequence allow markov markov appropriate several recent propose satisfied ergodicity ergodicity scheme two know two ergodicity space metropolis within involve specify proposal involve comprise adaptively line explore proposal empirical chain motivation present sir mutation process tn four subsection subsection accuracy synthetic comparing set recursively rao subsection bayes evidence final sub use real proposal propose additionally markov stage involve posterior increase construction particle stage involve burn non parameter sir particle mcmc proposal use sir mixture metropolis proposal subsequently use one combination chain careful bias overcome maintain would perform one inclusion static adaptation consider involve challenging datum set methodology estimate equation begin analysis number average entire state dimension sampler increase low estimate equation ultimately improve acceptance reasonable produce mix autocorrelation panel diagnostic diagnostic chain diagnostic derive non guide still produce constant occur chain sufficiently trace path sample demonstrate handle initialization parameter sample improvement include vertical dotted involve stage dotted dash top present true observation panel burn sample mmse posterior mmse estimate mmse grey predictive scatter plot pairwise parameter low density parameter triangular ht plot smooth static follow estimated coefficient static sample first follow six static accuracy estimate underlie give rao demonstrate set model recursively er static ip property condition derive modify integrate uncertainty marginalization utilize exist model mmse provide bind correspond matrix marginally static expression resort approximation filter new proposal obtain modify decomposition model assumption gaussian state observation expectation common filter filtering avoid marginal degeneracy hence time current chain particle exponential recursion classic filter particle kalman evaluation bayesian mmse parameter realization mse report average estimate provide methodology burn report per explore vary block calculation effect reflect observation level table increase generate parameter average realization block methodology mmse methodology estimate rmse estimate noise decrease estimator produce large rmse average estimate square block ideal choose integration respect parameterize ratio prior bayes test likely two datum bayes representative consider b b noise variance detailed population trajectory capacity challenge ambiguity actual true consider potentially capture form growth behaviour presence result switch realization bayes ambiguity flexible reason effect highlight challenge realistic observation count importance sampler highlight standard observation decrease exercise factor explore observation ratio focus factor capture vary degree factor process decrease order magnitude datum bayes level demonstrate several relate setting see majority strong significantly reduce reduction incorrect mark decrease perhaps distinguish confirm clear appropriate mix diagnostic parameter ensure study population establish world population abundance census select whereas observation add model strong figure plausible accord l post burn anneal component suboptimal model subsequently adjustment anneal mmse plot solid mmse demonstrate evidence presence comment mmse stable equilibrium go estimate ci capacity suggest fitting contrast aic suggest bic difference aic allow effect analyse time series south well study density dependence show surface process noise estimate surface comparison control equivalent parameter global likelihood local find bayes setting l burn suggest fitting model logistic five sampler diagnostic post require accurately weight proportion sampler mix mode enable assume plot estimate marginal posterior static estimate static factor bayes development sophisticated methodology particle enable robust jointly observation static parameter state model version several base dynamic cite place statistical selection realistic set accounting estimating latent consume design easily sophisticated algorithm population typically correlation
et typically efficient allow nonparametric result estimator major reproduce transaction adequate round li popular additive follow round end model discretize volatility filter series stochastic round round bid ask level book round market maker way past round previous book estimate although condition surely concrete specification g continue surely choose reason also easy less particle need cover pt round book market maker opinion frequency bid bid spread return transaction price produce particle assume support bid maker display axis transaction support log price real gray filter efficient price black maker see state unobserved continuous assumption develop consistency would asymptotic carry filter volatility efficient price specification example simple round ii py j jx x book long bid ask spread transaction observation assume transaction provide limit book bid ask level level really level k immediately transaction unknown past set price small price close book course guarantee realistic bid ask correspond ii lead small round transaction explanation book therefore stock heavily investigate situation contains implicitly trade large execute book large ask small market thick line support bid spread line width line width pt width width pt width width gray gray gray gray circle circle pt pt pt cycle cycle cycle cycle black black black maker available book bid ask satisfy either round view seem adequate need detail behavior market maker market automatically execute price make jump furthermore market maker efficient efficient choice reasonable parsimonious also demonstrate specify image book datum market maker example bid ask spread belong difficult mapping conditionally replace bad ask practical specification prefer finally partial compare round transaction round scheme show price price generate volatility deterministic automatically introduce log similar round bad superiority round section paragraph transaction bid bid evidence round stochastic round maker transaction volatility round include stochastic round end volatility hold formulate section security price aware fact trading present trading process nonlinear form distribute constitute slightly generalize give cause difficulty estimation nonlinear noise know lead log price transaction efficient particle localize estimator mention filter filter round noise localize volatility method volatility carlo et particle j approximation dirac particle filter use sampling technique know importance necessary j j conditional mention early particle filter distribution subsequently sample minimize particle normalize importance weight np suffer degeneracy particle particle I resample carry effective al resampling discuss particle specify show truncate weight model eq sample give difficult constant maximize sufficient modify towards kernel side recursive bt filter end lead I approximation result write new use standard algorithm variant every approximate describe step apply algorithm constant global old particle weight situation carefully investigate similar line eq furthermore turn prefer choice relate decay volatility locally smoothness either time dependent procedure run estimate estimation transaction volatility j line light gray line show red dark see plot new market transaction sequentially price price aspect back particle propagate em particle particle discussion end section merely tt realization process transaction dependence variance unit transaction per continuous volatility deep volatility use relation mention identify time fulfil fill change cf process accumulate obvious intensity estimation inverse eq recursion q estimate define fact bc var often require reason size classical almost change need line estimate j ci ci green gray line green gray log trading denote heuristic stepsize since ct j sign different curve become adaptive first volatility gray gray log trading volatility volatility I gray curve reveal typical volatility trading transaction visible trading intensity transaction decrease volatility trading worth green gray coincide use small period lag comment return lag reason noise small average filter calculation volatility investigate price high frequency hand still still return accomplish think happen consequence lag necessary new need carry correctly particular correctly get lag lag may quality lag however improve estimate presence pattern decrease begin create look estimator stay critical poor day block yield adjusted datum local back modification volatility two curve decomposition curve example analyze shape trading intensity trade volatility special care need affect mean volatility diag observation rescale unobserved negligible whole exactly weight proposition replace volatility finally rescale simple intensity pattern estimator recursion eq correspond bias decreasing propose empirically justify dependent practice line experience set approximately minimize approximate use adaptively select chen set volatility resulting imply computational minimal recursion recursion j critical univariate detailed description chen implement filter step normalize n nc particle resample easy line computationally complexity rarely proposal resample iteration particle quantity particle suffice proposal importance nontrivial truncate discuss extensively relevant reference problem approach sampling multivariate rectangular efficient propose quickly reasonable use g volatility magnitude close start simulate uniformly exclude effect start particle available run volatility consider volatility log volatility initial transaction obtain round volatility apply particle benchmark algorithm benchmark j start later term use price employ price plot plot also sufficient suitable addition large variance estimator volatility compare volatility efficient price generate dash upper lower realistic volatility transaction see transaction price obtain price nearest transaction day stock particle size analogous estimator eq noise function volatility outcome volatility plot low suboptimal describe calculate range plot estimator omit try use give surprisingly j respectively plot plot outperform benchmark general estimate bit nonparametric volatility mention filter covariance jump interval evidence stock price rare compound jump volatility index volatility pure jump acknowledge jump local volatility fine volatility transaction trading intensity volatility large would ds solely explain trading jump also occur although seems jump defer future red dark gray investigate jump take time return show volatility quickly recover jump see recover due adaptive stepsize modification stock transaction maker symbol c extract analysis line transaction trading period transaction transaction condition guide occur transaction constitute single transaction normally use trading inspection multiple transaction reveal transaction preserve decide datum transaction trading transaction correspond recursion recursion number trade fully time almost filter maker work maker match adjustment time transaction trade efficient price transaction price support filtering shown see skew consecutive zero uninformative filter transaction time volatility estimate maker round dark benchmark estimator gray gray middle plot transaction volatility show begin day vary almost constant advantageous practically almost experience feature transaction stock feature u end blue gray show round large realize volatility volatility come stochastic round obvious volatility price first indicator round preferable round volatility time display estimator calculate volatility transaction space transaction price c green plot trading duration stepsize minimize jt despite addition figure compare transaction estimate period occurrence volatility volatility new transaction efficient implementation filter contribution nonlinear bid ask bid price real treat filter third sequential type line estimation vary volatility make distinction volatility transaction model time total transaction volatility transaction turn transaction decrease begin day merely result trading intensity increase shape trading transaction technique model g drift noise likewise decomposition time volatility transaction volatility trading intensity course mathematical think hard achieve mathematically volatility optimal filter determine distribution price recursive em wang asymptotic recursive type present result properly rescale rao property attain estimator side datum even derive volatility grateful anonymous paper considerably express author reference em zhang presence review financial study volatility forecasting market noise manuscript particle partially journal normality asset journal finance induce security journal finance journal financial economic realize study design post j variance jump ti line latent datum journal statistical series chen estimation robust manuscript presence market finance stochastic central limit normalize increment presence round maximum incomplete journal e comparison resample particle international image analysis sequential method ed carlo practice journal machine research fan interface asymptotic power financial fluctuation nature journal graphical rectangular bivariate compute j student
choose replace member operation together produce member member enter fitness evolutionary maintain interact optimization prescribe number complete run provides expect represent formation rule query algorithm account voting relevance make programming follow implement htp initialize randomly determine query fitness initialize individual generation counter prescribed increment generation counter counter population otherwise increment population counter counter individual randomly winner member mutation increment counter count combine population pool member pool copy go count output number population query document relevant irrelevant wish query decision fitness represent either query metric take overfitte avoid benefit use es structured begin es system outline section es compete present benefit term corpus news remainder topic specific r relevance statement available adaptive none available topic description close histogram evaluation document balanced frequency bar set irrelevant variation topic topic large also relevant hundred irrelevant topic extreme r document role proportion well relevant topic provide evaluate efficacy htp implement software gate platform tool preprocesse software implementation es wikipedia wikipedia built publish al suitably integrate framework entity recognition carry field co algorithm toolbox al svm benchmark implement genetic population mutation h name population sub mutation depth maximum depth parameter terminal building concept could single query limit result wikipedia algorithm word comparison evaluate benefit es performance measure contribution integration wikipedia ask retrieval quantify effect wikipedia bag eliminate effect summarize token bag topic es improvement token interestingly compare result recall score well es precision measure query well relate ignore word token expression likely score es idea f es figure indicate topic semantic beneficial score precision splitting evaluation stem topic represent expression es rule tend complicate particular gate topic look death occur due mining first exactly difficult edge token topic favor improvement across htp wikipedia substantially percentage topic r r suggest achieve f previously essentially topic examine token gp precision topic es improve wikipedia concept appear performance es improvement stem ability high turn es strength include constraint narrow topic definition document characterize considerably overall token representation refer svm svm algorithms token es consistently benchmark term cause es rule appear quite relatively es token token svm algorithm es token token es token difference sake completeness token gp quick wikipedia document effect wikipedia es token comparison token vs svm token vs concept irrelevant substantially classical document information document purpose help user relevant concept token difficult retrieval suggest inherently feature beyond concept wikipedia fast technique human retrieval wikipedia database frequently require wikipedia construct query exist information exist framework improve genetic early es perform counterpart promise generic emphasis towards concept knowledge definition equip common world improve efficiency search direction query attempt wikipedia semantics present shift conventional token enhance information retrieval handle query evolutionary es learn co evolve evolutionary procedure base query evolutionary significant extensive study retrieval system system justify token query wikipedia indexing expert systems retrieval ir need significant knowledge knowledge basis expect semantic text close human extensive people concept unit dependency process extract semantic text collection semantic quite currently way document consume engineering trend develop resource semantic consider wikipedia world knowledge purpose user need term wikipedia concept instead query paradigm chen et incorporate semantics wikipedia principle user relevant document learn relevant imply wikipedia structure evolutionary boolean query integrate level information retrieval ir ir enable detect relevance concept identify concept relate contribute towards wikipedia concept base learn co evolve genetic gp call wikipedia semantics automate represent document bag paradigm focus evolutionary c ia development classical boolean ir broadly considerable demand learn run boolean platform restrict level leverage human note concept user want user search newly trade ask human reader economic due relationship appear benefit instead bag es corpus realistic news stream well fit conjunction evolutionary replace token wikipedia concepts precision considerable machine svm well robust furthermore compare produce es gp find query main contribution automate source wikipedia semantics es co evolutionary summarize key summarize document topic retrieve generate token decide perform token document analyze recognize interested matching token document concept token towards rather concept contain human level knowledge provide wikipedia semantics behind author utilize semantic efficacy concept document query search often available document mark entirely avoid relevance document query learn task contribute co evolve evolutionary training document document base single query multiple voting fit happen produce fitness objective space towards relevant irrelevant widely query produce fit token framework evaluate present concept framework evaluation result outperform idea wikipedia utilize briefly leverage wikipedia link development automate query particular genetic paradigm chen knowledge information provide way share despite limited engineering cost manually build resource keep resource know extensive limited need expensive recognize motivated research towards resource speak wikipedia thanks wikipedia rapidly maintain arrive daily wikipedia research use comprehensive review acknowledge et automate close reveal many similarity wikipedia article overall include hyper category wikipedia large al wikipedia solid rare mix recent wikipedia feature wikipedia considerably internal link concept bit closely instance wikipedia wikipedia article broader belong category unite broad category title article addition link represent concept refer concept article connect article bank page semantic wikipedia treat wikipedia concept model query notation wikipedia wikipedia representative certain concept follow recognition several resolve present trivial commonly automatic es question concern concept pair idea wikipedia provide need semantic evaluating measure corpus around quite wikipedia background take technique exist well wikipedia soon measure text wikipedia recent proposal propose internal link wikipedia approach correlation adopted essentially google wikipedia link define uniquely identify wikipedia concept gram concept principle article lot link likely highly link percent financial bank article course article thereby essential reasonably reliable way measure concept far semantic conventional setup concept perhaps relevant ask concept purpose link wikipedia wikipedia term collection wikipedia detect far rather intend document query allow mask concept illustrate document receive large general economic car model still paragraph trade car prototype prevent strongly link evaluate automate drive query need give learn help definition recall grow become solution formulation roughly weight query relevance feedback system feedback remove adjust system document represent current surface examine step learn become range genetic es query seek pick one process step see figure generation topic concern boolean irrelevant I algorithm es account appear one strongly gp es retrieval give evaluate incoming framework task match es pass name entity express term name entity matching rule rule module whether match currently es concept document rule return filter find match es return retrieve document terminate datum use match step ht rough overview closely retrieval wikipedia document continue rule evaluate detail document identify build al et recognize term act concept extend category wikipedia name entity concept name entity modification consider name entity general discuss bank name explicitly mention say sufficient concept exact concept name clearly account specify es document pair collection name entity wikipedia document space document document set name entity concept find wikipedia document document name entity recognition usual first wikipedia separation name concept sequence detect link correspond step recognition fields crf classifier model information construction wikipedia name examine pick recognize rule composition es provide picture es voting query vote weight represent relevance presentation es queries es rule evaluate individual voting system generate es rule summarize es discuss system begin boolean distinguish concept query hereafter ordinary boolean query consist part query utilize wikipedia match document expression wikipedia concept syntactic second document replace query dr k concept purpose threshold threshold function name entity sensitivity purpose name entity threshold name entity definition query request document concern receive lift repeat mr document car car document query point evaluation entity problematic sensitivity consider whether examine recognize strongly relate therefore concept equal acceptance provide recognize name entity reasonably strict level name entity concept high link mixing would serious able entity wikipedia tool due high deduce irrelevant ready fitness measure quality individual voting output fitness query admissible fitness evaluation precision set respectively denote define query benefit resolve document fitness evaluate quality deal contribute voting let finite collection query vote fitness respect vote base relevance leave research voting relevance several alternative query consider relevant irrelevant weighted take account help overfitte document set discussion es es denote admissible boolean query form wikipedia voting evaluate document relevance rule es query es provide query query learn search possible query optimization admissible es maximize collection document let denote
eigenvector eigenvalue corresponding eigenvalue come low neighborhood propose manifold algorithm explicit name preserve nonlinear precede find linear reconstruction reconstruct near weight following represent reconstruct neighbor form non entry th one entry preserve achieve problem show come simply section get matrix eigen consume new sample generating take operation exponentially extremely consume simplify remove hadamard reduce simplified summarize output summarize kernel method vary computational inner product obvious complexity explain linear may taylor nonlinear polynomial assumption approximation mapping test surface embed come space near underlying manifold image person intrinsic rotation sample nearest neighbor right sample randomly recover underlie satisfactory cost support computational image handwritten digit resolution intrinsic freedom sample neighbor train left column respectively randomly successfully underlie shown fig explicit manifold representation apply generic furthermore dimensionality reduction technique explicit preserve locality geometry provide sample simply test real effectiveness algorithm stand bar generate result stand datum versus test f f stand versus testing sample plot dot circle learn learning dot red circle versus number theorem remark zhang wang zhang state mathematics chinese china mail zhang ac cn field science drawback application practical linearity may restrictive explicit dimensional representation far locally derive name experimental result effective nonlinear geometry sample previous work manifold nonlinear draw interest manifold basic assumption sample smooth embed ambient around object high equal pixel aim intrinsic freedom input high sample draw linear laplacian le dm tangent alignment riemannian great success low meanwhile manifold face compressed expression hyper spectral shape appearance main manifold representation output embed procedure contain repeatedly extremely consume sequentially limit manifold linear projection assume datum low dimensional representation locality projection neighborhood preserve projection graph although linearity restrictive mapping manifold learning utilize space mapping implicit depend extremely polynomial datum mapping learning method implicit sample embed mapping specific kernel find projection high dimensional representation space clearly mapping handle sample lie meanwhile experiment combine nonlinear manifold paper concentrate manifold polynomial world illustrate validity review detail demonstrate conclusion exist algorithm linear nonlinear manifold notation use vector normal capital letter represent letter datum euclidean sample dimensional matrix norm vector preserve manifold cast two global preserve locally preserve pairwise distance unfold local maximize alignment preserve dimensional le preserve adjacency mapping preserve geodesic representation extend preserve representation assume representation denote project linear coordinate basis locality projection provide mapping laplacian le training le train dimensional representation preserve optimization algebraic diagonal whose entry eigenvalue solution small projection provide sample high find representation independently projection compute linear training weight solve problem ij neighbor eigenvalue projection high easily map locality preserve neighborhood preserve respectively projection unlike lead easy eigenvector reader besides manifold extension representation unseen manifold base common method manifold technique employ come et unify extend le eigenfunction dependent dependent implicitly le
write definition univariate cdf define many marginal application learn arbitrarily random variable marginal varie deviation unobserve deviation process monotonic parametrize latent shorthand standard univariate multivariate equivalently express model model learn subtle assume time period period condition know normally common assumption relaxed central ultimately wish hyperparameter transform kronecker delta maximize function intractable circumstance laplace integrate make find marginal relate g j kt approximation separately eq deviation goal treatment logarithm unnormalize laplace use taylor maximize entry problematic expectation iterate order cholesky laplace near numerical instability small finding laplace likelihood numerically stable maximum definite since decrease newton size always furthermore condition eigenvalue let newton update expression approximate likelihood evaluate numerically conjugate initialize vary make find approximation integrating give take function iteration newton method converge mcmc later make gaussian elliptical sampling extremely effective posterior correlate update element axis univariate prediction gaussian eq weight exact draw gp take non real deviation monotonic positive infinitely towards practice small extremely certain input wish literature assessment volatility variance follow observation proxy truth g rank compete derive use simulate call make ahead forecast observe make forecast volatility predict time safe comparing use panel true volatility laplace exp laplace historical volatility table result forecast panel accurately dependencie manually decrease replicate behaviour quickly exponential tend peak volatility extremely reconstruct forecast dramatically exponential region low also peak suffer mostly change daily dm great become assess refer return calculate dm day window return day volatility trading day historical mse historical unlike historical assessed see ahead forecast forecast operate suited datum vary exchange learn therefore convenient implement copula develop volatility outperform separate dependency distribution arguably far gaussian marginally cdf marginal probability density learn pdf show parameter stationary attractive advantage rich brownian mat ern periodic periodic learn function volatility copula rapidly become popular copula copula arbitrarily encourage copula bring machine historical exp la mcmc prediction historical dash la ahead volatility forecast dash blue learn marginal thank ar grant definition cm ex copula describe random distribution volatility copula process predict deviation inference laplace approximation monte carlo alternative comparable outperform financial unlike miss incorporate covariate rich structure measure measurement minute minute learn separate distribution separate marginal copula copula recently financial copula role play statistic open copula correlation compare describe volatility bayesian distribution dependency sequence leave get away important index economic economic time volatility arguably volatility index outperform discuss dependency distribution intuitively n cumulative cdf uniform random transform formally cdf nu intuition let exist uniquely determined f copula
unit state conditional conditional expectation dimensional denote simplex segment triangle take policy belong stationary policy stop decision terminate alarm choose variance markov variance penalty stop uncertainty alarm represent change alarm let alarm stop pick state alarm alarm state constant informally multiplier seek cost ii allow choice cost take change state time denote constant depict delay delay e sensor maker need thereby denote speak affect penalty cost incur eq due therefore reduce stop cost belief measurable endowed product algebra expectation ce adapt sequence eq economic discount non guarantee time delay easily delay kolmogorov namely consider solution bellman programming equation amenable c compatible compatibility decrease respect course stop coordinate albeit q bellman program q empty value methodology comment iteration confusion iteration bellman sup banach generate sup metric bellman equation belief practical indeed nonlinearity formulation although value iteration exploit structure switching devise algorithm solve cost stop problem penalty cost possibly observation ph sec namely sec discuss implication main sec approximation stochastic compute formulate ph distribution indistinguishable probability satisfy eq q choice generality translation notation order lattice denote penalty threshold theorem show stop stop choose equally alarm become delay cost assumption discuss sec observation appendix non maker assumption ph detection cost ph ex line threshold switch curve individually connect policy union nonempty threshold line ii distribute threshold trivially appendix intuition behind sec give delay cost stop continue decision maker false alarm f p e programming solver delay stop cost hold ph penalty say e stopping size ph matrix observation describe sec even characterized detection ph stop geometrically coincide total geometric say classical continue nonlinear alarm false alarm kolmogorov additional equivalent geometric threshold kolmogorov criterion trivially convexity threshold optimal case sufficient tp pp maker interior boundary determine theorem determine portion curve lie interior simplex portion lie simplex comprise conceptually eliminate ensure always following sufficient transition element likelihood satisfy subsequent state follow straightforwardly belief compute curve empty discuss sec outline ex maker stop cost assumption decrease decrease condition refer lattice programming policy discussion decrease submodular establish condition ex since decrease ex ph change variance construction ex delay monotone submodular ex preserve update increase iff detection book exponential poisson etc preserve theorem show iff increase sufficient tp order kernel detail satisfied class transition necessary comprise meta involve show ii decrease spirit convexity require monotone update belief stochastic see submodular require line chain restrictive entire line policy belief prove simplex cover line threshold curve illustrate theorem line connect simplex say curve intersect intersect say convex recall state triangle insight visualize say monotone almost almost everywhere decision trivial include belief decision though region mention give region policy increase curve intersect ph curve intersect theorem lie say lot penalty fig satisfy monotone go optimal condition happen hold numerical ex construct region assumption subsection assume ex hold estimating threshold need essential parametrize optimal apply optimal approximation threshold simplex attractive condition necessity linear linear policy parameter hyperplane parametrize linear threshold approximation curve ex hold linear set mean belief states line iff ii iff theorem solution constrained remark constraint necessary increase line define increase threshold yield clearly threshold conclusion slope become intercept lie imply fig slope lie fig show decrease segment start region requirement increase x estimate resort denote aim linearly constrain evaluated I optimization convert parametrization trivially constraint equivalent unconstraine simultaneous perturbation threshold ex social switching initial iteration coefficient pick evaluate number optima try initial condition dimension local one straightforward markovian give score process reinforcement applicable thereby yield threshold policy likelihood completely specify hold reinforcement tracking stochastic observation vary second deal threshold curve transition start state finally jump course since delay delay delay give visit process reach convenience penalty choose expect stop cost cost decision stop tp e sec delay cost conclusion observation degenerate ph time jump ph ph distribute period jump continue model distribute denote state start jump ph jump sec ph start delay decrease cost conclusion maker design satisfy maker solver assume state indistinguishable obviously unable transition matrix long tp follow penalty region deal exponential geometric change consider exponential penalty ph time formulation involve control motivation show risk stochastic state risk ph distribute show switching interpret false alarm delay cost control parameter let geometric cx e cx cx identical delay delay get delay cost additional rest time detection ph denote threshold function threshold nonconvex stop would think continue belief belief policy public continue stop global stop belief local main reason behavior cause update concave necessarily function concave explicitly action irrespective irrespective therefore social action nothing learn take belief comment interval modification composition filter localize interval behind theorem optimum formulation stop threshold optimal action constrain optimum formulation motivate optimum due sequential aid learn act optimize sec ignore cascade rule become denote action adapt sigma detect state second term involve pick action accord rule specifie agent chooses mode immediate cost pick reveal thereby social learning stop state equivalently terminology observation choose belief pa py remains incur equivalent final term define sequential seek policy achieve tradeoff incur act analogy characterize threshold require ce ce ce xy iy ce iy ce ce ex imply cost decrease costly decrease intuitively make accurate private ex satisfie structural switching union social threshold pick pricing implementation view protocol need individual agent store example finally convex stop model markov jump sec jump target evolve observation jump decision stop evolution belief control mention sec bound achievable main theorem ph agent act sequential index belief state choose mode depend mode obtain distribution mode observation obtain mode dependent scheduling accurate cost mode tradeoff observation mode model view confusion communication channel choose mode incur choose since mode optimal overall cost incur mode agent affect subsequent agent problem determine intractable however programming let belief pick low policy appendix usefulness stem rigorous low coincide region incur achievable trivial imply coincide apply dynamical concavity particular proof appendix comprise first concavity trivial cost apply detection scheduling distinct matrix markov question transition matrix phase large optimal type pose problem belief explicit transition matrix I explicit large incur cost optimal see distribution prove dynamic making verify tp ii iid uncertainty say total example change transition small ph model matrix ph change tp show transition order namely theorem transition model conjecture iid tp kolmogorov reason replace order paper assumption appendix illustrate structural theorem detection ph operational form dimensional unit grid length fig satisfie individually connect theorem say optimal stopping assumption hold recall condition fig satisfy condition appendix stop long monotone policy paper stop empty simplex c satisfy stop feasible choice matlab choose assumption illustrate ph probability transition geometric region mark structural distribute variance prove existence switch appendix give threshold several consider social scheduling penalty sensitive stochastic switching order simplex structural belong hold robustness since still useful various detector paper prove detection monotonically clear compare belief ratio order specialized order restrict simplex preserve definition subsequently order belief vector greater replace stochastically dominate I dimensional vector iff coincide state space partially always state simplex state lie segment connect comprise form order great respect chain e use great element comparable l variate lattice operator tp ordering variate univariate say tp p scalar statement row submodular chain e submodular argument decision f assumption major stochastic tp generic iff hold ex sec ex sec ex sec decrease assumption ex ex sec sec sec sec part prove condition decrease family first dominate belief parametrize light follow straightforwardly negative yield condition nothing decrease suffice choose yield ex ii suffice ex mathematical induction v decrease form ex belief e show omit condition respectively monotone increase respectively ii equivalent sec inequality e f iii theorem sufficient decrease imply submodular show induction decrease submodular imply pointwise limit establish submodular policy increase key characterization switching segment connect lemma segment moving segment always threshold note simplex cover consider region segment region intersect region action requirement case convex assume nothing prove line leave intersect action optimal consider iii follow first follow lemma comprise concave uniformly concave belief construct piecewise easily piecewise function compose concave function via concavity iteration fix arbitrary vi converge choice see piecewise see positively I k concave composition concave since linear concave concave concave follow preserve complete step concavity imply concave intuitively segment converge lemma argument repeat demonstrate convexity inequality iff e iff iff difference compare update include continue necessarily concave concave verify dynamic value decrease straightforwardly since concave concavity straightforwardly piecewise interval abuse recall straightforwardly sufficient theorem statement straightforwardly statement return yield since linear lemma interval I prove directly theorem ac c argument union region connect establish condition therefore concave jensen eq decrease introduce proof return rest q since follow take first recall monotone theorem optimal type threshold switch curve distribution order stochastic gradient optimal linear curve consider change threshold switch curve arise detection sensitive penalty stop time agent scheduling change make low optimal varie change impose monotone ratio exponential delay lattice programming making monitor finance deterministic goal delay subject alarm see formulation variable random distribution detection involve need tradeoff alarm frequency delay change model geometric change realize chain policy function belief px u u exist generalization framework dependent markov main generalization type goal lattice programming threshold policy time phase change ph ph use discrete change form find ph uniformly describe ph ph ph states multidimensional generalization comprise alarm variance penalty stop formulate quadratic existence quasi prove exist optimal threshold distribute ordered detection characterize policy stochastic lattice policy govern belief ph design policy threshold
security much want player classical fp good fp strategy utility function eq response give max nash point good mapping static subsection fp fp version fp mix fp calculate player basically exponential time player weight security game assumption payoff decrease attack payoff attack attack decrease attack payoff htp update opponent completely play accord strategy fp employ algorithm evolution fp equations evolution empirical static game subsection player action game admit nash ht response monotonicity fix unique scalar independent similarly pair completely specify mapping suffice write mapping mapping detail similarly transformation mapping constitute static game respect independent strictly increase distinct nash equilibria coincide generality assume strictly two curve equilibrium static proposition fp weight action estimate fp fp calculate writing estimate fp dynamic equation see equation discrete invariant system fix examine stability linearize see jacobian similarly local stability eigenvalue jacobian estimate frequency frequency converge nash equilibrium good response nash empirical frequency run examine hereafter fp invariant frequency size decrease algorithm step either keep fix base previous window initial minimum window frequency opponent strategy randomly play action window decrease compare nash static threshold rhs frequency result dynamic fp figure frequency limitation ne converge thus worth frequency ne fp fluctuation process graph limitation ne fp entropy choose empirical frequency fp fp fp however possible incorporate originally fp process converge adaptive adaptive fp higher less opponent play nash equilibrium dynamic update adaptively player converge fast fp exist yet research extension conjecture department electrical laboratory university st il edu edu security view nonzero sum game play evolution game play opponent vary alternative scheme update examine dynamic play stability equilibrium player theory recently tool security payoff player equilibrium play gain minimize player payoff game play player learn opponent fp current either fast equilibrium strategy motivated examine tool applicable game extensively paper survey
easy obtain bin unfolding calculate especially nuclear consume often calculate element source instability solve difficulty unfold formulate paper place propose unfold unfold training priori contain calculate configuration grid approximation transformation linear demonstrate possibility unfold whole transformation distribution create error component calculate bias bias unfold unfold validate wide nuclear model monte unfold true experimental simultaneous identification priori previous carlo training transformation unfold minimal bias validate unfold deconvolution robustness boost kf experimentally measure distribution differ true particular unfold unfold approach solve priori unfold unfold idea develop identifying solve unfold measure create sample calculate approximation minimize bias restriction shape linearization multidimensional unfold paper organize follow solve unfold formal method unfold unfold system basic unfold example elsewhere unfold run unfold method training investigation reveal bias unfold confirm statistical case identification transform true histogram histogram residual linear majority physics approximate distribution identification determine transform physical experimentally simulation use identification impulse impulse bin impulse input row contain histogram reconstruct output matrix q statistical solution type instability priori identification use impulse use experimental present write content reconstruct output variance statistical reconstruct generate formally square q calculation column reconstruct parameterize use row row copy example sample coincide impulse combine fs include new eq transformation element exclude satisfy maximum c calculation row row rather reconstruct bin row bin criterion relate minimize unfold minimum ellipsoid possibility improve introduce criterion distribution training goodness goodness reconstruct experimentally achieve training satisfy test statistic reconstruct distribution threshold reconstruct threshold level error run unfold boost training sample involve independent realization histogram description illustration elsewhere take parameter interval cm experimentally measure distribution detector acceptance resolution function histogram distribution bin distribution histogram choose bin previous identification comprise parameter simulate represent used identification histogram calculation great element transformation h cm xx basically reflect
implicit define discover behind attack vector briefly comprise sequence current adjust match output profile structure detail subsection form attack token attack finite hmm attack enough describe attack entire view production assign attack represent non terminal path possible production product transition emission capturing obtain implicit generating attack express posteriori maximized dirichlet token sequence calculate emission token bayes balance represent attack structure generalization relation build token come model structure structure merge merge maximum posteriori posteriori merge current merged model repeat token process build attack profile attack profile describe parameter token emission record token attack follow generate attack simulate attack evaluate possible phase start token subsequently token seek viterbi find token hmm purpose generation viterbi attack attack generator able attack attack embed attack attack technique code experiment attack concept subsection metric discover study finally measure attack identify web neither space attack state attack successful effective attack question effectiveness attack configuration environment configuration generate attack attack third attack web allow attack attack application manually check attack report attack really dataset collect collection totally attack several obvious characteristic symbol favor totally token unique token sequence list elementary high school school source attack list generate attack fp worth knowledge attack effective aspect attack interesting attack table attack attack http com prefer double attribute specification attribute separate attack format recognize internet web site attribute attack width height tag attack serious attack exploit body show attack tag second show page microsoft internet google think could social attack attack digital characteristic end tag indicator style attack body text reader third attack tag indicator attack string well third attack form original form block enter separate denote window attack attack hand ever test public rule attack attack detect character attack false still attack expert phase technique include box hybrid strength analysis variable code without however result false positive box input generally tool equip lot test user effective test serious utilize response server attack ignore hand attack box attack leverage usage modify attack back attack generate attack hybrid static verify identify test automate attack work maintain attack public source expert specification attack generate source attack identify web manner attack take attack hide structure thus generation mechanism attack experiment technique effectively attack help provide attack consideration web acknowledgement security center national acknowledge comment suggestion web department engineering national university institute science mail edu tw suffer site attack attack study attack art potential web application attack aim generate attack automatic generation attack hide attack implicit bayes determine model generalizing automatically attack practical attack ability testing identify helpful collect public attack web application security project already web application page attack embed specification tag event situation attack attack body code invoke technique beyond attack device like efficient attack attack page attribute double attack double critical character introduce attack attack vector embed page previous study focus create attack generate path equip attack exploit element attack structural learning mechanism attack vector generating attack present generate attack public attack attack structural module build generalize generate viterbi capability tool attack element structure attack internal string function attack many possible way invoke specification attack attack generation different web aim implicit present attack thereby public handle structure vector attack contribution follow automatically analysis attack ability helpful approach generating attack section conclusion structure generate attack mechanism attack attack attack generator phase phase attack generating phase attack attempt extract attack attack profile relation attack generation mutation attack token attack generator attack attack attack automatic attack attack overview structural depicted profile relative attack preprocesse module decode identification
introduction suitable latent variable present augmentation posterior standard extension model allow home tie em sample popular generalization section apply competition statistically comparison associate datum seek admit enjoy pair distribution rate low time allow pair arrival instead simple shape inverse information augmentation algorithm proceed similarly summarize introduce variable complete assign prior eq log proceed independent first map coincide simple augmentation z conditional augmentation proceed comparison home home disadvantage times home home times play home home total use maximize algorithm flat mm gibbs iteration allow comparison log introduce latent log tie improper maximize update perform win team recently likelihood parameter otherwise update q variable proceed iteration involve individual individual introduce follow ranking position event individual receive augmentation augmentation gibbs sampler model q write rescale dirichlet improve mix add normalize current drastically value log follow update rely parameter suffer augmentation slow arise hyperparameter assigning interesting variable implement conditional give propose accept relate random suppose capture formalize sequence sufficient graph log give require sake brevity augmentation whose density argument feature vector element importance object gibbs derive consider categorical multinomial logit em posterior rescaling influence ensure henceforth algorithm datum study compare property gibbs sampler slightly latter propose update iteration ranking dataset gibb run lag four confidence reasonably sample consider poorly sample car auto united involve find place need predict one base map etc test figure initialize sampler flat improper sample run burn detailed identifiable rating four place effectively ml mmse minimum square together deviation ten h small rating strong restrict international adopt model historical consideration system game either section predict outcome number walk bayesian sampler outcome different hyperparameter em sampler sampler iteration report benefit full report autocorrelation associate large display mix generalization arise numerous special straightforwardly algorithm elegant mix experimentally outperform grateful comment axiom conjecture remark em france mm pour du de dans les mod de inf est l carlo es des de metropolis es des maximization des de pour inf de
associated dpp corollary establishes induce rl convergence rl initial policy rule dpp rl unlike incremental rl involve decay know incremental rl learn step bad choice lead dpp rl seem suffer dpp rule rule dpp dpp rl fast rate dpp basis preference linear column vector preference nan nan case dpp operator project nan minimize expectation equation infeasible estimate q draw also nan nan rx nx x fitting least present dpp operator h n na n kx aa kx x kx aa nan n rx nx nan nan empirically examine property dpp rl algorithm several discrete action iteration vi limit per replacement source algorithm mdps benchmark interior proportional text problem consist cx arrange end chain state nx px na px k l k transition interior select state correspond formally state cx nx probability transition end interior associate reach soon consider stochastic state cx k arrange possible action state state otherwise automatically state intermediate reward upon take wrong proportional difficult goal transition transition mdp leave cx nx center assign remain reward move state long avoid right corner avoid state rl compare like dpp rl action pair iteration vi vi batch learn whole loss iteration consistent discount optimal value iteration polynomial linear otherwise asymptotically dpp rl fix fit make numerically variant subsequently infinite horizon discount measure accumulate product variable current depending keep optimal accumulate replace unique map space radial basis span fix definition next happen replace step compare select unlike action induce action soft policy algorithm implement matlab execute hardware specification average begin run interval area deviation start uniformly figure error observe reach near optimal error estimation solution comparison variance suggest complexity remarkably use increase conclude reduce rely incremental ac actor guide policy ac extend ac actor unbiased estimate incremental rl variant optimistic value concern presence incremental incremental rl problem relative restrict bellman instead approach bellman likewise formalism return fashion low kl another rely relative policy difference actor dpp inverse dpp optimal solution convergence dpp convergence policy programming dpp compute policy mdps theoretically prove performance loss bound dpp express supremum accumulate oppose free rl dpp rl estimate policy prove dpp rl numerically mdps show dpp rl unlike rl therefore suffer decay stochastic policy exploration pac mdp policy different kind dpp rely ac action approximate rely error natural search kind performance bind rely approximate dpp value dpp research pac dpp previous theoretical fit iteration analysis action rule computing idea monte soft lagrangian lagrangian function respect x hilbert science hilbert department definition nc novel iteration policy dpp infinite horizon process performance loss presence bound accumulate error oppose norm policy dpp cause monte compare variant dpp dpp rl reinforcement markov carlo dp action bellman systems tractable value monte programming iteration real world derive guarantee discount w r action policy approximation however supremum variance accumulate consider asymptotically error mathematically justify dynamic dpp prove dpp asymptotic dpp accumulate suggest dependency naturally incremental dpp update error article notation dpp investigate property dpp carlo convergent rl rl estimate optimal present replacement review mdps reinforcement rl standard notation euclidean norm norm ny ny discount mdp denote discount transition next upon state real value reward joint space mdp assume value immediate reward cx determine action call action independent last stationary policy policy action denote discount reward associate therefore discount define notice nan bellman equation eq likewise cx nx likewise contraction supremum norm factor value define linear define na qx cx bellman bellman operator eq derive bellman add entropy induce double loop reduce double dpp purpose motivate characterization theoretically investigate iteration asymptotic dpp new nx penalty term policy expect x bellman cx modify maximize reward policy baseline policy form cx x x cx respectively satisfy function compute maximize interested quantify mdp idea improve policy towards base newly compute regard repeat lead outer loop inner loop consist follow follow derive dpp update preference cx deduce na dpp analytically tractable dpp equation soft max na entropy obtain converge analysis somewhat simpler replace deduce dpp preference soft policy iteration gradually move policy greedy policy finally dpp result solve dpp regardless always incremental whereas double policy policy difference double loop c x kx ax x x prove dpp dpp uniformly cx nx policy dpp soft policy soft greedy policy immediate consequence q induce converge mild mdp unique ca na cx dpp appendix dpp problem generalize dpp problem technique dpp
mean refer method normally involve rejection well old may preferable processor implement fast number uniform generator normal require cycle good generalise require cycle four performance wide new pseudo generators generators stream pseudo evaluation elementary log normally number another normally thus pool normally another normally distribute multiplication inner loop implementation processor processor normally proportional scalar speed comparable fast uniform generator processor aspect processor discuss conclusion idea generator keep pool pseudo form uniformity counting occur bin time seed etc random number sample fourth repeat observed fourth moment sometimes significantly confidence explanation value pool consider pool q correlation successive return reduce eliminate pseudo generator idea processor generalise generator cycle however satisfactory property provide distinct describe appropriately orthogonal transformation satisfactory pool subject inner loop streams pseudo generate processor require thank I also discussion regard generalise research implementation generator propose new class pseudo generator generator require stream loop essentially matrix multiplication
average split conclusion draw figure go optimal well simple space linear wrong feature recursive look kernel specify boost automatic kernel capability linear feature enough produce relax linearity go automatic parameter vertical axis performance ap auc top task c essentially argue put emphasis feature space wrong automatic formula important empirically high high level alone reject hard development abuse notation usual classification training classify long regardless still implicit feature space kernel graph remain describe deep machine conduct commonly often enough apply trick implicit repeat deep recursive correction capability report deep architecture literature science engineering research remark example inspire grow interest datum study classification focus linear classifier kernel produce consider classifier space call architecture machine deep diffusion notation subset unknown respectively study predict protein interaction protein know associate g predict activity product assess idea apply call machine focus end really reason study conventional kernel suit left limitation alone else general instead independent standard problem wide variety simple neighbor near centroid sophisticated machine simplicity predict neighbor distance due machine learn label machine sparse solve cause observation shall simple kernel machine use analyze graph theoretic need adjacent notion often diffusion go roughly similarity kernel short paths adjacency kernel kernel matrix eigenvector compute j mi net respectively average class subscript summation general machine shall mostly kernel easily procedure determine coefficient kernel coefficient emphasize framework svms key machine regard space centroid imply base diffusion intuitively appeal way neighbor nearest centroid machine feature svms possible nothing prevent flexible use kernel kernel density serve distance write radial therefore estimate subscript notation emphasize estimate predict decision kernel whereas summarize say feature use kernel linear sufficient relax linearity choose nonlinear distance metric rise kernel use kernel update put discuss however space machine linearity construct via imply yet except j machine generate recursive deep machine level one kernel present general focused base one certainly base svms appear return return else accord return certainly go reasonable heuristic pair heuristic usa serious accounting attention energy email make publicly unique email account send email account status create email account never email another label status unbalance balanced task l svm collaborative social interaction law ask regard law issue status flexible third work e work rather adjacency friend status two management class highly unbalanced one
model smooth part correspond probability effectively combine generally suffer two drawback computational especially two kernel double tree accelerate drawback invariant potentially since ideally throughout mixture b b table synthetic dataset dataset mixture variable time old perform increase dataset obtain conditional train log double fold kernel subset fold compare double perform mixture cover note use wishart part secondly require wide possible almost though disadvantage bad widely dominate cover small dataset mainly robot less evenly match initially bad kernel high perform mostly case asymptotic tree partition approach markov model fundamental close incremental conditional density double structure result well kernel ability incremental using could obtain invariant would problem conditioning variable cover paper possible tree extremely structure application lattice structure remain tractable cover create distribution cover maintain spirit p thank point gr extremely thank go anonymous provide circle draw mm enable incremental parametric rely upon create sequence cover maintain within remain tractable specify cover fundamental approach incremental conditional cover approach dominate fast tractable incremental bayesian random walk simple variable construction tractable flexibility example estimation obtain conditional informally evidence wish denote borel idea tackle cover sequence cover contain sequence set combine obtain variable appropriately allow state measurable respect algebra contain sequence sequence cover subset partition tree cover refinement alphabet subset partition variable unit generalise construction shall focus conditional estimation structure measure index context context context cover countable sequence cover compose local index specifie conditional though local efficient whereby stage firstly sample derive sequence construct two bernoulli draw rely structure observation random cover stage stop observation context contexts context bound proof geometric markov estimator require walk appropriately markov start refinement let fact k specifically stop parameter contain prior conditional cover tree easily applicable start correspond part tree probability multiple context classical estimator secondly straightforwardly extend density via walk perform thing model context tractable similar nevertheless trivially apart separately ratio hand incorporate estimation context context estimation probability proceed method tree structure tree estimation maintain tree suggest finite set method double
multiple infinitely matter thresholding converge j regression estimator estimate unless specify satisfie mention huber poorly outli detection never outlier leverage reject outlier little shrinkage nonconvex cutoff yield outlier design robust penalty loss tune illustrate threshold follow soft thresholding px x px q threshold j j cost regression qr cost maintain dense give ii outli wavelet problem wavelet regularize step simplified thresholding shrinkage gram therefore recovery reveal outli easy involve operation inversion observation outlier difficult desirable robust I mild ignore clean contamination tune via bic relate suggest estimate square define I estimate known multiplicative help bad iteratively update method indicate characterize observation observation case correspond leverage change outlier leverage combination decrease stands approximately half encounter take criterion second narrow range counter smoothing choose neighborhood determine maxima spline way counter carry way highly outlier affine regression without intercept include outli tune compound procedure behave poorly report library refer provide default point experiment resample py procedure outperform mm term robustness theoretically r robust r version library apart cutoff outlier fully situation time report identification measure fraction label outli simulation outli detection serious cause latter lose ideally simulation table good similar co ht co co mm mm library nevertheless mm estimator identification outlier outlier situation experiment behave outli work unfortunately four low bad large dominate significance mc pair replicate numerator outlier minus large versus acceptable tradeoff cause scale identifie also apply simultaneous outli surprisingly soft thresholding fail challenge sparsity elastic net net severe adopt thresholding ridge refer qx thresholding portion remove influence remove partially extent control helpful shrinkage play bring challenge coefficient choose component j candidate run predictor getting fit later marginal fdr sis run choose selection outli detection concern solution concentrate predictor derivative nm nm nm outside range analyze fitting robust model also screen via proportional ridge ridge ordinary parameter minimize combination rich reasonable experience improve prediction c r satisfie joint kkt equation estimate nc q exactly huber joint estimation two yield huber moderate improvement replace claim argument choice characterization actually norm penalty huber completely algorithm design give thresholde p ft u monotone u ft lemma corollary address university email department stanford university stanford stanford partially nsf grant dms dms grateful anonymous helpful remark literature plus one apply criterion fail correspond soft thresholding identify hard connection bic outlier various iteratively reweighte least square cost avoid extend keyword sparsity analysis refer one routine often go serious selection ordinary least ol outlier regression robust p commonly look well residual detect leverage well way outli estimate base leave value suggest observation outli threshold reasonable n I remove one look residual method fail phenomenon th remaining outli outlier may mask go outlier one serious take observation parameter trivial meaningful thresholding denote exception note regression run good current method preliminary regression point surveys robust regression convex section soft criterion outlier section iteratively reweighte investigate discuss problem detection extend technique dimensional conclusion identification plus goal outlier suppose sparse great paper outlier penalize convex suggest alternate ol estimate soft match detect outlier fit regression soft soft work minimax optimal predictor residual residual probability conservative outlier prefer tune initialize must use discussion initial pseudo outline soft defer converge even section lar shall generalize nonconvex penalize outlier moderate fail remove illustration identify true matter plot soft relevant choose sure
system solution give ill apply symmetric definite subject convex program compatible although submatrix unique kkt length ill solve always condition alternatively monotonically eliminate also regularize equivalent define block limit construction dependent general method incomplete cholesky method return method access might improve condition solve dominant magnitude dominant choosing scale diagonal side column unit see converge condition incomplete cholesky preserve ic ic ic simple cholesky infinity expect small exclude enhance accuracy eigenvector condition small although reasonably accurate system lemma computation machine ghz intel gb ram solver system decrease even desirable rule cause termination accurate similar ls approximate illustrate compatible linear ls v effect round plot residual show terminate directly compute differ monotonically near iteration smaller excellent reach ill condition end mean reach satisfactory claim throughout therefore graph compute throughout iteration condition last residual differ eventually even bad reach soon iteration figure system involve author al retain iterate estimate exceed readily hermitian matrix precision recurrence formulate approximation method numerically future research mention source available acknowledgement van thank support grateful anonymous suggestion manuscript v e bb converge l matlab example c cg solve system singular least square cg solution understanding motivate design compute qr rotation condition system rule residual residual matrix f f iterative l directly complex hermitian ab pose aa ill condition solve system return solver type reliably compatible return solution ill condition illustrate system discretize value problem involve error toeplitz l letter denote cosine angle th unit letter vector letter denote quantity symbol shorthand l review section estimate process arithmetic stop unlikely define ab following keep change scalar shift becomes shift inverse rank might exactly strict singular see subproblem expand qr form later solution substitution obtain nothing else place choose solve compatible singular singular handle describe qr factorization full effect later k tend compatible singular decrease cf r bp ap ap x x unique ar ar v ls solution solution require useful recurrence consider framework relation norm note version take negligible v k ar orthogonal range phase ratio criterion solver large form van round plane good norm keep decrease compute become increase want compatible eventually even long focus involve triangular solve triangular solve final store shall propose consecutive qr column nonnegative element demonstrate extreme important main purpose ensure develop extra approach decomposition obtain process different round see retain perhaps convergence involve product idea understand right additional help show sort red circle plot circle subproblem eq figure remainder solve last update recurrence apply qr stop solve bl ax bx p u v z ax ax ax condition behave ill estimate k transfer w still short recurrence exceed estimate kp factorization e b l estimate ex ex ex k derive several summarize formulate stop condition r l regularization attempt ar recurrence relation norm residual otherwise zero recurrence relations ar ar essentially two follow directly lemma monotonically improve could extend diagonal inspire chen vector figure matrix vary scheme accurate scheme try add little iterative accurate need readily tb recurrence compute last therefore maintain update sometimes simple recurrence ax ax ask construct
receive unit user aim throughput channel slot designing policy could policy know channel reward obtain ts denote channel depend positively correlate channel positively whenever switch channel visit channel correlated channel switch soon channel visit visit case positively correlate channel channel horizon special general base concept policy specifically algorithm treat arm armed bandit one question operate present next desirable slowly duration step arbitrarily slowly channel reward play policy reward ji ji ji jk ji k discrete step q non sequence grow algorithm close dynamic channel regret constant relate chernoff hoeffding difference reveal expect either steady constant matrix omit expectation state prove optimal strong claim logarithmic know optimal chain case correlate open infinite offer bandit arm reward arm chain know seek reward obtain play hard consider hard parameter problem depend value solution prescribe optimal policy suitable treat different non bayesian armed arm optimal demonstrate spectrum channel expect reward aware average bandit access multi armed mab problem tool make dynamic uncertain environment multi armed arm stochastic seek arm reward classify know mab regret know policy desirable maximum particularly variant armed bandit arm evolve chain solve approach regime always tractable index parameter efficient new channel cognitive dynamic prove gap time grow decrease much sublinear average reward arbitrarily option markovian optimal hardness indeed option structure exact true high expect reward contain arm armed bandit operate minimize regret done handle adopt policy length play unknown
plausible finding boundary separate topology show bootstrappe way cluster microarray array effect effect simulate incorporate realistic generate simulated compare author gene percent bootstrap match way provide differently estimate presence absence rich develop tree posterior assess distribution proceed implementation tree combine pick distance multivariate generating geodesic metric arise path path cat unique geodesic tree trees geodesic origin connect path consist segment origin geodesic easily path call split transition transition introduce new meet geodesic tree let root label label formally root though trick include computation label convention consider two edge form uniquely identifies represent compatible add third geodesic present valid valid path geodesic partition property polynomial check uniquely form tree logical edge root compatibility intersection preprocesse classify edge serves share two tree drop think tree disjoint aid division subproblem division share treat part large shared edge classify tree edge work reach take share edge tree store reasonably second bin contain disjoint bin pairwise tree implementation theoretical computation practice dataset algorithm reduce check follow code check form computing form add weighting problem flow equal every vertex use runtime subtree property search flow drop edge subset happen minimum vertex cover point need bin initialize drop subtree path iteratively run follow max max flow represent geodesic give programming bipartite use set computationally efficient since minimum geodesic path final path implementation replicate minute second ghz processor curvature excellent length positively htbp illustrate triangle type geodesic vertex geodesic thin side geodesic thin see colored property close green triangle long actually triangle arbitrarily question try address space whether tree choice spatial representation one clearly pose year context dissimilarity measure quantitative computing section show long multidimensional scaling dissimilarity statistical method zero q offer summary give approximate center extract function come retain scaling see later distance nearly point apart sometimes modification distance tree simply resample overall variability conjunction dna h rr count rr tree rr type count branch simple shannon diversity get tree quite use index type project element project star length analogous interval difference tree region probability coverage package delta uniformly data set balanced bethe run simulation distance max sd bethe bethe raw split tree leave uniform split tree leave split leave ratio calibration configuration would approximate rough diameter scale implemented recently geometry follow could subject comprehensive review evolutionary align pose tree recommend sequence coherent process first proposal mixture tree various bayesian posterior tree bootstrappe mcmc generating make linkage picture combine tree single linkage available tree evaluate pick random pick discard approximated show see explain tree character branch tree run tuple topology colored representation focus far apart point however negative apart exponential dissimilarity sensitive noise around represent neighborhood reason think give ask question meaningful similarity ask combine information similar parametric notion account rescaling case dna far natural ask well branch tree close number sure differential neighborhood sense small induce sense closeness us stability perturbation bootstrappe look tree perturbation change bootstrap tree topology infer compete neighbor original estimate particularly interested ie tree boundary compete tree data star original exponentially contiguous tree close bootstrap topology respective share compete tree sequence interested sequence lead topology give estimate weight boundary position position maintain pair dna dna still proposal reject smaller great accept great accept minima introduce temperature position final tree hierarchical display root evaluate tree bayesian provide variability estimate substitute spread question remain notion issue way quantify variability diversity leave shannon available statistic evaluate approximated tree tree briefly correct statistic choice argue high price diameter geodesic triangle max approximation know distributional statistic dependence valid inferential become evaluate local curvature finite show close tree tree acknowledgement careful reading early pde lrr f pde cx cr r dna est resolve dna est tree tree tree combine tree tree combine bin tree tree bin dna est trees tree tree tree bin bin tree bins tree tree combine bin combine tree bin boundary tree combine boundary bin error bins boundary bin combine tree bins boundary boundary delta generate tree tree tree tree tree tree theorem theorem axiom title inferential summary tree set biology bioinformatic mining mathematical object stability summary distance develop equally show cat compare approximation multidimensional distance tree distance cluster evaluate whether array like cart lead analysis display natural computation advance report describe begin variability microarray plot patient microarray gene patient cross validate gene validate validate tree triangular validate cluster plot tree popular graphical evolutionary biology common standard root tree leave call root build organize leaf occur branch frequency compete tree categorical procedure tree tree tree variation coarse tree take value call refinement come account place question evaluate tree estimation overview current analysis come example include multidimensional scale tree use multidimensional compare validate influential hierarchical cluster detect mixture quantitative tell appropriate tree find branching order path anneal na I become case must together explain complexity evolutionary satisfactory simple evolutionary may characterize represent character familiar matrix tree h root represent tree process process one character generate character root random
optimal know cd procedure would goal exchangeability permutation lose exchangeability explanatory assume take take apply permutation invariant explanatory lost procedure account explanatory manner group exchangeable group explanatory observation cd relate realization aim approximate decision give emphasize cd often familiar terminology accounting covariate spirit currently section suggest naturally explanatory section suggest census involve certain explanatory statistical historical idea setup estimating proportion applying bring normal setup denote find risk motivation ideally sample unknown belong large parametric family np moderate curse belong curse may good estimator respect obtain suitable initially collection affine transformation distribute ty ty define trivial implicitly goal usual follow minor rt opt many harmonic start transform follow outline transform representation transform collection wavelet basis find see chen et hard special parametric ty bb p particular explanatory conditional sequel assume joint covariate follow useful represent confusion later establish follow estimate subsection appropriate affine depend explanatory basis arrive easy handle rough procedure prefer hence indeed account explanatory bring accounting explanatory suitable nearly procedure transform bayes asymptotic induce stein estimator consideration wide correspond consistent estimating stein method examine adapt n every case trivial minimize span square development see bayes follow triangular stage sequence q bandwidth given appropriate three adapt replace estimate x plausible least residual far demonstrate follow minimize equivalent favorable task find favorable reasonable vc regularization may require beyond scope residual inefficient might risk could cause transform structure transform separate original conclude advantage trivial square converge correspond trivial converge point treatment purpose parametric computationally intensive good sequence estimator obtain coupling dominate couple permutation give mean rt sub triangular array assumption equal condition symmetric replace permutation invariant though equivalence remark interested denote mind may explanatory interested may depend conditional ix heavy still try approximate among upon rate slow heavy might however city divide call belong sub include additional proportion area among live area adjust estimate recent census mean shape simulation number area area simulate simulate covariate area temporal analyze binomial setup binomial explain sub entry table sequel parametric may et simulate think suppose use record covariate number people area although well binomial relation response explanatory covariate year scenario specifically change year change roughly shape section transformation square transformation although couple compare risk risk naive estimator apply stage estimate correspond covariate worst moderately naive additional helpful see spatial statistical sub area neighborhood statistical area area belong sub base census proportion people area treat people neighborhood well temporal transformation simulation procedure explanation temporal strong induce isolate extremely smoothing behave roughly naive non parametric e transform option bad simulated risk section spatial introduce transformation also try transformation covariate temporal cause unnecessary among seven estimation kernel kernel denote straightforward apply estimating point close consequently truncation paper bandwidth validation et together improvement sensitive choice em j simple ann parametric compound ann decision chen atomic pursuit issue compound
explicitly whereas generative observable hmm observable embed large translate condition hmm hmm mixture gaussian number mixture invert diagonal case case simultaneously case wherein value use gaussian base k also linear chain hmm px evaluate discrete gaussian factorize computation n hmms hadamard index align rely upon distance computation address sequence computation draw stochastic generation use operator measure dynamic eq estimator previously implicitly learn advantage kde rbf observation evaluation gaussian isotropic gaussian admit integral kde dependency empirical include include rich bayesian network map maximal clique maximal clique observable clique distribution former clique compute empirical clique due variable latent empirical respect conditional expectation embed space clique eq variable clique distribution latent end compute many svm even intractable trick product without yield admits truncate taylor expansion empirically low rkhs feature even distributional space embedding give accomplish alternate kernel learn induce variable identify early maximal clique yield former latter clique backward notation clique cx cx likewise use encode treat identically gaussian latent symbol approach hmms plain kernel hmms fisher identity special kernel special mean exponential gx analog operator derivation require computable clique appear appear clique another one fix isotropic q map explore generalize scalar positive however learn complexity quantify term amenable generalization empirical expectation focus kernel rather high type covering exist cover formally mild abuse b e take correspond operator banach number convolutional convolutional decay exponentially cover act radius ball eq depend indeed decay exponentially briefly summarize decay kernel convolutional transform make fx p fx gx gx ft operator exponentially independent decay rapidly even hence gaussian decay fourier spectrum transform regularization class distribution show svm make cover define risk theorem simplify denote compact measure probability pf restriction cover bind human dna five continuous dataset series repository explore svms discrete observation hmms latent class initialization transition probability initialization emission initialization via viterbi execution rbf map spaced setting linearly spaced length space maintain fisher markov feature string consist manually state symbol generator binary hmms show classifier whereas fisher perform bad loss possibly high run vary symbol formula hmm learn low fisher state restrict low perform slightly outperform poorly sequence subsample l synthetic vs principal triangle circle competitive point series interestingly future plan explore apply visualize relationship sample consist grid grid median temperature specie species kde integrate square form separate certain specie visualization kernel separation toward toward generative distribution theoretic embedding generative operate spectra compare hope distribution discrete outperform bayes technique expressive factorial markov field highly expensive testing rkh letter measure generative kernel measure hilbert incorporate beneficial generalization present model generalization kernel offer elegant clustering particularly iid incorporate statistical dependence information rich machine svms principal component learn particular iid classification observation svms use nonlinear variety sequential sequence use manifold english reproduce kernel I generalize special structural kernel describe distribution provide naturally extend hilbert space embedding empirical mean lack accept mean map provide generative observation generative accomplish measure observation distribution map map widely use extension connection section conclude result species datum concept phrase al observation iid rkhs induce rbf
maximum quadrature confidence root evaluating simulation require estimate core ghz intel mb precision yield van remark number use compute infinitely simple natural goodness statistic draw although present focus straightforwardly parameterization scalar handle usual advantage square computer value simulation suitable thank fan observation support nsf nsf research fellowship fellowship support nsf fellowship extension article level root square goodness fit involve associate black box algorithm level classic test circumstance level monte simulation identically draw member specify draw value accordance terminology bin category work single fully focus parameterized extend parameterization scalar equivalently parameterization natural come use square construct draw empirical page section arise specify root square confident quantify confident let square construct draw arise complement different square various obtain statistic availability computer simple feasible convenient number via monte practical advantage structure review develop utilize level associate rapid section summarize cumulative independent suppose unit cumulative square root branch square root therefore evaluate attain domain integration whole low order double precision root mean square draw fact distribution determine confidence draw root square construct come distribution value distribution draw please simple maximum canonical formulae goodness respective bin particular choose obviously maximum estimate actual determine monotonically confidence level convenience focus classic replace statistic likelihood differentiable occur domain almost surely large draw remark limit joint limit proportional dirac delta concentrated hyperplane hyperplane restriction generalize intersection hyperplane restrict multivariate principal axis algorithm sum square level draw multivariate axis whose vector consist normal hyperplane maximum example take estimate orthonormal basis column construct orthonormal construction van maximum parameter multiply right onto hyperplane obtain eigenvalue adjoint eigenvalue desire axis low computing require appropriate condition sketch proof multinomial distribution maximize define combine desire draw proceed give necessarily number cumulative distribution function obtain algorithm center random variance remark numerically via first detail interpretation model third poisson table poisson test algorithm conduct simulation draw paper concern
generation well policy map context action define convenience payoff multi armed problem devote develop total formally may strategy trial special case contextual bandit since refer type news recommendation view article pool th visit trial arm serve article precisely maximize click turn payoff bandit balance past experience world suboptimal bandit process exploration suboptimal arm payoff refine payoff clearly neither explore purely need roughly class attempt regret grow extensively bandit algorithms bayes index bayesian competitive prohibitive without describe noted method learn goal base algorithm want trial action context interactive nature run likely infeasible serious risk rather offline available payoff arm choose likely evaluate evaluation reinforcement bandit efficient bandit process simulator introduce simulator hard justify reliability contrast sound choose uniformly although considerably allow decrease datum unknown tuple draw I form consist context payoff event crucially partially label bandit mapping history current therefore people evaluate allow life note focus contextual exposition arm old arm may event investigate set simplicity long stream event datum stream one history retain different take entirely ignore input stream stream policy choose retain independent everything else retain distribution evaluate world event evaluate policy stream intuition context payoff event stream say history identical estimate per trial return therefore estimate respective quantity repeat multiple average trial payoff trial payoff respective guarantee event statement mathematical induction event stream retain expect retain event probability expect accurate increase unfortunately situation policy contain per detailed stream similar valid event number reason unbiased trial payoff next show final long enough accurate policy choose independent emphasize fix contexts payoff return close value order decrease case application notation let event return choose side since chernoff bind side statement equation inequality might bandit concentration impossible general contextual coin flip operate choose choose trial payoff always choose style deviation bind dependent history though provable furthermore always repeat evaluation estimate empirically return highly try position offline method world validate methodology specifically evidence guarantee iii effectiveness methodology effectiveness evaluation provide relate offline production yahoo page contextual evaluation unbiased valid event offline algorithms relationship offline three bandit module yahoo front page visit page internet snapshot highlight news article pool maintain human illustrate four article picture title four article highlight picture title click highlight article read click article highlight attractive article naturally model contextual bandit click article age article infer bandit article pool click otherwise payoff click user turn maximize payoff letter randomly generate specify example user letter fall unless offline evaluation million random random article article pool user offline article pool focus position visit click business ratio offline methodology verify compare metric another spatio article high winner serve extract article offline ensure set available note conclusion present differ typical event old article leave enter business rule course visit site still win repeatedly article visit due article click effect assumption fortunately consider consecutive visit winner article whole offline winner online article winner article view treat ground metric offline overall policy article impossible simulator base evaluation evaluation provide accurate offline online decrease evaluation article respectively upper suggest practice rate predict stable algorithm f appendix illustrate offline technique deterministic variant deterministic contextual estimate arm payoff may event run return table summarize deviation find specific deviation demonstrate give natural quite std min section give accuracy method static section view policy particular bandit available article user via serve user article segment age gender note policy fine grain user base age maintain predict user article click estimation model three collect state period exploration run period trial payoff note business online module article fortunately report business roughly multiplicative impact across day cause estimate online business time remain present scatter view correlation slope regression almost policy bandit contextual estimate use maintain article scatter day plot respectively offline business rule retrieval delay daily offline historical presence business rule paper bandit algorithm datum simulator method arm ideally show evaluation method unbiased estimate total complexity yahoo front challenge article verify encouraging suggest usefulness relate online refinement rank ad evaluation however fraction use expensive application uniformly practical constraint evaluation collect reduce exploit specific avoid evaluation progress could policy iteration much baseline policy suppose collect datum act implicit collect new trace choose average trace agree policy iteration score superior superiority superiority although may capture construct appendix simple payoff payoff clearly limit decay regret pick exploration arm suboptimal explore balance exploration trial payoff confidence achieve upper bind ucb parameter uncertainty refine estimate vanishe behave appropriately total bandit extensively understand contextual largely open use achieve payoff contextual shrink assume algorithm empirical weak special strong regret various modeling contextual payoff ucb
hazard hazard leave space take hazard transition impossible prove intensity hazard function underlie ignore product normality hazard rate bound interval support interval show truncation would hazard readily eq cox terminology cox difference situation truncation sec extension convergence terminology translate underlie unbiased may see al convergence cox variation observation scheme censor count inefficient product delay right censor et cox censor hoc rich relationship stationary line assume observe independently nonparametric soon chapter area assumption simple model involve sometimes calculate less alternative possibility four censor version recurrence censor complete right censor observation product situation cox account pe al start define give comprehensive asymptotic limit estimator alternative build particular martingale al available al study robustness gap non discussion start claim pe recurrence ignore al cf fact goodness estimator stationary intensity think axis poisson call individual suppose birth corresponding intersection segment axis observational particular age exactly within death occur time nonparametric maximum appropriate nonparametric inverse easy root estimation possible problem inefficient computable limit type likelihood basically process inefficient easy combine formal censor survival count observation twice censor discard doubly doubly censor theory delta fairly formula covariance easily resample partially estimate extension surface observe observational plane likelihood maximize seem root inefficient easy type van study line segment merely stationary poisson basis estimating equation efficient quasi van fine come full idea early process line quasi case intensity life whether inefficient ad hoc difficult option soon compute inefficient research institute natural inference reference analysis b censor unconditional f york cox new york processes nonparametric censor process survival analysis classical assess huber cox claim life r centre mathematics survival monotone measure length appendix cox pe na weak convergence york soon nonparametric van efficiency segment van van line bias censor deconvolution decrease density estimation problem analysis sample inefficient analysis recurrent event stationary backward recurrence recurrence time cox cox gap propose
prior produce rule horizon specify play capture remain budget horizon feedback value execution trajectory realization goal approximately maximize specify possible dag initial observation obtain rule play state additional observation comparable arm programming computing budget help ignore policy term regard arm lose arm policy feedback horizon I previous change iv feedback time encode single play show single policy reduce statement seem policy martingale proceed via stop truncation trace horizon make ii characterized policy play affect outcome execute induce stop refer edge reward pass rp pr claim induce path policy stop truncate statement preserved accounting integrate result truncation prove lp arm ensemble lp round variant appear arm play point play algebraic sequence curve line part convenience dominate areas jk rhs follow horizon bayesian mab instantaneous feedback reward number play playing play less truncation consequence independence r jt factor arm arm arm distribution lp play optimum observe ever reward time horizon increase optimum prefer play arm optimum keep play optimum ap pa slightly separate delay delay feedback lp collection policy polynomial delay policy enable happen policy step applied policy horizon policy lp scheduling approximation phase size policy order first structured policy horizon behind proof delay every convert policy eliminate know step ensure know outcome next execute play adaptive know outcome observe horizon observe prove structured play make horizon conservative time feedback policy reward factor initially make play subsequent eliminate instead outcome play crucially martingale property arm define structured policy play stop intuitively delay policy play significantly instantaneous reality feedback play policy make delay block pi play continuously contiguous play make outcome simulate play stop decision play pi p pi step compact policy far retain constant block well block play continuously delay free structured free pi block play delay execution couple path decision path play lie couple consider increase root play block eliminate extra outcome plays fix stochastically store follow execution stochastically eliminate block root procedure terminate play eliminate decision number since procedure plays path constant factor arm identical stop step denote path delay block play delay free policy stop contiguous therefore play reduced truncation term terminate truncated lemma approximation policy formulate lp relaxation explicitly truncate meaning begin block step delay block feedback couple two state easy omit consecutive end play make beginning formulate lp randomize truncated block arm state lp relaxation simply structured arm play lp describe htbp begin block play obtain feedback arm collection policy arm policy globally solution remain active aspect novel base play passive complete feedback passive final arm initially active among arm active current play accord allocation become passive follow delay feedback expect contribution observe impact play make make complete since high without interference least linearity conclude pt fact problem definition problem immediately naturally study extensively albeit absence delay set delay provable bandit delay feedback force significant ability significantly prior structural carry feedback available show improve generalize iterate allocation resource uncertainty problem decision outcome seminal develop reference bandit agent compete uncertain reward play arm update focus scenario instantaneous negligible horizon early delay decision make delay obtain upon delay increase bayesian armed delay bandit arm reward specify arm feedback far accord reward reward arm feedback play decision policy concrete framework online determination web present visit page historical page display resolve however characterization goodness visit delay delay system issue budget page display present heuristic base bandit problem stochastic immediate difficulty uncertain controller computable analyze policy output policy rest paper focus mab generalize generalization fact use truncation presence constraint partially execute arm policy original argument space martingale property policy yield approximation horizon absence presence improve lp relaxation
extreme value attribute differently generative permutation object recursive position selection object prefer object distinct propose assign logit refer logit reason axiom show preference utility maximization start early statistical develop necessarily study utilize area primarily example application operation management address generalization notable generalization avoid increase see interested reader refer article generalize sort attractive generally appropriate risk mis model valuable scenario generality applicability maximum impose parameter effective underlie cf scaling dimension maximum impose imply flip learning model summarize either restriction structure limitation choice certainly permutation potentially parametric impose criterion hand approach consider paper simplicity permutation select marginal fraction issue principle distinct permutation need choice information think linear space doubly sparse consistent observation solution computational identify signature show signature noiseless exactly computational scale linearly indeed excellent condition datum available randomly choice signature condition summary signature randomly reality free importantly arise main problem realistic potentially corrupted specifically problem equivalently datum cast linear restrict primarily defer efficient manner describe answer question understand elaborate doubly explain section consistent convex doubly radius von doubly stochastic polytope permutation doubly extreme point observation doubly near choice somewhat establish concerned approximated choice sparsity value doubly next efficiently mention signature play natural ask recover consistent answer question identify original choice approximation sparse signature establish dense exponential family approximate signature designing give noisy leverage structural signature adaptation multiplicative exist signature family approximate well existence approximation search would computation well approximated signature bound polynomial ambient equal model framework compressed programming polynomially effectiveness sparse association rank interestingly basic look projection vote datum capture structural work streaming cf compressive literature mean measurement linear coding correspond receive streaming algorithm correspond maintain algorithmic similarity compressive sense establish generic condition design put view provide non sense efficiently permutation organize precise condition introduce exponential state establish benchmark learn discuss relevance provide search choice permutation component sum choice primarily restrict point marginal doubly item observation generality doubly stochastic else approximate von early exist program guarantee relaxation near put doubly significantly small geometrically translates span extreme point significantly small search run class signature establish signature appropriately dense thereby restrict signature family recall choice say family pair equivalently rank permutation rank support signature question later version model value describe family parametrize doubly family precisely interested reader refer correspondence exponential family moment answer question answer sparse doubly allow explain observation choice emphasize tight dependence require sparsity doubly stochastic zero convex relaxation doubly tolerance choice model precisely doubly pick consistent certainly ignore efficiently restrict improve exponent precisely establish signature family choice remark proof constructive sense propose sparse result signature factor exponent time find sparsity reduction introduce factor worth theorem choice scale scale ignore polynomial ambient scaling sense recover solve polynomial ambient existence signature fit restrictive specifically doubly possibility signature fit precision signature secondly end signature sparsity like scenario happen establish signature family long observation order marginal parameter tuple integer far replace appropriately clarity somewhat weak establish rich exponential approximated marginal thing care marginal sparse signature ignore distribution existence establish restrict signature signature next present theorem prove tuple ease exposition dimension order iff column represent signature family suppose probabilitie signature family permutation signature marginal summation entry column identify signature correspond tuple permutation consistent signature q enforce summary signature index signature choice signature first remainder devote signature tuple signature pick tuple among satisfy essentially approach optimize describe equation justify suffice hull satisfy von establish signature computation cost use algorithm utilize pack possible order sparse signature near signature signature permutation check j tuple signature component towards find signature choice put way interested set satisfy collection gap feasible find provide program signature consistent signature precise notation think satisfy interested essentially try lagrangian relaxation manner lagrangian iteratively initially inequalitie lp problem one infeasible choice relaxed lagrange multipli program negative weight signature feasible solution penalty proportion else impose slack constraint note doubly summation subset sum multiplicative weight solution n permutation signature imply signature component polynomial cost overall priori exist maximum increase value first succeed would well inherently effectively rich describe empirical support question american association choice collect preference candidate candidate vote ranking vote vote win determine common cognitive candidate candidate fact cast complete candidate member simplicity information discuss marginal retain underlie like candidate lot first position vote position vote vote vote vote preferred ranking course manner question however flexibility aggregation retain aggregation choice rank sparse therefore understanding let denote underlying distribution ranking candidate first marginal run algorithm roughly try manner model signature approximate however unable guarantee keep exposition simply refer heuristic description digit candidate position list rank position rank position candidate position full size approximation order average define measure successfully find huge approximate try order marginal draw rich cdf sparse along order nearby permutation sense pairwise need visually well model determine winner determine rank certain functional rich datum winner determination preferred ranking winner vote substantially rich vote permutation approximately additional vote candidate apply win candidate aggregation winner also yield aggregate represent aggregate fair apply turn remarkable conclusion high vote particular order datum fraction position account distinct candidate fourth candidate middle go conclusion line fall behind candidate group wherein least something candidate remarkably somewhat relate model structure candidate vote effectively exhibit preference primarily distinct preference explanation perfect split model important underlie true base make provide nonparametric towards main show first order choice significant expect efficient choice signature oppose force complexity free work robustness signature appropriately recently literature efficient sparse via program effectively linear programming signature another choice efficient result first information theorem order marginal hence reasonably believe family choice extend belief result rely von extend type possibly feasibility respect high direction overcome computational develop approximation marginal inspire exact signature utilize early strongly believe heuristic likely computationally signature family significant alternative speedup relative marginal e doubly accuracy signature sparsity approximate order model reasonable cost bad factor signature describe alternate heuristic outside order contrast signature many employ guarantee research provide algorithm section specifically try permutation marginal unknown permutation represent linear relaxation imply signature however signature option search search efficiently
advantageous mutation fitness increase fitness structure dynamic sampling organization evolutionary equilibria equilibrium correspond rapid innovation organization connection evolutionary innovation organization extend mutation theoretical use sequentially generator capture aim organization crucial dynamic drift innovation architecture subspace question remain jump drift well structural drift organization drift fortunately correspondence precede process within evidence reason optimistic structural drift predict lead acknowledgment physical project via view represent imply department corollary sequential inference learner estimate pass extend neutral structured diffusion organization drift control fidelity lead sequential memory phrase advance illustrative transmission happen knowledge sequentially chain fidelity information lose occur causal teacher pass student quickly motivate sequential game phrase player player phrase win game match original typically interest derive evolve surprising impossible less education human communication error phrase merely product make phrase phrase accumulate set analytical progress sequential select language operate extension evolutionary population neutral drift process population substantial population evolutionary biology require notion diffusion several drift demonstrate drift organization fidelity evolutionary without application effort briefly drift seem available mathematical introduce examine complexity diffusion subspace decomposition structural thereby close structural consequence evolutionary familiar neutral theory skip drift change frequency phenomenon evolution evolve neither phenotype play modern molecular evolutionary neutral evolution mention neutral track population individual space random reduce round biased coin sample memory drift simply bias summarize state drift measurable member state respectively variation vanish state purpose drift homogeneous sampling fact stochastic develop early advance substantially handle realistic effect mutation allele genetic explores relax restriction walk force step identically stochastic iid time probabilistic sampling large learning exhibit like behavior phenomenon organization drift balance structure precise description drift theory begin several alternate gene early color either white parent population mutation allele pass may change individual varie generation generation evolutionary neither mutation assume moreover successive population overlap fisher theory reduce drift version familiar walk receive allele parent previous generation copy generation generation specify drift state drift discrete coin population coin coin generation coin reflect realize walk capture allele individual allele none allele copy population random fluctuation bias represent copy copy expect copy number eventually reach allele states allele drift rate allele selective case mean individual candidate initially plot allele sampling solid line structural sd coin versus mc estimate realization one consequence variation vanish homogeneous population population word establish stop showing theoretically predict predict reference average realization realization binomial random allele count initial allele allele simply generation mc method simulation allele fisher process generalize drift drift consider symbol individual allele drift generation allele replacement contrast structural produce sample string give spatial individual within appear difference evolutionary biology collection individual identically structure mind interpretation life return sampling stochastic drift one lose brief reader value process markov transition give causal allele state leave eigenvector maintain population think generator length string I compact representation string alternate ap ij take allele allele transition one transition generate simple generate symbol enforce alternate period positive transition ba ready describe begin follow first individual infer allele repeat allele drift generation reach population vary net stochastically terminate stop population produce new model infer population generator previous highlight generator finite create sequential closely genetic except analogous biased coin newly bias coin eventually bias towards drift coin replace allow memory transition number presence absence capture essential aspect structural dynamic produce string infer sufficiently mean generation consider allele neither allele current nonetheless alternate prevent despite equilibrium notion say structural correspond become prevent drift however structural drift represent via internal population alternate bit single state suitable biased coin drift single biased coin process generation bias coin become coin becomes alternate variance allele condition vanish population eq allele process call bit per allele entropy bit close asymptotic population diversity form sampling sampling periodic lose randomness generate branching criterion notion occur formally statement vanish structural drift memory population sampling branching recurrent I variation infer lack variation drift process stop entropy vanish allow structural drift stochastic process say analytically drift present structural nontrivial genetic drift coin process process population binomial process coin free sequential inference property explore drift bias h occur final completely occur right drift rapid drop population transition produce later discussion run biased biased coin fig carlo drift modify terminate illustrate allow genetic drift surprisingly interpret genetic case follow low proximity close initial similarly fair coin bias coin represent iid reach consecutive initialize drift library ref alternate transition generation fig remove mean reach transform coin right transition towards reach transform alternate alternate coin pathway biased coin subspace color structural behavior population block consecutive compare coin observe biased coin reach reach different note even process biased coin time setting one substantially drift understand affect ce diagram produce several realization ce diagram vary complexity shannon internal small memory correspondence ce coordinate capture intrinsic population two ce diagram reach transform fix coin alternating reach begin upper left transition merge force bias coin coin stay reach track internal coin process require diagram broad view population structure roughly current transition stochastically since cause correspond subspace fig ref curve subspace process quasi structural drift force change remove state transition quasi invariance break topological shift reflect ce movement often topology innovation return mean biased coin highly mutation drift elsewhere right turn behavior coin weight coin fc ap gm coin bc pass realization reach drift sum visit pathway pathway subspace reached spend structure population break independent time transition subspace jump analytically future drift weight connected pathway spend show bias line along confidence middle sum coin fc alternate pathway transition unlikely pathway fc pathway high pathway frequently grow begin influence fc pathway likely total dominate ap high ap pathway subspace pathway bc spend spend diffusion pathway gm bc maximally far typically jump result small time bc subspace fc calculate know ap fc gm bc jump population realization infer transition pseudo drift drift control trade structural innovation instead infer infer generation generation state create noise transition represent likelihood maximum map retain edge generator way allow innovation zero merge state simplification consider select topology aic correct penalize penalize however may well expense several version drift qualitatively introduce loss panel structural innovation wide diversity sampling create quite produce high variance period period innovation leave drift none phenomenon occur population size jump subspace drift kind behavior select emphasize toward neutral evolution work return example game temporal structure pass language ref ref effort structural drift capture dynamically change semantic drive fluctuation organize
density trick concerned although affect accelerate temperature keep simulation choose schedule minimize distance computational nonlinearity potential curse make convergent include purely method metropolis importance solely purpose therein comparison molecular langevin external carlo introduce walk refer molecular purely langevin dynamic add energy classical molecular dynamic dynamic convergence since langevin multiscale langevin sim anneal introduce anneal accelerate dynamic temperature system occur intercept approach calculate free auxiliary collective stay global annealing perspective distinct accelerate method replica md sampler review temperature general temperature mcmc langevin tune base geometric standard tuning convergent tune accelerate multiscale algorithm optimal acceleration achieved propose handle case negative curvature equal annealing describe bound convergence situation final one sample seek minimize b temperature distance transition convergence rate near schedule refer number langevin detail gibbs need temperature barrier simplicity intuitively interpret landscape tune accelerated accelerated variation however worth st dimensional nonlinear molecular consist fix atom degree system landscape local barrier two right atom start initial momentum long marginal indicator markov converge tune tune anneal individually effect propose fast convergence addition choice investigate grant grateful fast follow system assume system ergodic admit solution fundamental define ode dt cb calculate matrix recall follow discussion achieve minimize hence converge fast opposed immediately difficulty method square could work instance molecular system schedule g ergodicity step repeat statistical total variation triangle equation h energy harmonic minimize nonlinearity temperature g step employ schedule cite truncate inverse fix schedule serve free eq ease optimization optimize purpose error typical temperature linearly ensure number unless inverse error optimal well schedule trivial schedule temperature accelerate type logarithmic popular schedule c inverse shift limit temperature concentrated result show approximated empirical time exp shift inverse setting total schedule increment free investigate schedule rank total agree total accumulation addition discussion dependent error total simulation performance consistent rigorously towards
left plot rsc vertical estimator experiment tuning rsc correct value htp adapt general optimization alternatively bregman et specifically simulation present theorem value tuning calculate calibrate estimator outline selection performance rsc rsc optimally rsc optimally tune validation understand true adaptive rsc version rsc construct matrix generating row normal matrix generate denote th noise experiment r px qx setup require simulated variable strength experiment varied combination result summarize varied correlation mse mean recovery mse median rank percentage p rsc rsc mse mse c mse mse mse mse mse mse mse mse mse mm mean square recovery p rsc rsc mse mse mse mse mse find rsc adaptive excellent behave rsc use large rsc signal moderate rsc excellent mean rsc covariate accurate rsc regularization tune correct experiment improvement support moderate tuning much rsc accurately time regularization threshold mild snr moderate continue true noise build parsimonious via rsc recommend tuning coincide select computationally rsc snr extension rsc penalty induce offset conjecture investigate carefully research suppose much large requirement verify ideal interval give solution construct pair follow outline subsection path recover plot pair conclude tight htp display f iii b xy achieve generalize singular limited generalize page diagonal singular regular singular usual let large notation consist last vector n generalized claim restrict section metric subgaussian inequality subgaussian large subgaussian entry directly subgaussian subgaussian q note element kolmogorov discretization give write subgaussian subgaussian subgaussian subgaussian bind fix obtain claim claim tail bind string self dx x x dt degree freedom et page would like paris thank constructive dms criterion rank matrix multivariate rsc estimator frobenius reduce rsc consistent number value target appropriately classic asymptotic regime stay bound grow either much square rsc computational linear candidate nuclear penalize square inherently high complexity rsc multivariate rsc albeit parsimonious rsc consistent finding extensive class type observation response assume relate unknown equivalent predictor separately multivariate correlate discussion phenomenon restrict equal history back reduce model follow include rao regression excellent comprehensive date asymptotic asymptotic two work develop class estimator adaptively predictor historical latter py k projection onto column singular reveal prominent final describe analysis section sparsity general need count singular turn tuning prove expect bound propose bound ideal penalty penalize proportional estimator al al involve map investigate noiseless study include general spirit comparable albeit condition propose important nuclear less parsimonious offer yield result suggest strongly situation preferable always penalize least denote parameter computationally calculate suggest inverse let column eigenvector correspond arrange small final row obtain first characterize square exceed exceed lemma fit let eq appendix x denote singular reduce py see py consist conclude g k moreover property instance consequence predictor much guarantee describe theorem control singular value control probability consequence pe pe x b pe pe b pe obtain pe appendix state achieve minimum equal assume j j mean inequality hold multiplicative first claim follow pe claim error decrease time expectation bind constant pe free interpret corollary ii large estimator estimator square bias trade restrict f pe pe pe pe rx f inequality pe obtain x appendix evaluate matrix equal proof remark square bias trade rank coincide effective ideal last easier interpret index reduce clearly accuracy small follow eq c recall b pe k immediately I remainder fast ii assume rsc penalty restrict rip plan equivalent assume small large error notice rest subgaussian enough invoke iv error state fact multivariate write
graphical structure significantly difficult necessary response input input snps output phenotype regression relevant covariate correlate output leverage response dependency assume preprocesse step study response group gene identify parsimonious input dense subgraph lasso penalty employ fusion penalty regression across connectivity response guide property success statistical optimization side optimization adopt manuscript response fuse lasso however dimension cone programming qp interior moderate solver exploit introduce proximal qp fusion line rest adopt discuss fuse lasso preliminary fuse dataset conclude discussion instance input intercept denote solve frobenius control recently norm joint across set encourage relevant share across find zero follow q denote combine input incorporate account dependency structure represent computing correlation node edge correlation threshold easily focus denote represent connected edge influence relevant constraint standard follow absolute pair correlate opposite sign subset densely tend fusion effect pair propagation level assume share covariate output penalty qp approach qp newton moderate propose proximal utilize low precisely fusion max bind optimize original adopt accelerate method proximal quite simple recently solve optimization include completion smooth penalty loss handle non smooth find complicated penalty rewrite fusion edge graph fusion penalty wise rewrite penalty eq entry inner matrix auxiliary provide penalty smooth introduce approximation positive strongly original view easily jk control accuracy rate key lipschitz continuous smoothness derivation subdifferential detail continuously adjoint gradient constant bl operator zero project solution kk proximal gradient substitute smooth obtain large descent nesterov minimize th accuracy gradient obtain stepsize combination combination intuitively superior current information current optimize prove present iteration b key decompose part iii gap ii accuracy minimize balance present fast subgradient depend good problem ratio accord assume calculate complexity per edge large comparison complexity iteration accord cost thus cubic edge linear iteration require memory efficiently problem fusion fuse fuse emphasize fusion graph solve univariate norm vector coefficient graph input chain straightforward c relatively applicable analogous fuse estimator function c ml provide demonstrate real superiority gradient exist accuracy scale cross code terminate simulation compare regularize multi scenario mapping simulate input parent panel individual individual regression correlate coefficient relevant select relevant group relevant input induce output assume another correlation subgraph phenotype coefficient simulate entire final select row column input cccc cccc figure apparent f positive reveal clear block structure information across correlate relevant systematically compute specificity average affect element coefficient matrix curve use generate outperform ratio examine sensitivity include purpose spurious include exhibit great threshold correlation close effectively performance approach lasso roc curve entirely overall method output relevant input offer become empty compare prox qp formulation package choose well cpu select qp vary second intel ghz ram vary generate unable qp error due prox substantially qp remove introduce addition increase computation significantly affect cccc c clinical program compare one phenotype agglomerative highly phenotype cluster block row column phenotype phenotype accord give agglomerative cluster figure align phenotype see bar phenotype structure much tend span phenotype regularize e structured graph guide set algorithm problem involve fusion arbitrary simulate demonstrate rich improve performance discover relevant input magnitude scalable qp appendix convex let want differentiable everywhere subdifferential singleton inequality equality conjugate chapter equivalent ad eq function subdifferential everywhere rewrite utilize continuously take reader definition therefore bind find upper formulation eq summation change order bound arbitrary convex smooth minimize lemma present optimize eq b utilize term accord equal b plug side achieve constant minimize notice dimensional holding follow assumption pt output complex graph structured exploit encourage share set input gradient problem smooth fusion structure fusion fast scalable widely cone programming programming provide demonstrate superiority efficiency scalability multi learn greatly advantageous task learn effectively small input covariate relevant furthermore assume related manner closely relate tend relevant goal recover structured coefficient task relate input norm relevant
observation long b inference location even isolate relative yx yx hold requirement isolate strongly derivative linearly intuitive connection strong selective shrinkage previous setting improve isolated observation transform measurement transform measurement selective intuitive nonparametric keep kx lastly carry motivate tail however derivation lose prior jointly let k follow construction view transformation parameter relaxation liu interpretation define could across nc keep constant n label ultimately c hope selective occur pf provide heavy tail lie predictive use laplace gaussian approximation pz combine pz give pz non log pz write nc stack diagonal w resort predictive arrange straightforwardly approximate respect pf x convert average match b laplace log take need account mode evaluate angle discrete angle class covariate pair angular proposition taylor optimize either similarly regularize algorithm regularization objective train ten fold ten fold ten dense laplace marginal perform evaluate predefine sparse contain cross fold dense sparse outperform laplace marginal laplace marginal sparse dense lie region growth eventually marginally report correspond dark copula allow modelling multivariate early song gaussian copula popularity copula literature li pair marginals gaussian modify et al approximate transformation define ng extended work marginally bind copula focus copula give also liu away matrix consistency scenario literature heavy gps elegant region base importantly tail base favorable computational predictive inference utilize tailed idea tail easily build gps heavy tailed selective gp currently covariate discussion give access mm mm science division computer division california berkeley berkeley cs berkeley cs berkeley edu enhance often show tail gaussian via copula shrink dense heavy tailed sufficiently improvement produce competitive result dense region provide predictive region protein particular protein free function protein explore region poor guide view covariate shift differ investigate address overfitte shrink selective strategy end readily could reflect regression augment partitioning advanced kernel problem modify observe uniformity interpret cause address present selective shrinkage replace process tail student tail view outlier I shrinkage notion view outlier I selective far theoretical heavy precisely kind shrinkage structure present construction heavy tailed computation process present biological heavy tailed process substantially dense overview research final mean usually semidefinite location kx kx label observation could towards shrink close mass paper intuition replace underlie section stochastic transform treat kx heavy tailed q center heavy tailed origin note distribution al monotonic transformation structure kx x fx fx take observe kx tail recover process tail easy transform satisfies tail shrinkage selective vary specifically interested shrink isolate dense section marginal heavy induce selective construction induce selective shrinkage investigate additional b gp special shrinkage check build top gp analysis selective obtain focus
indeed covariance large problematic poorly condition instead sample factored exist likelihood elliptical prior weak important regime elliptical moderately obvious much sampling classification hamiltonian monte carlo require alternative elliptical elliptical sampling apply evaluation typically sampler crucially carefully parameter generic slice update although suffer slow gibbs slice assume linear seem role elliptical potentially way achieve search multivariate update sampling update work expect base sampler perform thin straight line point method covariance begin resample control resample move accept reject use processing cost one update roughly elliptical update use likelihood although minor cost might reason give subset proposal conditional prior operator gaussian prior variable elliptical slice alternative standard sampling probabilistic modeling task brief reproduce material model covariance overall covariance apply unchanged likelihood ten dataset input explore application noise shift intensity inference perform intensity bin contribute explain bin distribution mean offset log poisson process cox process bin day day range offset match move trace elliptical slice different make valid make difficult control bar row rescale group cpu elliptical bar height quantitative quality estimate figure take choose maximize provide implementation bit problematic specific detail number primarily low space control comparable elliptical slice sampling less variable fail dimension run unable elliptical increase elliptical slice effective synthetic huge method would elliptical slice along straight involve perform slice sampler many evaluation partly reduce likelihood completely possible elliptical ease human time advantage exchange fix potentially dramatically optimize elliptical simple performance relate scheme wide application acknowledgement suggestion institute advance strong dependency gaussian carlo properties applicable free work well variety property need derive update complex commonly specify priori belief probabilistic gaussian markov marginal gaussian express generally review inference simple carlo apply generally circumstance probabilistic one among poorly contain simple also remove preliminary sampler use metropolis method gp proportional multivariate likelihood tie observe indicate latent also shift p mcmc likelihood point hasting introduce conservative next report fast implement wide drawback need appropriately mix preliminary run usually parameter update desirable step maintain half equivalent rich update slice provide step proposal also select value transition construct state slice adaptively intuitively augment probabilistic slice adjust probabilistic replace still however first operator invariant unchanged effective variable generic implement slice routine cm pt sample update slice make link slice easy validity approach range precise connection technical present include detail slice within update lie plane purpose analysis angle identify distribution quantity vertical stop algorithm likelihood angle jacobian leave probability invariant verify substituting value useful mcmc make initial final rotation return pair rotation reproduce visit initial equilibrium probability draw reverse illustrate ensure angle probable reversible angle acceptable angle draw select use intermediate include initial reverse involve shrink decision reverse transition imply markov chain stationary deterministic unit jacobian probability density obtain variable generate generate remain quantity use probability angle distribution non non enough chain unique stationary distribution repeat elliptical slice start choice sampler
divergence cd ml update hide visible hide respective variable gibbs variable plus cd learn boltzmann machine rbm energy direct interaction unit q visible connection unit bipartite figure different visible unit therefore long show rbms state rbm kl rbm conditionally hide versa logistic column parallel fast rbms integrate variable unnormalized employ visible keep somewhat definition allow individual machine proceed approximation determine isotropic constraint unit connection unconstraine importantly analytic expression available furthermore factorial efficiently independence important visible due generative compose rbms generalization layer multi perceptron become attractive two density rbms visible hide state interestingly result machine undirecte connection evident gibbs top definition easily extend layer another additional prior replace case layer rbms visible unit possibility allow interaction restrictive model exponential rbm unit define replace train log layer log second approximate cd evaluation involve integrate factorial optimize variational likelihood additional approximate conditionally turn discuss layer visible estimate see later density rbms two difficulty arise analytically unnormalized integration hide previous difficulty unbiased contribution estimator applicability partition boltzmann estimate expectation whenever get estimate partition function obtain draw average result point partition minimize distribution well close try proposal approximate act proposal distribution far leave invariant e right integration introduce third hence order simple intermediate precede rbms intermediate rbm whose weighted find estimate factorial entropy still draw carefully chain construct unbiased inverse partition tend short run markov estimate slow bind share conceptually apply consistent equation obtain choice visible wish assign adjust depend layer know unbiased estimate expectation necessary already multiplicative generally rbm unbiased unnormalize second consequence jensen estimate consistency asymptotically unbiased partition tend limit still heavily skewed question whether term reliable formulation become evident unnormalized normalization mention bind readily marginal density rbms state layer state layer generate feed multiply rbms p p unnormalized importance weight use estimate proposal importance easy importance estimate hidden manner however introduce weight take value optimal unnormalized marginal unfortunately importance reliable consider scenario idea likelihood achieve layer usefulness become already rbm generalization boltzmann serve thereby distribution greedy reach optimize log kl kl rbms principle approximate potentially rbm height cm limit none xlabel ylabel legend style anchor east cell west exp exp axis cs lower replace state give conditionally unit bind loss suboptimal log likelihood however rather depict order patch choose thorough analysis effect parameter experiment apply include log center whiten dc patch layer suggest modeling patch model third layer propose possess statistical intensity pair orient edge far analyse likelihood evaluation training initialize layer visible marginal second consist train initialized likelihood likelihood consequence likelihood partition marginal performance measure component evaluation appendix true avg est avg loss investigate likelihood still hide unit third layer train estimate close observation unable visible height cm width xlabel ylabel log bit legend cell south east plot marker width xlabel ylabel bit true legend style cell anchor west blue coordinate evaluate hide blue sample procedure intermediate annealing use sensitive show change loss even unnormalized layer third make observation almost observe estimate log however reason observation likely size third contribute nothing explanation sample require many need satisfactory marginal lead ylabel bit height bar style color ica anchor east plot experiment performance large ica layer cd perhaps interpretation add affect scale gaussian suggest patch unit hand outperform estimate performance optimistic also might improve true improve rbms decrease train parameter connection log indeed ylabel log xlabel number layer minor legend anchor south west anchor west list solid solid mark line dash coordinate approximation ml cd lead improve cd add add lead even worse train cd learn process converge bit cd ask improvement add log likelihood represents achieve greedy additional layer log loss involve integral nevertheless optimistic infer something capability integral thereby encourage optimistic log likelihood log although reconstruction might valid become loss train cd width xlabel ylabel legend anchor east cell mark plot bar cd plot bar cd cd layer log loss minimize perfect learning lead might cd plot could confirm lead well log show reliable estimator important tool evaluation model task effect optimize likelihood toolbox search well effect intractable unnormalized layer still marginal require large one article rbms readily applicable provide evidence suit add improve especially log layer learn low layer would unlikely suggest possible might potential achieve learn greedy might replace different future research improvement natural several multi model substantial improvement beyond layer architecture prove apparent overcome create patch whether due task natural image xlabel ylabel legend west marker color marker plot dash marker width color line width coordinate relevant well experiment layer epoch decrease largely converge epoch visible unit treat momentum unit layer rough encourage layer share parallel used marginal number intermediate distribution anneal schedule anneal determine intermediate equally spaced schedule theoretical effect take layer second estimate optimistic xlabel ylabel bits legend west plot marker line coordinate plot color marker line width width coordinate plot marker color marker width coordinate take unit reveal continue decrease increase code evaluate rgb provide field machine learn way compare model statistical gain increase popularity apply variety complex deep belief limit qualitative analysis due
family concave sense exist functional log function automatically arbitrary complete convex close three nonempty interior satisfie moment inequality satisfy uniqueness density satisfy inequality verify arbitrary moreover theorem affine distribution elementary consideration convexity profile arbitrary side infimum function denote small closed suppose follow concave characterize term function consequence rescale version student concave laplace distribution respectively suffice equal expression everywhere remark apply otherwise log concave interval differentiable concave convex dot mixture f blue interval application understand first mapping respect turn however strong eq infimum also wasserstein due couple quantile present main mention useful fact support distribution moreover imply respect topology supremum depth ready converge entail continuity continuity let density nx fy nx fx dx mode recent result latter contain sublinear describe function estimate maximizer unchanged dx mx define write aim state jensen inequality elementary existence suppose dirac would plausible consist affine design subspace maximizer unless consist function close cone maximum general regularity condition equal equality triangular fix unknown nn zero distribution basic subset write maximizer empirical residual expectation furthermore write thm distance distribution family consistency satisfy far nf n asymptotic linear assumption size point satisfy let let additional refer likelihood turn maxima strength different stochastic search profile mean simulation provide moderately large skewed distribution consider simple linear regression design shape square versus also curve important instance height apply survey united show scatter enhance add seem appropriate neither cubic degree cubic spline say seem quite exact monte fit function kolmogorov residual reveal estimate quantile additive regression curve plus leave hand side curve similar quantile proof elementary lebesgue ingredient pointwise nonempty subsequence second hyperplane suppose nonempty interior sec accord entail small equal supremum next maximizer establishe combine deduce qx b nk boundary lebesgue follow lem nonnegative functions uniqueness maximizer essentially strict convexity concave everywhere theorem nonempty infimum real constant pointwise dt ft dt dt eq whenever dt define analogously suppose log q continue dt ds f ds one equal half dt half line deduce display concavity theorem consequence nn satisfy nx fy fy elementary unbounded converge statement hold equality maxima satisfie similar closed convex lem simplex ccc nu nc nu eq virtue deduce exist pt replace subsequence constant function condition meet moreover dx convergence b h h x equal maximizer approximation smaller dominate satisfy nx fy everywhere lebesgue measure establish essentially cover continuity long vector via side arbitrarily proof theorem equip none dirac maximizer explain estimation attention iw x infinity lem eq combine continuity existence jensen suppose rx entail compact set entail imply n specify view define sufficiently note nx know mx mx maximizer exist large consistency eps nx eq satisfy sufficiently verify subsequence deduce theorem remain argument reveal r side coincide utilize bound distance
exponentially requirement sequence simple indicator general detailed visit expert guarantee policy convex well policy tell upper case trajectory let trajectory follow least policy refined strong convexity j property strongly pick policy reduction treat batch trajectory online concept policy incur loss observe next unknown adversarial fashion go many algorithm find distribution policy analysis need bind variation continue call pick trivial large let ns rl policy surrogate guarantee find limit choose negligible strong classification require strongly surrogate albeit observe induce current observe trajectory true trajectory observe trajectory n ny difference loss random bound martingale order mn l mn mn generalization negligible leverage convexity lead tight demonstrate efficacy scalability problem recognition similar popular current human correct analog linear learner hz b b track star track number per point method total observe baseline training similar help learner recover mistake obtain fall track twice able obtain never train though never fall track significantly baseline smooth look qualitatively video qualitative behavior super platform game character avoid gap run use simulator ai competition stage difficulty gap goal train scenario near simulate consequence action leave hz vector iw performance average complete generate difficulty complete around vary roughly difficulty type gap gap stage hard compare sup choose choice indicator slow iteration report se interval per observe supervise approach controller reason supervise get controller obstacle expert distance next obstacle iterative eventually outperform consider significantly slow improve slightly well indicator potentially due datum generate indicator location collect wide show qualitative behavior obtain efficacy handwritten sequence image character adopt degenerate pass early prediction future compare roughly total character fold experiment fold fold repeat fold term character fold word predict character order predict character predict greedy method well approach simply predict character character always correctly label iteration fold character use svm character character character performance pure influence current character unstable reinforcement small significantly greedy decode benefit respect pass decode state include provide strong guarantee prediction sophisticated strategy decode structured prediction base classifier inverse aid technique present leverage estimate provide understand success online setting acknowledgement support reduced information online incur observe iteration vary adversarial fashion policy regret well regret sequence policy sequence rl regret also regret provide convex use find state policy policy result task enough pick note simply guarantee want show state execute thus rl last inequality fact assume policy n form choose require required online observe infinite I trajectory induce current trajectory would finite trajectory go assume proceed trajectory trajectory guarantee policy induce average iteration random hoeffding mn n rl mn finite policy result n require convexity loss inequality fast corollary list sequential future depend prediction lead poor practice number paper train deterministic set combine policy labeling prediction commonly practice sequence reveal fail must resort prediction learn controller art variety regressor predict expert encounter observation perform expert learner affect crucial assumption statistical ignore make mistake encounter mistake expectation classifier induce soon learner mistake expert guarantee mistake error several input expert learn stationary unfortunately impractical ill train stationary finite policy add practical policy mixture controller mistake cost grows linearly take exist except sub routine nearly expert additionally learner guarantee establish method policy forward induce improve one useful let step initial guarantee follow execute expert j j upper increase bad supervise sub improve
value improper source firstly lasso may covariate step select turn column column produce spurious fit covariate noiseless case exactly expand fit require fail influence produce full fit fit secondly number unlike size high fit simultaneously noise fit vanish higher spurious fit vanish pattern work true pattern recover sparsity attempt predictive illustrate calculate cv fold fold suggest examine thick perfect result however unlike lar individual cv note use calculation nearby tuning value solution extension procedure may may circumstance bag aggregate bootstrap method usually succeed smoothing especially smoothed bag reliably great study aggregate fit pattern select individual bag sparsity treat individually optimisation fit fit covariate great deal term sense natural implicitly effect incorporate though greatly increase shall conduct ordinary initial fit f conduct covariate analogy fit shrinkage covariate fit entirely reweighte encourage optimisation large step covariate mean true fit irrelevant concavity optimisation enhance reject straight repeat iteration recommend cross pose variant scad calculate scad scad adaptive hence procedure conventional covariate know covariate contribute mostly monotonic way know covariate covariate perhaps amongst focus univariate possible check monotonic covariate effect contribution relaxation selective monotonicity case relaxed square variation fit univariate mean follow monotonicity q one monotonic increase fit let knot knot similarly non monotonic theorem k decomposition thresholding style slow however covariate compare check former alternative sign conduct fit treat negative component fit separately decompose fit apply decomposition fit sign p adaptive shrinkage small fit negative positive increase decrease design handle study fit implementation grid dataset estimator plus house price censor version library though discard covariate presentation suggest effect suggest since sign know sign monotonic cross q monotonic monotonically ht variable explain require cross choose involve though greatly characteristic create often contrast fit spam mind sense difficult original covariate include spam covariate finding similar covariate select fairly effect though fit clearly however spam mostly aside monotonicity seem interpretation covariate characteristic residual reasonable monotonicity constraint variable vary generate r zero variance mean take subset replacement correctly amongst quantify look replication covariate reduce additional attempt give lasso recovery slowly thus imply possible fit arise line correct shrinkage shrinkage exhibit end fit see univariate case component fit perform black improve keep correct sparsity recovery add compare contexts repetition plus strong covariate rescale algorithm random forest aggregation tree multiple regression backward shape model spam lasso thresholding penalty sparsity algorithm separate validation separate plug average smoothness tuning run mse response scale approximately scad spam rf rf continue study spurious seem excess shrinkage clearly correction provide implementation thing extent set stick hence handle amongst perhaps adaptive spam though basic perhaps enforce amongst original covariate fairly explanation penalty moderate amount greatly algorithm complex covariate small shift generate l rf scad equally outperform spam rf power covariate variation range spam unable sharp without variability end fit previously method fit sort improve relative scenario save spurious covariate l adaptive scad spam rf case preserve superiority high dimensionality except adaptive increase pick relevant predictor present extend linear behaviour empirically term memory precise success require style oracle addition adaptive improve monotonicity monotonicity know knowledge type penalty effective calculation boundary unconstraine problem univariate first induction suppose far prove proceed full note fit convex function constraint combination solution optimisation contradiction ga would satisfy restricted residual sum applying contradict optimality versus make satisfying imply restrict contradict solution proof corollary strictly optimisation hence solution optimisation domain convex strictly away neighbourhood exist square bind least square proof optimum satisfie thresholded seek note piecewise monotone solution otherwise solution property go independent increase exist component iteration pick cycle continuously differentiable compact subdifferential plane zero plane subdifferential converge observation sort order observation either mean function monotonically monotonically decrease mean zero constraint break q gx gx hx hx therefore solution lie within step function decomposition purpose decompose map decompose vice versa hence one covariate lemma remark department university additive attempt determine multi additive sum monotonically high dimensional versus carry monotonicity keyword describe many parametric may restrictive well know available variable vector covariate identically distribute fit smooth constraint normal justified modelling produce array suggest additive specify restriction knowledge survey turn pool first rapid problem restriction alone retain covariate strictly monotone transformation apart monotonicity smoothing generalize consider cyclic style procedure build around covariate repeatedly fail square convexity allow component fit combine training replicate component impossible different achieve sparse conduct pattern prohibitive context optimisation resolve good amongst identify empirical calculate additive modelling work describe group lasso additive modelling smoothness penalty apply modification restrict solution negative particular calculate correlation mean requirement must
allow incomplete simulation aim matrix low figure completion performance opt fast high quality reconstruction subsequently compare fast achieve excellent reconstruction twice c c pass e simplicity theoretical estimate possible set yield bound main algorithm start subspace global yet investigate automatically performance technique automatically residual grant fa college complete department computer engineering department subspace highly incomplete basic linear subspace gradient slight modification incremental miss entry subspace online system traffic origin flow monitor water activity subspace aforementione first dimensional using identify dimensional even infeasible measurement scale subsample full sense redundant provide intuition algorithm identify subspace highly incomplete traffic spike could efficiency gain consumption large tracking build high sample incremental procedure dimension streaming entry remarkably enable update entry time maintain preference collaborative filter track evolve denote select coordinate error subspace natural euclidean matrix column span submatrix index full rank minimum time subspace average cost steady behavior point evolve horizon equivalence eq write precisely analysis author minimize cost present online algorithm art subspace dimension compact form derive incremental descent short geodesic program partial derivative respect denote use gradient partial derivative equality verify definition step geodesic singular trivial value one respectively orthonormal gradient onto rule remarkable modification current basis along direction taylor approximately serve keep surprisingly additive maintain change vanish converge static tracking rate steady filter explore understand qr svd batch subspace rely eigenvalue counterpart comprehensive find increment estimate orthogonal would span incremental modification exactly subspace work compute excellent proxy energy lie subspace q rotation algorithm compute subspace mixture determine stepsize identifying follow subspace unless series vector accord q whose spam subspace iid variable denote gaussian snr implement stepsize steady subspace vary yield small flat range converge result yet identical stepsize successful show confirm prove residual norm htb cc c except stepsize htb ccc norm track stepsize signature anomaly ability adapt scenario three break select random depend second evolve ordinary sample skew symmetric normally result effectiveness tracking instant red c sensor track density
threshold maintain fit another determine completely fashion lasso improve ols fitness thresholded ols article ols commonly design fit lasso ols thresholded ols lasso ols post corollary ols fitness thresholded lasso near lasso ols ols lasso rate strict ol post fit outperform term sparsity ensure hold lasso oracle c thresholded thresholding component threshold arguably ol suppose c c mn specify f provide characterize lasso characterize level result relative model bad lasso ols post thresholded goodness article sa ols lasso ols post thresholded obeys nonparametric bound suggest sound guarantees thresholded achieve implement practice large inferior parametric lasso ol post thresholded strictly thresholded hold ol post thresholded achieve appendix derivation supplement least result prove definition c take bound step cn x x jk argument eq r ct x r x r x thm x definition nc nc nm condition turn result proof theorem definition follow combine relation n c n b proof lemma bound schwarz master sparse proof modify note proof divide supremum van respect obeys nf cumulative nf substitution nf nm nf nm step step n bind proof nb q apply theorem participant mit comment thank lee numerous suggestion improve thank point usefulness eigenvalue gram acknowledge national science foundation regard oracle sparse eigenvalue monte access theorem remark apply ol estimator lasso regression nearly ol post least remarkably occur lasso fail component true oracle strictly convergence base selection achieve extreme perfectly ols oracle important ingredient sparsity nonparametric act selector apply estimator thresholded rate cover new practical induce maintain certain goodness latter scheme study high dimensional covariate many area analysis author investigate focus act b demonstrate achievable true similar excellent rapid regressor addition obtain interesting article ordinary least ol penalize nearly rate least lasso nice occur sense miss well choose oracle ol post sense rate convergence perfectly select true estimator oracle importantly step selector modification generic step bind generic thresholded perform thresholde lasso lasso thresholding scheme drive estimator finally confirm finding provide summary showing estimator covariate equal similarly intermediate ol post fit outperform post lasso ols post ols post produce occur low large ols lasso improve lasso ols lasso ol post lot kp simulation covariate establish aforementioned ol lasso fitness thresholded problem build establish also build result cite estimator important ingredient sparsity guarantee dimension build reason median large rely maximal inequality primitive sharp sparsity organize benchmark drive nonparametric derive generic post estimator thresholde auxiliary experiment make statement allow parameter index size omit confusion norm denote vector symbol denote standard x bc indexed event say occur increase fix regressor error index namely square square oracle value integer oracle equal achieve risk post selection square accounting mistake select detailed idea oracle previous use problem bound special ol lasso remark later mention balanced say able balance norm square large dominate estimation oracle simplify regressor covariate need obtain task later condition lead post estimator set empirical thresholding fitness near performance post selection prefer lasso estimator fit near well search ol lasso eigenvalue gram give restrict eigenvalue introduce quite post estimator use restrict gram matrix mt isometry nonzero eigenvalue matrix realization primitive well restrict away def high probability primitive condition bound dictionary polynomial similar regressor well plausibility gram equal namely eigenvalue gram q probability estimator model property estimator crucially property lasso dependent extend nonparametric generalizing drive moreover hold drive extend drive derive subsequent function tucker lasso similar imply performance asymptotic performance q remain relation drive drive second generalize obtain general rate compare lasso post first perfectly common post like model possible oracle nonparametric separate parametric thresholded lasso perfect separation near orthogonality perfect model phenomenon possible perfect selection oracle levels supplement parametric sharp begin preliminary achieve oracle sparsity yield sharp sharp sparse two give sparsity bind penalty cn nc main implication cs valid design consequently design sparsity consider bind comparable depend unknown selection post act selector estimator incorrect regressor three implication prediction state ol step rate estimator regressor selector note performance determine
acceptance concern scale mcmc form field literature scaling limit adopt framework author target hilbert absolutely detail form functional normalize precise framework capture arise practice mathematical make amenable dimension product study highlight mathematical generalize hilbert separable full densely adjoint pt trace eigenvalue arrange decrease function paper move particular respect expansion realization independent variable coordinate study straightforward absolutely sure property consequence law behave typical large coordinate preserve change offer hope well however prevent prove consequence product inherent measure sde expect dimensional stochastic thus fact attractive view limit paper hilbert value brownian covariance candidate rwm covariance moreover maximize enable idea limit originally arise nonparametric statistic bayesian see numerically finite target factor coincide respect lebesgue walk enable us sample facilitate clean dimensional walk coordinate pt function change rwm proposal affect identify exercise localize propose move explore rapidly rapidly small inverse poor negligible acceptance critical value rwm distance acceptance rwm proposal acceptance discussion rwm proposal target stationarity show stationarity scale metropolis take explore view home rwm approximation target measure form behave optimally adjust proposal covariance measure stationarity extend high target rwm hasting rwm isotropic scaling proposal conjecture change rely measure pde isotropic rwm start attempt scale question hope future paper practitioner function aware generalization walk standard walk analyze trade explore cost higher several method literature prove invariance rwm utilize dirichlet form appeal another convergence generator process use correct scaling step nearly euler wiener trajectory noise allow often use show increment converge value wiener central limit weak preserve rwm diffusion rwm necessarily weak convergence mala fix investigate mathematical rwm present heuristic detailed outline state theorem high proof invariance principle process proof drift unless letter context explain technique introduce state concern later readily separable space class operator respectively complete assume eigenvalue arrange decrease expansion product norm rr us alternate hilbert space identity operator eq l orthonormal trace hence gaussian cx write form condition may center hilbert measure property frequently subset exponent instance act continuous space decay exist entire distinguished assume eigenvalue order measure q orthonormal x namely hellinger density invertible later impose explain evolve coordinate acceptance chain one conditional random independently see way introduce bernoulli project coordinate random metropolis recall target goal start stationarity convergence dimension outline emphasis prove skeleton outline argument evolve hilbert note time scale result precisely let rwm let rwm explain stationarity explore invariant scale demonstrate measure stationarity maximize acceptance probability implication markov require explore implication three remarkably also study sigma algebra generate expectation one drift notational proposition drift rwm satisfy time markov rwm drift subsequent martingale term converge hilbert state appropriate invariance time mesh step large enough resemble euler simulate finite dimensional drift function covariance underlie chain look like weak euler approximation note important analyze clearly euler scheme identically metropolis preserve stationarity invariance martingale central rescale motion operator appropriate introduce value martingale weakly tend brownian operator converge weakly brownian condition invariance outline sc piecewise randomness stationarity reveal coincide closeness desire drift continuity show weak independent article state precisely argue rwm euler modify brownian go rwm rwm appropriate perform process heart metropolis rest derive drift recall setup start algebra random still define operator drift key accept point lead remainder stay since rx returning suggests concerned one step rwm understand careful critical source drift drift invariance z z ab calculation calculation drift need prove long key give argument orthonormal span denote notational set letting easy involve term hence ix ix see imply lemma identify observe deduce coordinate drift metropolis achieve rigorous control error quantify quantify thereby prove heuristic expect diffusion use simple calculation evaluate expect drift replace fact drift dominate large word drift section make term quantify approximate assumption main decay deviation prior measure derivative measurable ensuring application addition twice limit linearly derivative trace realization assumption ensure weak second derivative assumption globally result inequality thus c three operator imply assume throughout paper handle may show impose give equation constant straightforward imply see candidate furthermore constant functional assumption measurable dx q given repeatedly sequel let rwm rwm weakly diffusion throughout assumption without state proposition prove fix euler let returning produce define sum nu nu du term complete proof r nu nu nu ci nu nu nu nu nu nu tu u r u show rr cp nx prove claim conclude straightforward application pt n generate stationarity show converge motion pt converge weakly deduce weakly since independent precisely law consider lipschitz application contraction unique uniqueness solve subtract globally w z w w space explicit follow concern triangular martingale increment array triangular array independent increment pt array self adjoint limit weakly brownian convergence distribution hypothesis equation imply orthonormal fix give proposition proposition proposition sigma since error require condition proposition lipschitz also n tc fact condition remark stationarity use n b b b b follow x establish verify nt convergence theorem since thus verify hypothesis weakly limit enough converge independent statement adapt furthermore x calculation nearly minor notational e taylor expansion proceeding recursively obtain ns stationarity follow fact e proof lemma pair operator verify notice u I r deduce stationary lem verify corollary weakly brownian operator corollary complete drift proposition lemma preliminary respectively quantity take independent estimate error replacement recall start imply qx rx n lemma variable q lemma qx notice preserve derivative independently first recall deduce since gaussian write hence conclude almost surely strong number surely sure limit preserve cauchy schwarz c n c degree give follow applying follow q calculation lead fix realization converge limit go argument derive obtain wasserstein variable give wasserstein surely respect variable classical estimate triangle claim lipschitz conclude later q could publish fix interpolation interpolation since z difference optimize yield equation drift proof lemma derive control well approximate ix n x
iteration ff practical consequence es biased choice start distribution iteration bias graph node appropriately run rw start even rw follow reduce burn configuration case technique cover component consequently pool independently actual equivalent uniform replacement rw finally expected degree exploration node estimator original sample therefore proportional allow similarly rw trivially degree c rw rw sample sample degree sampling generate heavy tail real estimate technique technique behave average degree cover c b degree apply change degree iteratively left time consequently evaluate turn iteratively fraction cover repeat find traversal coverage k implement simulate technique rw confirm analytical importantly simulation topological capture formulae analytical expectation plain line plot line lie top match theory traversal technique curve initially coincide weight exhibit exactly apply depend type connect similar social detail summarize table user uniformly allocate version rejection uniform regardless actual mainly truth various technique run consists small es randomly allow thought randomly rw straightforward ccccc sample avg degree correct expect correct present facebook average sample contrast rw high degree three rw large drop confirm finding long significantly bias row average distribution expect work rw reality facebook cluster coefficient incorporate model appropriately row apply rw rw work unfortunately reason paragraph hold show facebook rw log take fall toward possible bias precisely practice graph reasonably size rw fig extreme small correction rw actual recommend rw rw method rw really graph diameter short produce view densely calculate carefully affect diameter usually drop grow node conclusion paper traversal technique static model traversal technique walk practice future plan theoretical topological technique question social empirically toward walk characterize date quantify calculate node expect covered random traversal correct broad perspective compare exploration random study easy property bias demonstrate facebook degree bias focus various level web www social network network entire researcher collect www exploration operation two category replacement random replacement traversal first classic web walk either bias rw correct paper baseline c l average fraction sample rw traversal technique c forest full degree distribution rw reference average rw graph traversal technique curve depend decrease second traversal technique completion method example depth forest ff large www well technique understand incomplete particular sometimes g lattice lattice unfortunately intuition introduce towards confirm facebook average popularity surprising return replacement introduce dependency explanation bias date understanding fraction graph central relate traversal technique depth forest ff traversal demonstrate exploration fraction monotonically decrease complete derive addition graph network scope hold discuss graph graph degree various technique correct facebook widely follow al social interaction facebook facebook facebook technique rw empirically incomplete confirm facebook inspire paper analyze far os enyi degrees match random tractable propose assign et operation node union path form include edge traversal discover node origin complete process visit complete trivial body replacement promise path knowledge applicable problem speak term later require propose heuristic degree moderately cover random walks walks web general walk correct walk paper graph unknown begin recursively visit neighbor simply follow classic example classic start uniformly among rw introduce bias node degree uniformly probability depend loop node transition bias rw exploit sampling terminology select randomly typically node visit thus argue compare confirm degree rw tested visit classic traversal seed explore new explore visit select within seed explore visit ff every flip coin probability ff case ff forest inspire name classic sampling visit visit l node fraction fraction node equal depend degree range regular concentrated os enyi well balance heavily skewed decade www internet system assume completely use classic generate call match ignore rectangular interval leave analyze bias degree bias technique run degree node begin walk rw serve reference excellent node stationary degree calculation fix connect eq average square degree show rw make analyze chain dependency handle elegant bias however differs begin define graph traversal interested node integral queue keep follow node discover discover add nonempty discover add remove scheduling implement schedule last forest edge never opposite c initially assign value start queue system already match
observe state easy energy reach observable determine state equal ignore generalize nonlinear advance energy balanced need equation method impulse response balance truncate principal control effectively projection despite balancing assume nonlinear unbalanced coordinate transformation approximation could computational gray mention study define operator rkh provide advantageous though langevin eigenfunction combinatorial coordinate reduced principal component gaussian reduction study outside review balance linear define matrix view instance past energy minimal reach span subspace coincide subspace key also lyapunov q several method solve equation representation align state formally find transformation transform coordinate balanced look singular occur responsible system make contribution idea cholesky note energy solve lyapunov nonlinear hypothesis linearization origin asymptotically equilibrium neighborhood origin solution smooth solution origin equilibrium system neighborhood origin singular sort singular reduction proceed least balance system numerically compute systematic tool solve problem taylor expansions white noise balance differential approach aspect approach analytic balance suitable form q rx xt zero observable linearization also origin asymptotically stable estimating reproduce kernel empirical need particular estimate original idea drive sample trajectory extend generalize lift vector reproduce endow bound use follow important kx reproduce rkh x kx rkhs always rotation suggest xt respective response within finite assume empirical fixing measure response initial great clarity follow generalize high dimensional space taking consider feature working denote center however eigenvalue normalize unit space normalization convention magnitude eigenvalue order nc qx show space kernel kx essence collect simultaneous rkhs rkhs using become simultaneously strictly speak pca favorable continue refer turn collection scale index consist pairwise rank number equal dimension matrix restrictive assumption equal take svd c discard project system well approximate set apply rule map rkh empty difficulty rkhs reduce approximate high degree square taylor need second order reproduce well fact jacobian map capture value q ignore avoid reduce example j pair map training seek function square take form ik z rkhs coordinate function general choose learn validation notational convenience ix j broad separable one function turn approximate appear compute taylor q desire good return kernel derivative improve perhaps use family two kernel respectively case recall dx yx invertible approximation reasonable problem consistent formal rt reduce counterpart original virtue way correspond recognize kernel estimate rkh notion write dynamical eq expression jacobian closed system immediately true situation system essential precise taylor map familiar ix j ik jk rkhs note give compare dynamic controller control input appear pg impulse interval kernel counterpart impose balance transformation cholesky reduction map space next variable follow section choose wave solve polynomial dependent trajectory dimension offset fast leave validation computation remark dynamic explicitly rd degree rkhs appear improve however special version learn wave control match dynamic hyperparameter select system wave panel output bottom leave panel closely system main wave reduce
utilize computation statistic comment show particularly strong geometric commonly occur exploit geometry surface ridge advance sampling constant equally create identifiability namely decrease ridge ridge strong hmc demonstrate ability follow length hmc differ sensitivity step describe hmc suffer presence jump hamiltonian follow step generalize momentum parameter theoretical derivative choose converge size standard vary identifiability step nan subsequent admit normally algorithm affect behavior bring insight adapting provide adaptively devise direction move introduction technical report contribution present journal second comment riemannian manifold present ability adapt proposal proposal case control utilize covariance metropolis mala surveys remain unchanged shape widely shape comment utilize strength adaptive step combine riemannian involve center equivalent additional mc mcmc highly advantageous another curvature updating dependence structure compare
module order shorthand level illustrate module compactly represent diagram infinitely major diagram point sample plane define diagram range persistence bottleneck persistence map diagram figure diagram persistence module strongly q cloud persistence diagram curve show cloud leave persistence diagram wish compute persistence associate module convenient substitute term computable persistence two persistence module suppose module map module family map module diagram pair number rise new persistence module persistence module module homology module arise topological sum way give module family produce topological ambient formally x ambient grow rise persistence module persistence diagram module diagram embed plane along persistence diagram diagram bottleneck indeed diagram upper family space solid line ball circle restrict pair pair persistence module homology group persistence dimension appear dot diagram module fx u r p acknowledgment thank useful discussion thank david discussion sm acknowledge support grant acknowledge nsf dms xt lemma proposition theorem topology great basic cloud ambient space limitation statement manifold consider decompose vary fit together cloud sample clear belong large space unclear everything noisy small scale notion algebraic topology particular equivalence homology attempt persistent homology manifold homology statement cloud infer homology converge homology nature large homology space approach develop state probability belong compute correctness review need background topological geometric statement correctness contain main body discussion necessary persistent homology mostly adapt present material simplify first module mainly
encoding let correspond encoding endow discrete topology encoding easily natural element induce hence follow diagram remove contain cycle cycle definition consistent chinese restaurant crp crp marginal totally analogy encoding regard interpret element ad form group metric model commonly literature consider intersection eq model hence j uniformly independence preserve uniform model coincide distance metric neutral permutation decomposable metric define adjacent transform neutral permutation position discount choose differ discount cycle adequate construction cycle verify construct chinese element place cycle respective create prior projective family projective family application permutation generate guarantee suppose concentrate borel embed canonical inclusion less permutation borel lemma cf proof let projective bayesian conditional parametrize hyperparameter sequence satisfy almost surely countable theorem projective hyperparameter j update describe wise anti permutation deviation decrease closely term sufficient relation may useful parameter instead require eventually occur model represent counterpart statistic update projective respective function analogous give statistic measure exponential prior mean projective limit many construction projective construction field mathematic marginal suitable projective existence expect representation formalize sure conjugacy involve process establish intuition since ab consistently arguably measure restriction inclusion map measurable assign intersection conversely definition function compatible mapping preserve integral proof let projective construct projective h kernels ny q projective limit guarantee theorem projective index projective mapping conjugacy marginal conjugacy projective mean continuous open closed exist measurable draw concept selector function assignment map axiom regularity underlie existence theorem admit selector weakly empty correspondence aa make correspondence measurable measurable measurable analogous index projective form projective posterior p h since model sampling satisfy form mapping mapping proof lemma topology space separable countable countable statement whenever singleton one da measurable mapping simply measurable hand since verify generate compact ball point subset ni I dc nx lemma projective family lemma permutation thus group equivalently image establish lemma projective conditional prior partition establish projective conditional lemma hence borel embed regard first subset sequence tail algebra borel image canonical inclusion composition virtual finite entry cycle cyclic set borel sum converge hence random q either jt jj j example note projective limit conjugate nonparametric conjugate class study include dirichlet infinite projective limit projective respective counterpart application model construction permutation nonparametric effectively dirichlet beta process conjugate nonparametric fundamental conjugacy characterize conjugate posterior infinite priori process conjugacy dirichlet existence marginals exponential dirichlet connection conjugacy conjugacy dirichlet prior show intuitively conjugate posterior family marginal construct conjugacy marginal guarantee analysis projective limit finite use evy process stick break construction transform completely come price technical choice representation probability measure characteristic projective translation invariant infinite process characteristic ill suited question live mathematical object projective limit kolmogorov extension generalization construction infinite dimensional represent finite projective important path projective mapping account path term projective measure mathematical structure limit associate obtain nonparametric bayesian property parametric marginal posterior finite counterpart establish large dirichlet regard nonparametric analogue precise model wide domain regard projective limit mathematical projective use construction topological algebraic discuss construction random construction limit limit projective process obtain applicable unified projective intuitively projective result projective projective limit hold probability fact projective projective infinite theorem generalize conditional conditional projective model bayesian projective limit sec structure sec operation computation posterior diagram word directly marginal analogous projective projective limit applicable projective sufficient marginal resp algebra projective limit sec marginal minimal minimal projective utility bayesian carry marginal projective conjugate projective limit update mapping marginal model obtain analogue model example sec infinite statistical parametric average beyond exchangeable observation projective observation nonetheless projective field projective analogue projective parametric nonparametric bayes applicable conjugacy dirichlet amenable gibbs process law conjugate conjugacy analyze property notable exception family discuss posterior invoke relate process process purpose projective application representation nonparametric posterior projective fact variable share frequently etc mapping field space index denote measure indicate outer assume exchangeable topological space I separable unless refer borel space probability survey stochastic process definition terminology projective limit sense projective limit appendix let topological borel generalize projective mapping induce family projection mapping projective limit family kolmogorov projective family probability dp context represent continuous real represent stochastic collection random distribute index principle projective suitably construction projective address two technical countable word unless countable projective projective space projective projective negativity monotonicity function projective measure function function imply projective construction consist finitely additive contain projective problem address elegant manner space possibly measurable construct stochastic process specifically embed map let mapping call analogously constitute image suitable countable direct projective asymptotically identifiable countable freedom path countable think dimension degree many degree suitably projective projective construct euclidean space subset separability separable modification subset dense see path uniquely establish borel suppose uniquely representation countable mention representation measure algebra borel suppose choose projective countable algebra measure restriction admit parametrization dirichlet fix form projective satisfie projective construct inclusion projective p gaussian borel since restriction algebra induce coincide algebra hence requirement existence marginal parameter kolmogorov obtain satisfy white probability mean theorem projective limit manner projective theorem projective borel space p immediately generalize measure act projective family argument assume order accordance projective projective conditional pa projective conditionals projective converse hold additional condition projective family family effectively parametrize projective limit countable let p projective projective measurable exist denote p projective measurable analogy conditional limit continuity measurable measurable projective projective family projective limit w df regular p regard family measurable mapping projective projective however mapping mapping induce measurable mapping projective index np measurable projective first due projective limit observe limit countable x almost everywhere projective limit generalize regular regular borel hausdorff random variable therefore involve usually image variable parameter kolmogorov continuity function subset sec since algebra wise regular construction construct x measurable namely embed measurable case well projective limit theorem kolmogorov projective system space hausdorff p projective probability measure p compact set eq projective marginal derive result certain preserve projective limit construct applicable construct kernel effectively projective projective space satisfy kolmogorov extension projective probability remarkable requirement impose upon space countable need general regularity support induce marginal random projective random variable relation eq p x imply unconditional measure p sufficient converse generate x correspond projective similar projective conditional family address projective projective projective dp product conditional projective eq justify consist give inclusion p establish implication projective ax integrate section projective bayesian completely define combination projective limit borel embedding allow represent bayesian projective family parametric parametrize regardless briefly parameter exchangeable induce parametric image call call p regard family guarantee conditional probability validity number empirical well image parameter abstract event completely determine contain condition suitable condition asymptotically recover additionally represent parametrize whole purpose sufficient dominate applicable notably conjugacy sec suppose projective sample space parametrize object abstract equip projective mapping conditional index projective limit index diagram constitute model carry projective projective borel p x uniqueness equivalence projective conditional diagram n n infinite projective parametrize largely analogous projective limit restriction abstract assume measurable assume nan probability describe small induce represent canonical restriction respective restrict parametrize justify integrable random family outer turn induce explained infinite dimensional finite g gaussian dimensional quantity posterior censor general formalize bayesian explain measurement draw generate sample j formalism stochastically independent censor projection asymptotically either censor depend application projective parametrize admit projective statistic projective sufficient statistic regular probability kernel ii space mapping sufficient projective limit projective I system measurable x equivalent x draw firstly secondly eq assumption conditional projective x accord I obtain generate projective simply application projective since marginal latter make construction process parametrize borel show fine algebra satisfy conclude question small algebra closely statistic transformation algebra algebra capture inaccurate case point instead demand indistinguishable resolution minimal rather contrary sufficient theorem construct projective marginal admit algebra x algebra imply ac bayesian always exist define deduce posterior prior bayesian appeal nonparametric type g specify family measurable mapping posterior n refer system index loss contain also conjugacy conjugate marginal conjugacy projective projective posterior index projective n index projective model conjugate projective limit model conversely projective open consequently drop either assumption projective mapping lift restriction projective limit family always obtain etc projective mapping closure sampling mapping concentrate countable projective limit preserve conjugacy conjugate conditional restriction preserve nonetheless constitutes define regular ensure abstract immediate definition index parametric example widely nonparametric measurement concrete illustration brevity sec detailed construction record corrupt since space adequate function orthonormal separable hilbert space projective embed projective outer cf process realization terminology projective satisfy denote positive hermitian operator gaussian projective define f projective prior define latter operator marginal choose model eq formally nonparametric construct candidate straightforward verify ii family map coincide projective theorem conjugate assign outer statistic marginal trivially projective proposition projective conjugacy property interpret conjugate characterized canonical prior respect suitable define density sufficient topological contain parameter equip function normalization parametrize model hyperparameter update
subsequence history concatenation string description necessary describe say however lead high bayes try regard length describe resemble kolmogorov combinatorial notion kolmogorov combinatorial provide reduction description decide state still thus bit choose random sequence overlap redundancy sequence representation circular position still point mainly add redundancy somewhat improve datum leave base single bit first section extract sequence set character whose fall string zero character prediction length single character encode sequence character could symbol perspective convert transition entry repeating time symbol last half last encode character clearly redundancy yet obviously source good choice error value select score define instead represent sequence attempt code state transition symbol last transition candidate symbol start either concatenation candidate symbol example symbol agree symbol lead score extra need describe convert process string string encode start bit end candidate symbol sequence symbol form sequence remove right example several source encourage trial close enough even history second run time value length indistinguishable lead predict score compression minimal file never go matter next idea consider extend repeat repeat example qr v g v version string experiment lead improvement history size treat case source predictor source history time predict bit symbol represent predictor represent state bit representation two repeat predict qr next realistic allow order trial change history measure figure average run decrease prediction error start next character contain example figure decrease increase predictor prediction bad steady rich order reach happen predictor produce already problem low already order next inaccurate still slightly slightly pure history reach figure decrease albeit slow error depend increase steady enough combination bit long figure change incorporate dictionary predictive show initially dictionary exponential predictor sure compression encoding investigation predictor compression prediction error plot compression black measure amount score obtain maximize prediction agree say future prediction give compression next symbol text history current symbol skew shorter next opposite suppose predict symbol examine error introduction area information give picture movie data compression shorter search permit remove repetition exist phase first remove redundancy describe phase description encode produce next picture encode predict arithmetic compression compute arithmetic encode pose prediction black change predict symbol motivation practically instance model statistical predict symbol sequence compression principle approach compression question make source sequence incremental start bit add finish predict time predict bit error goodness estimate error predictor compression encode arithmetic permit set
bit distribution start step chain start chance leave walk start exponentially choose polynomially let walk hypercube weight xy xy xy n low hypercube enough complete testing chain computation much mix recall standard machine iff computable function polynomial polynomially every lemma transition chain approximate proof bit query least otherwise yes accept increase see chapter compute enough check vertex possible string run fraction stop amount due step exponential use additive bound use since second run dt ty take conclusion chain reversible reduction computable string string instance input define state space markov generality generality let accept reject either reach markov chain reversible walk weight type connect pair state role play machine transition depend machine particular graph chain polynomial reduction circuit specify polynomially bit polynomially bit secondly tm read weight connect edge complement furthermore weight edge transition loop connect bind take therefore q state acknowledgment thank helpful rgb rgb pt lemma assumption dms work department mathematic support program china grant cb grant dms grant ga problem algorithm iteration stationarity chain practitioner like carry assess lead whether far setting markov rapidly hard zero knowledge start stationarity far stationarity intersect give rapidly hard stationarity stationarity complete distinguish markov stationarity far markov chain monte distribution physics biology image important impractical determining far converge mixing time see carlo integration partition physics effective bound may state time rapid impractical require practitioner development variety try far stationarity survey majority practitioner multiple chain converge public domain diagnostic package behind copy compute various functional guarantee convergence formalize stationarity study main contribution show chain know distinguish much computational problem far stationarity assumption chain diagnostic computer science role study define time measure recall give tv follow mix mixing start formulate think rule circuit rule formalize imagine like diagnostic determine say within stationarity require total variation exactly practitioner diagnostic mixed variation stationarity mix least distance away weak diagnostic within distance stationarity approximate even easier provide correct actual denote realistic formalize diagnostic much strong requirement continue hardness discussion definition diagnostic circuit move chain unary theorem diagnostic unlikely version state input circuit specify markov ergodic yes informally mc start time mix away diagnostic give room term problem hard informally solve time hardness class second statistical see evidence sd completeness restriction put completeness conclusion cover parameter psd flip tail accept reject interactive completeness matter completeness notice inequality tight protocol want statistical case say forget distance via later two strategy long compute quantization compute representative estimate sure exact never run variant protocol proportional minus strategy detect give start identity side equal left hand equal right precisely protocol circuit fraction unary hash accept reject completeness yes protocol chebyshev getting fix number least chebyshev getting take expect chebyshev force second either generality assume number chebyshev conclude repeat sufficiently many consequence protocol protocol protocol circuit reduce constant constant result give claim protocol accept otherwise reject I n completeness rely q similar sum completeness claim none protocol case establish completeness protocol bind use restrict diagnostic start respective prove sd choose suppose eq case sd reduce show membership show sd condition small sd instance circuit output construct state state accord otherwise stationary yx variation
increase alm hour minute hour hour minute alm network breast cancer reference description set alm ht alm n cpu gap alm alm solution sparsity problem primal invert solution fair truncate version complementary th element row number produce method inverse datum alm alm vs vs alm alm alm vs alm table three produce solution exactly note roc positive parameter provide report support grant dms dms de er show follow definition generalization v subgradient subdifferential point subgradient set iteration complement let get let hold summing since instead similarly let get increase yield fy desire solution proposition theorem corollary department great structure maximum likelihood alternate linearization subproblem solution association show practical outperform multivariate analysis graphical gaussian markov meaningful vector gaussian network edge denote independence covariance thus pattern iy hard nature numerically cardinality envelope convex min problem primal problem strictly objective duality semidefinite per propose block descent method method update row programming base subproblem another descent et solving formulate subproblem prox method cd cd inferior project consider art iteration result variant apply apply nesterov primal smoothing nonsmooth obtain complexity optimal nesterov however since performance alternate augment wang impractical alternate linearization alm solve penalty sparsity inverse covariance closely primal plus approximation show method justify intuitive interpretation perspective alm algorithm outperform method organization briefly review linearization complexity alm intuition result synthetic section alm alternate linearization method alm solve convex function effective way introduce new rewrite alternate direction augment subproblem lagrange via jointly alternate version augment lagrangian e g update subproblem symmetric advantage subproblem substitute relation refer technique adopt fast improved complexity almost however apply define property challenge nevertheless apply directly implicitly fortunately proposition n formulate matrix operation need constraint instead impose constraint tight sufficiently iterate remain determine iteration claim hold neighborhood pick solve k x x k condition step ignore constraint change solve effort know z decomposition dominate result definite iterate eventually intuition find fit try propose address objective sparse length recall seek optimize good fit regularization impose prior current guess inverse update take seek solution impose prior whose mean inverse definite pick theory tell present result synthetic efficiency alm alm write intel cpu gb feasible long iteration gap primal objective function terminate alm since decomposition see
recommend segregation high power triangle replicate association nn computed segregation point respectively association solid association alternative replicate triangle estimate almost nan alternative large association imply small notice skew approximate ht leave solid line ht carlo power proportional triangle alternative row middle row right nn mc carlo case get segregation high association level triangle monte power relative density triangle value column association triangle parameter compute density base power estimate get triangle empirical power consider recommend association triangle degenerate association replicate observe imply small power depict density association dash present nan density power imply central similarity value row leave column middle monte carlo mild association small attain mild moderate severe association value recommend triangle middle function bottom multiple empirical present observe monte power tend decrease recommend correspond segregation mild severe power triangle segregation central association proportional consideration index versus efficacy investigation limit nan parameter condition pc equivalent function pc replace satisfy pc pc critical n satisfy quantity efficiency base terminology discussion segregation derivative minimum derivative respect segregation alternative r standard pc pc detail respect q parameter segregation association expansion one relative edge central dash left expansion segregation pe sr pe sr based suggest choose large segregation skewness density suggest test sequence notice explicit segregation respect get note r derivative pc continuity next per substitute numerator denominator present association pe ar pe pe ar pe asymptotic small moderate skewness moderate h calculate segregation case derivative follow continuity numerator denominator easily pc association choose testing moderate normal appropriate due skewness suggest association alternative pe pe relative go rate local hold provide convex segregation asymptotic asymptotic unlike asymptotic variance investigate function investigation tree south rectangular plot near contingency table live breast height record pattern tree specie four comprise stem middle rectangular plot interaction tree take hence tree segregation plot circle black datum tree inside outside hull convex hull proportion correction decrease magnitude raw alternative consider alternative sided alternative level segregation similarity hull correct test circle triangle connect dash line notice horizontal differently carlo randomization calculate value observe replacement trees value monte randomization value determine rank divide divide left side correction determine outside determine convex hull correct monte randomize none significance normality randomization randomization proportional correct solid triangle connect line horizontal horizontal differently segregation live tie nearest nn cell marginal segregation overall deviation tree test interaction vice versa situation test abundance specie cell size cc cc cnn sum right tree distance second modify bivariate rectangular estimating length rectangle since rectangular symmetric theory correction slightly asymmetric bivariate plot suggest significantly significantly bivariate tree line monte simulation independence article consider proximity edge similarity segregation association spatial randomness knowledge theoretic testing spatial expansion central similarity demonstrate express statistic arc thereby normality extensive carlo simulations proportional edge segregation segregation association central segregation association performance segregation relative relative family asymptotic relative proportional edge asymptotically segregation central similarity parameter respect empirical ordering let size point convex hull pattern independence circular point family test geometry invariance property triangle point imply imbalance relative class method appropriate imbalance test construction proportion inside inference ii correction recommend normal approximation simulation might parametrization distribution alternative geometry triangle alternative e might approximate parametrization segregation far away cluster around segregation parametrize expect regardless parametrization simulation execute laboratory conjecture mail tr parametrize namely proportional position density use segregation complete randomness scale properly statistic family theory statistic test monte simulation test alternative assess expansion segregation empirical power central similarity one segregation proportional edge well illustrate keyword complete consistency efficiency proximity relative segregation spatial receive attention proximity direct vertex point distinguish arc vertex vertex class give exact class observation minimum relatively dominating prototype minimum dominating generalize correspond define region triangle increase distance point increase parameterize class point triangle gain popularity analysis move suit connectivity introduce landscape utility explicitly reference introduce spatial integrate tie patch location spatial dimension however usually graph construct datum instance concept depend adjacency object express information allow vertex new reflect intra patch patch quantify among graph theoretic spatial class possibly correlation relative density segregation association one tend segregation observation segregation specie proximity region arc segregation cover many arc thus relative arc arc unfortunately precise dimension geometry neighborhood segregation purpose associate region proximity vertex edge triangle lie article parameterize family bivariate spatial pattern compare finite relative asymptotic alternative triangle propose term hull point family central similarity multiple alternative segregation normality version section performance assess asymptotic propose outside test derivation expression defer appendix proximity measurable consider x nx I identically region extensive treatment proximity cardinality ratio arcs arc complete order brevity asymmetric arcs variance however simplify central theorem stand normal variance asymptotic determine normal iid support illustrative composition scale triangle uniformity transform manner parameter proximity map segment vx xx fall vertex region arbitrarily let orientation vertex opposite xx r define central similarity proximity opposite segment center fall fall edge region assign arbitrarily similarity triangle triangle orientation similarity iii parametrize similarity proximity map notice degenerate density support vertex occur case boundary vx vx vx illustration construction proximity region iid triangle randomness poisson nan analysis relative geometry relative central provide geometry trivially likewise geometry invariance trivially degenerate base uniform nan vertice henceforth invariance proximity mass orthogonal projection edge vertex triangle orthogonal edge projection triangle hence depend need combination geometric mean asymptotic central normality nan proportional nan px n probability arc similarly h theorems degenerate derivation asymptotic arc leave expansion piecewise definition scale function depict monotonically sn pe sn pe explain become continuous monotonically x cs sn cs plot asymptotic function depict also r plot asymptotic nan expansion vertical piecewise function differently h arc expansion asymptotic asymptotic edge solid similarity vertical interval axis differently scale stand convergence equivalently skewness analytically much rr leave right h central line vertical interval definition observe successively condition hence calculation value hold density proportional indicate figure demonstrate severe skewness depicted distribution right histogram replicate density vertical axis histogram replicate indicate skewness extreme indicate approximation central skewness however skewness depict histogram monte approximate differently depict replicate middle indicate severe skewness extreme value present triangle suppose mm result triangle triangle segregation see present identically segregation right r proportional multiple triangle define similarly let theorem conditional rr equation mr p mat mat j furthermore pe pe pe nr gr p x pe rp pe mr pe x mp r mp pe nr ac pe nr p pe pe mr p pe gr q likewise r central jensen I rr phenomenon segregation occur member tendency type vice versa imply unlikely locate alternatively occur tendency member specie alternative parametrize without fp pattern segregation family basic alternative segregation association respectively let b segregation possibility association require occur around class alternative u u j segregation alternative segregation particular expansion association alternative expansion region triangle result alternative family segregation area line parallel segregation triangle away vertex segment opposite segregation similar available triangle segregation triangle case triangle transform thus segregation small segregation nan realization depict asymptotic density hypothesis segregation nan let expectation segregation r association l association sketch ii ii explicit piecewise explicit piecewise q alternative normality segregation alternative normality furthermore normality central segregation normality hold alternative triangle relative proportional replace segregation segregation since segregation association standardize segregation association test triangle multiple triangle one consistent hold multiple triangle multiple nan pattern replicate approximation estimate side segregation association expansion parameter r standardized replicate segregation alternative mc mc mc mc empirical smaller large conservative asymptotic proportion determine significance proportion test significance less conservative great limit plot get level approximation get segregation size nominal seem increase leave I relative association level much size normal base replicate segregation association bottom column column pattern horizontal line locate threshold nominal threshold nan iid conservative size together triangle one triangle nominal alternative segregation side nominal although empirical seem far level side conservative sided alternative ht triangle empirical proportional triangle replicate sided alternative relative segregation side relative bottom column pattern horizontal line threshold nominal low case generation replicate j standardized replicate segregation mc cs mc mc triangle similarity upper limit increase get well right sided nominal test seem conservative testing segregation side nominal size desire sample conservative recommend appropriate get wide large similarity replicate left segregation side association column triangle figure observe triangle level triangle furthermore sided test size recommend segregation yield seem appropriate recommend empirical relative central triangle carlo replicate segregation right e one triangle case estimate similarity close segregation nominal alternative performance statistic uniformly support segregation association alternatives carlo replicate segregation x nn carlo replicate compute segregation segregation degenerate almost rr rr proportional solid segregation dash
output odd pattern local stability pattern change analogy stability pattern stable flip pattern misclassifie flip double learn process strategy search strategy follow ref configuration generate perceptron correctly otherwise adjusted elementary weight flip double flip j correctly solution space reach fig allow construct flip misclassifie choose obvious configuration pattern protocol difference allow contain order integer configuration pattern stability order pair integer learn order th total satisfying pattern one elementary local change process stop classify visit isolate last exceed maximal overcome landscape perceptron remarkable pattern weight one misclassifie threshold process configuration weight achievable storage capacity process strategy ignore ignore pattern hope present work implement ising perceptron unable remove learn stage succeed remove sequential succeed proceed attempt fail stage high large length strategy input pattern strategy pattern stop number correctly classify pattern report appear storage law bad strategy storage less storage systems elementary pattern probably strategie regard walk storage indeed internal storage pattern run strategy strategy configuration measure overlap solution predict replica calculation suggest solution distance theoretical storage reduce hamming solution pattern pattern run histogram almost width ref ising perceptron demonstrate constraint increase shift suggest level solution typical hamming compatible observe double multiple peak histogram simulation consistent random member problem ise perceptron co suggest ising perceptron dominate vanish cluster compatible problem first pattern grow linearly community need walk community maximal process storage capacity make search space point process propose stochastic ise fig input pattern strategy add linearly input pattern work suggest learn random strategy exploit system neurons elementary separate typical capacity study reach strategy solution particularly field isolate space completely fortunately even allow configuration isolate capable separate isolated solution extent strategy replica symmetric give search structural solution ise influence solution ise perceptron input output long uncorrelated perceptron student learn teacher amount student perceptron student match teacher poor foundation china grant china cb laboratory theoretical physics physics sciences china institute china institute chinese sciences china laboratory theoretical institute physics chinese sciences china institute physics china theoretical physics chinese sciences china variants search perceptron study pattern ise configuration modify double weight reach storage length performance give solution construct typical hamming decrease constraint hamming decrease feed neuron perceptron build one learn memory perceptron neuron unit connect value correctly classified perceptron assignment compare ise perceptron two take state perceptron np need grow exponentially weight complete enumeration weight feasible small system pattern unable correctly classify matter weight transition phenomenon perceptron sample maximal storage capacity statistical perceptron perceptron correctly pattern predict step modification exploited expect sophisticated biological perceptron usually read sequential learn new pattern learn biological mechanism help prevent old ref motivated biological consideration sequential mechanism namely space sequential random double weight newly add pattern previously misclassifie later stage simple good neuron perceptron sec walk present
practice principal distinction particularly construct genetic variable repeatedly criterion aic stop path practice go reader main termination force solution deterministic sub optimize aic step arrive ensemble genetic algorithm stochastic optimizer description optimizer despite crucial section aic aic course mid simulation feasible subset show merely exhaustive aic selection article ensemble produce regression cox describe take three variable select st rr rr signal st st consider construction type variable signal ratio relatively low relatively true average type st fit table message experiment st term noise st slightly importantly st significantly weak signal st optimizer stochastic simply st optimizer desirable motivating frequency type notice stability idea start latter graphical shall limit ensemble present limitation dealing correlate present finite shall later control expense signal apply procedure level consist run repeatedly bootstrap aggregate regard randomized randomly select cv cv scale false proceed structure approach st st well ensemble genetic st variable variation lar scad elastic net section comment ensemble tackle describe structured variable selection selection add retain backward already include improve discard continue add add randomly decide traditional assess one often group select assess good contain detailed simply repeat forward step suppose want say group randomly candidate improve aic want number candidate assess size randomly without replacement assess group good objective aic bag random subspace ensemble classifier word increase unfortunately reduce principle ensemble importance correspond trial variance must carefully liu although strength diversity tradeoff ensemble tradeoff explain recall stochastic st st language st algorithm exercise balance diversity tradeoff specify tuning parameter give size say balance diversity variation measure variation objective path optimize strength percent improvement contain base fix quantity plot middle logarithmic also plot st simulation depict fairly typical tend greedy many candidate give evaluate good one high strength search become greedy reduce strength chance find subset explain greedy start parameter control easy reduce diversity middle reach start drop choose look peak course emphasize simulated example merely validation reality must alone choice select decrease desire increase far eventually drop panel st drop clear tradeoff move right corner along panel decrease along course emphasize panel simulate true mainly verification aforementioned ridge rely principle pair along aforementione would subset current marked ridge find strategy move examine size motivating create difficult allow see three type behave reasonably respect noise consistency setting motivate type variable strong noise st sample size smoothing spline fit well visualization motivate c example stability run stability detail control false widely I true signal noise almost major publish ever nonnegative lasso elastic net relaxed lasso group oracle signal scad cross tuning tuning c scad lasso hard subset good oracle stability together e competitive exclude signal together st st excluding signal say relatively behave true signal ability lasso variation elastic net much algorithm exclude performance relax widely time different noise median max lasso elastic net lasso lasso stability elastic relaxed lasso net lasso specifically highly coefficient correlate group correlation highly minimal maximal simulation st min lasso lasso elastic net relax lasso st stability elastic net relaxed lasso together clearly st lasso top much tendency selection quite discovery stability true signal relax st clearly demonstrate highly correlate significantly st robust diabetes example standardize package lar body index response result path decrease enter list diabetes angle st table st agree top least variable intermediate importance age rank middle large early certainly attribute highly correlate st bottom statement tc diabetes ranking map age tc age end discuss nice approach use thresholding ensemble opinion theoretic sparse quite objective imagine search useful medical think rank list provide prefer thresholding must active decision multiple rule produce else concern measure use potential variable repeatedly answer st lasso report paper bootstrapping alone within satisfactory explain randomization mechanism good
high randomness obvious theoretical distribution hamming weight recurrence satisfy thus occurrence hamming sequence lag statistical detect sufficiently serious generator generator since bit hamming recommend recurrence prefer generator different addition mod odd odd return early period period mod vector algebraic structure generator undesirable vanish use generator package generalise extremely long period greater easy align operation output ham simple combine generalise free software period generator generalise obtain long primitive choose minimal polynomial generator implementation shift shift exclusive pseudo number typically say present case shift base eight odd omit fast base state eventually desirable long length generator certain perhaps generator satisfy generator block consecutive bit generator detect relationship segment cycle processor seed processor want segment negligible important generator generator memory cache small bit discuss mainly extension suggest present choose problem sure generator maximal restrict power know q number generator relevant generators example generator mention possible improvement usually distinguish interpret exclusive computer regard shall identify bit treat x eq c recurrence circular array array mod unless recurrence regard block recurrence polynomial primitive primitive power mod verify case irreducible satisfie linear recurrence good number important recurrence full choice choice least transformation mix bit necessary associate could great assume depend want want choose generator parameter choose set characteristic polynomial table search criterion decrease criterion possibilitie consuming criterion repeat
uniform table tree intuitive indexing origin curse let interested indexing indexing partially property base indexing choose recursively value discard answer namely property irrelevant ht terminate discard condition come near two go infinity cf figure illustrate characteristic gaussian one normalize observation common effect moderate point outside region mark bar discard histogram randomly draw vertical mark fewer discard indexing repeatedly pp mention need lipschitz build indexing scheme domain highly concentrate variety geometric object jointly phenomenon link regard regard dataset low combinatorial average sense curse fact concentration explain projection hamming approach get curse entire indexing scheme still discuss briefly conclude article remark dimensionality black lee well base graph informally phenomenon state high every lipschitz different dispersion precise definition probability measure concentration upper complement neighbourhood subset cf drop regular hamming cube chernoff obtain combine chernoff see bind real real one variance function phenomenon admits illustration dimensional cube projection onto subspace cube choose concentrate centre high dimension red outline dimensional projection cube ht shape cube get ever disk difference unit cube diameter interesting essentially dimensional observer precise statement reason choice ball matter much book treat concentration reader comprehensive contain equip agree model characteristic fashion appear realistic metric space measure intrinsic develop slow assumption asymptotic indexing growth note draw dataset point amount randomly draw domain equip index infinite regard product parametrize dimension dimension single domain dataset near function every except cube space etc asymptotically result proof important subset vc dimension denote classical interval family hamming ball intersection member estimate vc hyperplane generally conjecture open banach finite result euclidean borel countable intersection ball borel sigma algebra call subset attention empirical regard normalize finite sample go matter formulation convergence measure whenever theorem uniform interval enter selection book treat mention write metric function lipschitz guarantee view projective viewpoint detail overhead know efficient cf every equip kf ip reduction sequential scan image analyze indexing nn similarity process stand calculation latter computation need separate false optimize correct hamming indexing measure cube addition ball consequently table half distance measure ht intersection spherical nearly high table deduce intersection vc whose near great least empty range query belong notice contain without exponential influential theorem parameter verify sphere parameter claim always fact euclidean domain becomes lead confusion h begin leave inner symbol partially metric type indexing structure consist rooted assignment pruning subset bin cover identify sub subset binary string lf f l xt label range child visit branch amount branching considerable curse metric type indexing scheme content cell read pass computed child follow leaf reach query branch requirement cell cell currently structure early namely say free reason dimensionality use indexing long lipschitz exhibit concentration price pay relevant distance get use exact nn hamming cube binary function space support normalize multiply course dataset point vc exceed accord coordinate hamming cube restriction mapping hamming cube normalize preserve cf coordinate appropriate ann possible range generalization develop project hamming cube transformation assume key suitable large away cube storing nearest answer discovery employ series projection ann indexing pair let draw value ht vast pair geometric explanation intuitive simple half sphere interval contain region projection projection direction subset identify z consist dimensional projection combinatorial vc expectation factor confidence randomly pairwise concentration instead dimension projection pairwise distance remain mean empirical mean lemma large mapping moreover high euclidean even normalize quite good distortion considerably map nn ok histogram concentrate explain search indexing scheme indexing nn article appear time cell probe asymptotic indexing artificial datum belief intrinsic image range area outside great measure concentration spirit paper impossible readily statistical interesting ball vc concentration behaviour really indexing dimension spirit box study lee instance preprocesse indexing similarity search call formally domain classical remarkable nn minimum every cover diameter diameter supremum parameter lee algorithm take query contrary exhibit curse contain unit convert black model set domain metric space uniquely ht finite satisfy remarkable obtain almost surely copy separable reason universal black box finite space box simple produce every query uniquely near initially point whose deterministic execute call distance admit nearest namely search step start define require follow underlie become subtle indexing still adjacent intersect suppose geodesic previously include near else adjacent strictly close short triangle turn indexing exact nearest q move return study metric algorithm efficient vertex observe general specifically prove element exist subspace connect graph translate universal two situation sphere hamming indexing curse concentration consideration would three look like highlight difficulty obtain uniform way bound possible indexing formalize even indexing successfully mine fact investigate two geodesic sense every triangle thin neighbourhood
dynamical together sample prior characterize encode exponentially thus jacobian accumulation fast construct module framework wherein module allow module capable interact module interaction module logic suggest identify resource module brain computational filter per capacity sequence capacity communication network measure per ability unit second quantify mathematically reference illustrative presentation idea statement algorithmic history network sense module module arrange learn ever version overview carlo bayesian planning abstract aim work rational make artificial physics mathematical rare sufficient reinforcement style resource effect resource limitation architecture keyword recurrent reinforcement planning sequence pareto computable monte plan universal model resource replace distribution gold research main difficulty universal obtain algorithmic computable proof hypothesis characterize main environmental must implicit parameterized frequency alternatively approximate bind representation defer equivalence machine network additional available parameterized straightforward sample statistically monte carlo meaningful operation present modify long performance continuously flip fed control straightforward perform way motivation prevent error signal gate directly computation first cell act accord input present output treat input bit distribution network concatenation input subset agent output output next predict predict generate plan specify aggregate reward overview universal prior class operate environmental recent estimate neural accord assign input current horizon generate consist output environmental choose obtain recursively update light prior face sample exponentially rare event technique seek class distribution predictor likelihood length kolmogorov state minimum bit certain precision multiple input go weight influence simplification omit rnn architecture weight prior formula recursively previously however previous paragraph machine bound transition initialize acceptance parameterize proportion hasting construct density repeat boundary level repeat quantile estimate normalization though quantile hasting repeatedly draw multivariate bound variable multivariate laplace covariance architecture previously upon literature especially dynamic block maintain common kalman filter neural latter use relax input bit weight current sample take bernoulli respect observe tw threshold replacement assign form b estimator sample assigning unit suggest kernel density towards mean kernel parameterize translate particle role state plan input joint sequence sequence appropriate stationary sufficiently execute time input sequence state step modify horizon sequence share next
triangle value define totally ultrametric induce various ultrametric non order metric ultrametric tree dx dy cf reading addition ultrametric ultrametric everything ultrametric triangle either shape ultrametric symmetry pattern property circle ultrametric ultrametric topology term iii ultrametric clear ultrametric intuitive notion keep mind ultrametric everything ultrametric permutation row column element decrease circumstance diagonal column ultrametric format dendrogram produce ultrametric read dendrogram visualization figure ccccc width normalization agglomerative figure ultrametric visible appropriate row priori ultrametric matrix one way mode oppose observation attribute column yield order near generalization visualization firstly column index attribute acting set generalize facilitate visualization optimize survey presence element consideration datum comprise cluster reciprocal neighbor dissimilarity map positive symmetric dx dx dy triangular dissimilarity endow metric map onto ultrametric practice need endow satisfactory binary term dendrogram embed object set strong subset index height ultrametric often article refer constructive neighbor chain end reciprocal far ultrametric join review relaxed similarity pair distance unless achieve dissimilarity attribute characterize individual index etc distance join denote characterize absence top dissimilarity matching could euclidean prefer component contribution latter lattice show middle f lattice lattice correspond da da db dc c define pairwise linkage linkage lattice base abstract partially cluster dissimilarity initially subsection usual ultrametric ultrametric generalize ultrametric order generalized ultrametric monotonic rigorous conditional sometimes logic e require monotonic certain important operator monotonic consequence applicable syntactic operator argument order include finally ultrametric ultrametric precisely complete due cluster produce reach requirement reciprocal neighbor requirement access parallel share memory architecture look hierarchical cluster difficulty application mining huge allow consideration agglomerative hierarchical find essential operation million boolean cluster I distance value traditional hierarchical also early proceed benefit metric long metric ultrametric ultrametric chemical chemical precision integer precision assume convenience arrange precision cardinality set generality hence digit left build word equal distance bound ultrametric set determine number st digit partition level nd digit reach fine find successive level pair distance identical number level level projection chemical onto design implicit digit seek impose precision separate equality sufficient w j well cf table immediate sum whether chemical million note identical project read random projection dimensional point equal work literature se guarantee well effectively hash md near hash thereby provide map identically value identically want avoid follow refer stable distribution limit sum distribution example tail virtue high dimensionality lie regular aspect perhaps find work normalize datum attribute small weight projection respectively near map work confirm hypothesis behave identical nearly always discussion remark search sort digit deep induce tree set yield partition member embed inclusion partition digits digit take operation read operation evaluation dendrogram widely hierarchical agglomerative induce article diverse dendrogram possibility map dendrogram important ultrametric space multiplication subsection exist terminal traversal dendrogram root specify terminal embed specify dendrogram specify comprise object traversal short assume repeat traversal terminal terminal choose avoid ambiguity among encode rank binary tree rank binary branch encoding code order ii unique express terminal terminal x branch right branch terminal transpose characteristic branch branching index dendrogram order increasingly node read form dendrogram branching code power fix equation might rank oppose partial consider deal must work expression p somewhat perfectly scale dimension feasible representation terminal arrange metric topology encode dendrogram identical identical look equal code example look look find metric topology string ultrametric infimum long infinite provide hierarchical hierarchical simultaneously ultrametric metric real number integer exponent non decomposition rank first non zero I infinity ultrametric similarity series symmetry operator provide tree consider object uniqueness code concern generality multiplication lose lowest lose tree remain operation level product bottom dendrogram mean cluster possibly singleton remain merged therefore relationship apply application singleton terminal end encoding equal nan element preserve implication take open term inductive inductive reasoning signal way dendrogram invariance wavelet invariant rotation alternatively right application seek shift equivalently permutation permutation cyclic node cyclic term full algorithm subtree subtree product give subtree level alternatively term look amount convenient node say object consider component scheme moving term hierarchical view term term new cluster initially motivate couple algorithm discuss take function wavelet support illustrate haar dendrogram discrete wavelet transform spatial respect work something show figure namely smooth forward vector dimensionality determine exactly read root reconstruct observation signal wavelet level smooth haar dendrogram base figure wavelet detail apply wavelet ultrametric wavelet find treat wavelet recent wavelet general ambient ultrametric say embed become immediate direct dimensionality increase quantify inherent structure experimentally latter gaussian cloud hull sized face convex would expect structure dimension appear firstly structure proximity implication high structure show cluster exploit considerable symmetry dominant picture topological ambient increase dimension ambient proportion triangle sample triangle proportion triangle triangle ultrametric financial exchange stream comprise bid price size case symbol bid report figure embedding window successive start step successive window length window point overlap window regard involve distance find concentration cf great embed dimension dimensional successive term distance figure follow criterion number effectively peak histogram appropriate number histogram gaussians bic approximate bayes outcome well mean respective peak relate histogram want correspond original cluster financial peak distance intra peak histogram peak inter peak histogram approximately co financial signal consistent cluster use multidimensional distance axis account chapter discuss mutually nonetheless relate fundamentally phenomenon case polynomial consist multiple explanation shape see pp cluster capable map curve show ultrametric totally order hierarchy datum know could agglomerative hierarchical see complete criterion sure come explain relate basis check embed read membership point segment dimensional embedding differ inversion dendrogram summarize peak histogram dimensionality expect provide coordinate either constrain clustering original complete imply determine signal volatility explanation average cover article permutation associate analyze approach ultrametric time signal symmetry hierarchy large collection thesis system way theory note symmetry many representation look member mutually similar
subdifferential appendix detail object characterize non involve characterization differentiable usefulness expression functional remarkably learn typically structure expression separable differential different often problem separability correspondence exist compute method cluster describe iterate accord regard equation choose second involve separately iterate characterization suitable coordinate optimality cluster loss element wise component iterate also iteration properly choose procedure give accuracy analyze iteration q emphasize separation role step involve computationally demand input relate fast training prediction speed desirable independently matrix vector multiplication need operation compute observe product operation easily order product significant overall feature significant memory computing might vector prediction input differentiable whose implement choose report supervise svm denote classic classical solve suitably accord q cluster problem strong positive differentiable converge fix matrix definite everywhere unique solution fix problem involve last affect equivalent kernel compute way generally normalize rule observe semidefinite spectral basis norm enforce separable rewrite general new observe memory analyze coordinate compute soon I c ii descent sub last year via coordinate become machine enjoy favorable property supervise functional allow efficiency competitive limitation solver scale supervise coordinate descent technique update depend component two section soon coefficient indeed view kernel coefficient let coefficient optimal th also point involve regard solving equation assume initialized correspondence remarkably implement row recursively accord iteration essentially cyclic group update index pick least report satisfy essentially cyclic macro pick macro notice smooth kernel indeed become positive soon positivity separable form differentiable derive let function continuous theorem compute search obtain hinge function modify subtract coordinate convex risk algorithm solve equation characterize advantage coordinate descent exploit information potential minimizer regularization functional stationary differentiable regularization functional regularization risk review algebra proof endow standard value maps associate point see singleton multi exist modulus lipschitz eq identity finite value function subdifferential value hold convex differentiable singleton whose gradient value function envelope min q remarkable convex modulus fp x x inverse generate modulus converge result page close b prove proof functional continuous bound show solution minimization kernel uniquely decompose belong matrix solution range eq exist introduce inverse equation multiply subtract introduce start subdifferential equivalently multiply side subtract thesis solve solution exist satisfy belong prove observe orthogonal far positive semidefinite hold nan least observe eq subsequence proof rewrite theorem contraction theorem eigenvalue semidefinite condition hold denote differentiable lipschitz easy differentiable lipschitz modulus monotone last consider differentiable contraction mapping theorem iteration converge shall apply descent macro map macro alphabet character observe finite union map union obtain close search direction equivalently subdifferential eq inequality change position macro sequence descent non increase converge size uniformly fix subsequence index th pick essentially observe recall algebra rewrite subdifferential decompose eigenvalue triangular term view imply subsequence consist macro iteration well macro regularization property multiply side finally thesis corollary algorithm kernel quadratic analyze coordinate exploit separable compute solution search close already characterization regularize problem sub differential calculus operators paper gauss development software heavily influence
singular instance entry corrupt function certain allow decomposition perturbation application principal analysis result analysis technique completion significantly purely structural price pay allow relative gap interesting follow study consider support largely complementary analysis review technical tool rank desire decomposition analyse incoherence property identifiability optimality formulation guarantee goal decompose convex optimization trace regularize entry consider norm add constraint image interest hence relaxation core rely constraint robustness recover application refinement characterize target support product eq inner b matrix orthonormal certain norm structural entry balance accommodate balance remark optimization involve spread vector align axis vector consequence take decomposition least conversely decomposition argue matrix non sufficiently spread low singular basis study roughly symmetric explicitly claim possible rectangular norm since must entry fraction zero allow condition vanishing guarantee recover decomposition mild condition recovery guarantee property eq approximately reference dimensional come maximum zero column far proceed satisfied satisfy note analysis formulate require chosen satisfy outlier contrast apply strong analysis fix either process clear recovery perturbation accuracy wise amount serve entry trace simplify choose error possible simply although may introduce slack finally second constraint post recovery guarantee regularize ne let section obtain exact recovery perturbation accuracy entry norm choose choose arbitrary uniformly family one previously enough word nearly entry bad guarantee gap generic probabilistic gap find narrow lie independent entry analysis model infinity z standard probability finally lemma simplify condition mn take mn mn mn mn outlier operator operator vector frobenius linear nn operators composition unique range two define th position also induce norm norm nuclear hybrid parametrize n associated lemma lemma lemma eq concern operator operator invertible invertible taylor expansion claim definition subspace follow proposition fix matrix claim rely x sm orthonormal induce use leave orthonormal non characterize subdifferential non smooth norm subdifferential subdifferential consequence x x x duality note duality give throughout singular singular balance quantity quantity coherence singular coordinate basis constant identifiable prove lemma imply operator complementary imply definition composition contraction norm principle simultaneously small conclusion incoherence construct central optimization system existence arbitrary recall recovery take satisfy constitute subgradient bound eq q throughout fix appropriately accurately recover target decomposition decompose let dual lemma condition eq separately side eq inequality either augment constraint first bound j throughout target constraint condition choose target exist g l lagrangian imply subgradient b j j hold prove dual equation eq subtracting inequality fourth inequality rearrange
system simulate roughly refer simulation monte update phase state parameter call negligible neighboring parameter exchange accord show implement markov chain whose enable system local minima may update desirable replica drift terminal space first merely allow suffice together precise show expand log k distribution determine initial feedback procedure merely overlap though neighboring exist high replica replica repeatedly bottleneck pass merely chain ensure replica round place briefly summarize insight replica visit replica terminal visit replica visit either contribute maximize replica flow fraction decrease old measure make run point tends move small slope towards drop algorithm appear define feedback pt optimize step define within interpolation stop meet result bottleneck present encounter simulate ise unfortunately initial unstable one measure drop number would wide chain replica visit break recursion bottleneck fact require accurately appear chain increase dramatically occurrence merely nonetheless make modification problem several necessary feedback set emphasize fundamentally original achieve line choose geometric circumstance practical experience substantial efficacy number chain space short exchange minimum swap example spin consider typically try would probably place however chain swap compute swap attempt equilibrium swap estimate short need add swap need chain need order routine proceed follow minimum initial swap uniform parameter chain meet practically list initial end sort parameter terminal swap unnecessary discuss begin tackle feedback instability problem severe concentrate replica next iteration implement weight w advantageous estimate become smoothing eliminate though routine swap using accept predict accept result low threshold thresholded result interval add continue grow find around minimize feedback swap parameter swap rate post process k old somewhat counter intuitive normalize smoothed rapid quickly instead implicit analogous properly incorrect reliable smoothing f use feedback implementation aid notice considerably ground state consider correct discuss quantum spin quantum partition classical ise couple slice draw equilibrium system quantum slice well approximation demand hamiltonian purely quantum hamiltonian respectively unnecessary ise appearing crucially spin I suffer issue pt distribution st turn serious transition method essential simulation interested tends concentrate precisely interested paper hundred spin correspond system result represent quantum spin slice perform illustration spin quantum ise slice instance extremely simulate quantum achieve swap bottleneck numerous fair routine rate show demonstrate call target swap chain estimator swap produce interval impossible achieve around achieve target rate htbp show fairly swap rate detail read essential recursion past run realistic length space take gap demonstrate efficacy
estimate likelihood htbp rate mh acceptance explain perfect acceptance diabetes record diabete associate variable logistic size consider transformation diabetes criterion tolerance dl pressure mm mm body mass index diabetes age year specific empirical use eq adjustment maximize likelihood weight calculate aic probability aic include include evidence prior qualitatively reduce inclusion except bic f cf six average fit lead fit mcmc result plot look note interval figure fractional polynomial systematic transformation ordinary polynomial take logarithm power choose collect collect fp multiplication logarithm model uniquely implement bayesian inference fp therein comprise automatic depend hyperparameter manual prior depend transformation equally implement jeffreys infeasible composition nan move successive modification accept mh acceptance ensure f million idea ghz implement package include efficient core author approach covariate comparison inclusion covariate interesting important transformation consider much cf examine marginal inclusion look transformation since map produce uncertainty variable inclusion varied transformation covariate panel strong diabetes odd linearly rare diabetes increase diabetes age diabetes middle participant qualitatively htbp marginal differ average probability visit get decrease also visible range analogously control rise prior could desirable also investigate hyper available difficult glm family important area thorough prior literature summarize performance perhaps explain bayesian fact motivate huge prior marginal likelihood yet suit replace slow careful automatic supervised replace property linear work generalise regression along line hyper possibility drop brevity rewrite score q hence laplace laplace efficiently compute cholesky hold modelling inference model choice acknowledgment sl em chapter priors institute develop classical model handle estimation free metropolis methodology automatic diabetes datum integrate fractional response come glm incorporate ti canonical family derivative coefficient identity iw iy assign distribution indicator covariate covariate vector also transformation indicate prior manual infeasible prior regression normal zero eq locally jeffreys n jeffreys conditional locally scale implement idea accounting observation covariate model nice invariance imply predictor rescale translation covariate automatic situation near act sensitive preference complex phenomenon jeffreys perfectly versus go infinity develop automatic specification correspond fully unfortunately closed hyper incomplete retain closed expression efficient inference handle generalize prior connection inference practical automatic fast estimation tuning monte carlo sampler variable selection fractional modelling discuss research design dispersion improper regression recognize consider appendix standard asymptotic expect g theorem especially size see bernoulli family identity logit mode length fisher information response estimate correlation distribution comparison prior original specific maximize likelihood fully avoid informative regression probit logit link factor improper regard degenerate shape improper factor generalize hyper prior conditional give unity improper prior obvious exception refer order explore accurate procedure laplace approximation plug integrate numerical integrated laplace generally ease gaussian iterative weighted preserve prior high taylor canonical convenient section detail use gauss quadrature approximate unnormalized density numerically routine derivative log gauss quadrature function seven deviation
close boundary support compare bias boundary section detect moment boundary define concentration coverage high ball around concentration choice partition size nn ball pool nn neighbor nn count arbitrary equivalently use fy fx clear imply apply threshold boundary show construct nn boundary point nn volume estimate precise consider let case interior rewrite probability consequently c n ok ok h ok bias nn boundary reduce bias boundary probability variance density correct central correct continue rate interior contribution cross moment boundary correct negligible interior result cross include f kx kx estimator derive concentrate expectation since turn imply therefore x ok bias boundary estimate oracle know boundary boundary know correction exist fx immediately handle f pz pz event conclude z z conjunction schwarz choice conjunction z fy show conclude dx logarithmic follow z g om ok om ok mean ok om logarithmic growth condition assumption observe g bc nn estimate identical set exact method ok lemma z conclude random stress variable process appeal define f fx x theorem borel component independent f f mf z z subsequently observe logarithmic kx derivation strictly parameterize covariance also show kx ig step observe sum conclude proof logarithmic since n bc follow absolute third bind appropriately normalize type main uniform plug appendix correct density density functional list bias plug give fy depend suppose condition plug c fy functional list university school department university lemma corollary neighbor bipartite non shannon r enyi assume estimator functional use piece respectively use integral statistical explicit estimator base optimal decrease square mse converge fast central allow confidence multivariate density application signal processing statistic include application match texture develop component analysis signal internet anomaly uniformity normality empirically densitie gap near neighbor estimator approximation plug nn bipartite near bipartite see sec split construct estimate nn functional integral bipartite approximating exploit geometry automatically knowledge support occur near boundary support set boundary correction attain support since convergence practical include estimator result select derive mse attain certain specific fast achieve nn shannon r enyi positive recently propose estimate unknown realization density wang author consistent author enyi variance distance lead propose normal estimator use paper bipartite estimator truncation near estimator mse consistent mse guarantee minimize specific confidence extended plug mutual estimation distinction estimator shannon enyi latter propose require shannon enyi fast mse fix hold regime remainder paper organize estimator asymptotic consequence proof appendix correction shannon enyi shannon briefly discuss numerically validate discuss section random face functional density realization ff estimator n n nm boundary realization subsequently basic bipartite estimator estimate bipartite graph part density bound lie strictly support however observation boundary intersect describe suitably volume nn distance near neighbor amongst realization radius center kx estimator ball center point intersect consequence ball tend high boundary significant nn estimator kx kk kx finally interior appendix motivate method done identify near show point correct density interior emphasize assume support density interior close realization draw nx u c l relate k x bias general functional specify establish mse z ok z generally appendix idea taylor expansions functional lead bounding remainder taylor series ignore term variance henceforth refer concentration event integral ic density functional estimate might principle estimate ok ok gradient addition variance appropriately asymptotic shorthand limit assumption asymptotic random variable key exchangeable random et exchangeable get et get idea rigorously treat nn r enyi experimental establish functional include enyi interval estimate functional imply unbiased likewise order polynomially plug size functional derivative iii equivalent minimize close define constant possibly opposite sign observe opt analysis require evaluate c k significant mse experimental optimal grow sample decrease explain observe entropy correspondingly functional increase near boundary grow bias decay decay asymptotic sum distance realization support estimator estimator estimator consistent suitably normalize contrast high condition continuity functional nn entropy require bias specify choice minimum restrict decay decay exponent oppose furthermore optimal rate overall optimal decay estimator fast mse propose enyi entropy al shannon correction analyze deduce establish shannon r enyi functional h shannon enyi entropy respectively contrast functional kk si correction incorporate addition list bias estimator iv specify iv specify variance mse convergence mse experimental mse apply functional bias correction incorporate next shannon entropies shannon enyi section reduce assumption note mp u mp k iii derivative great mse density give shannon mi estimate mi xy shannon mi entropy entropy nn estimated remain nn density neighbor estimation I obtain use q neighbor plugging define shannon mi iii require marginal manner estimator experimentally shannon sample draw bias agree well observation finite sample regime predict bc k shannon propose draw fix empirical agreement bias monotonically experimentally shannon density theoretically agree theory section cube uniform univariate beta set shannon bias iii determined theoretically minimize curve even though theory useful finite specify bandwidth significantly mse choosing bias iv bias agreement bias iv increase empirically determined variance express vary choice fix theoretically predict agree u vertical axis population axis shannon sample beta linearity central show fig iii limit length coverage predict confidence interval compare length interval plot bias sample size n bc experimentally shannon draw uniform density empirical agreement bias iv tb enyi entropy sample beta entropy validate iv iv use interval predict determined simulation range accurately enyi entropy define propose correction fast agreement estimator discover structural variable multivariate task recognition machine density parsimonious inherent dimensional configuration discover sample realization priori c six factor true false true entropy play surrogate enough sufficient estimating see list estimate utilize practice form adopt constant vs vs high vs exp vs vs vs constant always test towards entropy irrespective dimensional true elaborate false bias test statistic model false bias nearly surrogate statistic compare result correspondingly compare graph model constant positive bias towards introduce ball knn near relate r entropy unlike dimension e knn estimator functional guarantee among class translate performance realization intrinsic disjoint x follow partition near neighbor illustrated nn approximate density sample give volume nn ball implies rewrite linear respect estimating function different estimate relation relate recognize estimate plug report error property optimize value r give partition constant estimate highly correlate difference k k expectation remain bias modification term original distinct prediction serve upper excellent product beta project hyperplane transformation column orthonormal apply compute partition show experimental e bandwidth fig partition accord show consequence theoretically choice minimize e theoretically serve strict improve expression predict theory modify significantly outperform compare estimate al depend dimension unknown plug et optimal bandwidth selection estimation establish optimal suboptimal theoretically choice choice use bandwidth partition applicable suboptimal knn size choice indicate experiment original marginally inferior compare estimator improve performance superior correlate different anomaly anomaly monitor send pf traffic imply strong behaviour reflect al boundary plug estimation functional bound support bipartite boundary outperform nn term convergence expression derive dimension central develop validate theory finite sample estimator establish density plug density estimator rate density boundary thereby plug achieve optimize discovery intrinsic problem shannon furthermore intrinsic dimension reader use list description correction factor density support unit dimension x n x nz ni tn c constant appear theorems decay condition event kx kx throughout section rx support denote seek realization sphere dimension uniform region density estimator define illustrate density ht neighborhood small equivalent coverage taylor cx tr fx chernoff estimate binomial taylor expansion imply random density estimate since variable uniform integer distinct expect positive close far ht disjoint set complement ball intersect disjoint disjoint ball x r ball x therefore concern disjoint let positive mass chernoff l power density power condition bound f cauchy schwarz derive stand ball ball moment nn euclidean denote euclidean near neighbor amongst volume region volume coverage define beta follow identity chernoff binomial variable exp p satisfie similarly xx x aa arbitrary variable event neighborhood lie continuous neighborhood close boundary neighborhood intersect boundary term volume ball expand taylor write taylor around notation volume monotonic divide side get substitute rhs tuple positive cardinality finite cx fx h rx op x k event nn whenever density fix turn also explicitly finally nn let use turn give mp implie next identical let error define event note cx tx rx mx beta k turn q trivially express tx tx tx le bound concentrate form conclude prove seek nn ball define
magnitude require especially problematic since involve approximation original sampling operate column matrix study theoretical computer community matrix om particularly range crucial involve low extract subset explain negative instance extreme dimensional vector although approximated subset unless theoretical property often loose nonetheless deal kernel sampling recently characterize extract show theoretical evidence tie nystr om singular span capture field research compress sense completion use coherence motivated show class randomly generate matrix low coherence uniform arbitrary help basis determine attractive bound hand singular prohibitive cost calculate precisely motivation behind numerous theoretical related address number remainder paper introduce definition brief estimate formally section coherence experimental synthetic analysis provide algorithm wide excellent basis th th entry vector thin svd x orthogonal right singular correspond define u orthogonal onto projection orthogonal x semidefinite start accuracy use frobenius e briefly common form nystr om rectangular first x take contrast sampling nystr om deal without sample nystr om k complexity nystr om extent previously mention analyze sense robust pca om use variety notion column notion coherence follow coherence contain coherence matrix coherence u q coherence nystr om singular completion rectangular two provide moreover use deal noise completion pca coherence low deal x dependent coherence occur u statement corollary relate coherence rank theorem column method x unchanged second event lemma proof lemma orthogonal onto span relate projection assume column span view subspace span span orthogonal orthogonal onto statement l yield statement difference always inequality apply statement former x coherence event coherence dependent coherence matrix select imagine lead high coherence force completely illustrate synthetic generate matlab ht worst extensive empirical perform variety vary coherence suggest address match rarely encounter remainder f presence compare coherence level low spectra rank matrix decay value decay singular vary manually use vector induce singular singular value minimal coherence set use baseline mid vary column coherence deviation trial although coherence estimate coherence matrix recover note influence singular induce examine scenario matrix medium decay experiment create rank qr orthogonal left singular singular repeat small noisy deviation presence clearly estimate coherence fairly accurate function quite varied value need capture show coherence dataset converge slowly trial next nystr om reconstruction illustrate connection coherence exhibit significantly performance remain l c essential protein bag word nm robot coherence type error equal coherence estimate low tied choose algorithm efficiently
regret policy quantify degradation distribute policy provable accomplish note lower derive good result good distribute centralized access consider distribute bad centralized policy learn whereas sense use centralized regret different future user regret growth fix varied sum centralize decrease increase bound monotonically prove regret access increase hence suffice bad channel second consist increase extent simulation reveal actual regret centralized scheme channel since make bad exist far increase anomalous fail account among user compete bad channel increase regret central allocation pre allocation centralize eps bind l eps allocation low c central scheme allocation scheme low central allocation distribute centralized eps simulation scheme early vary availability characterize bernoulli evenly centralized scheme channel bind centralized insight incorporate cognitive open problem user sense relaxed unknown incorporate dynamically leave dynamic traffic secondary node formulation author anonymous valuable comment point liu extensive version manuscript sharing mit union bind regret break term n bind user slot eqn channel use fact mn channel sensing could modification realize slot new user original allocation coincide scheme orthogonality orthogonal slot reciprocal nothing way ball position allocation user configuration slot probability orthogonality increase absence reach orthogonality eq union chernoff imply h q eq bad eq slot good number good channel event run top channel hence hold bad good channel line define event event slot bad channel good correct rank hence decay choose edu channel access cognitive multiple availability channel initially secondary sense explicit exchange agreement secondary user policy cognitive throughput successful secondary learning policy total number distribute learn regret low regret unknown estimate asymptotic regret grow cognitive armed bandit distribute extensive cognitive decade resolve challenge encounter communication main challenge heterogeneous typical cognitive user primary user primary user secondary cognitive spectrum resource hardware give cognitive secondary channel transmission slot sense beneficial user channel high mean channel less channel availability priori secondary since secondary require sense estimate channel access transmission throughput designing goal paper require converge correct availability available sense strong throughput due perfectly desirable fine throughput sub regret respect throughput additionally framework exchange agreement secondary introduce throughput secondary competition channel channel policy hence unknown distribute primary user l secondary cognitive eps contribution two learn policy guarantee one achieve transmission policy requirement incorporate design distribute access secondary self secondary prove logarithmic transmission distribute policy regret also logarithmic secondary achieve secondary regret characterize verify good exploitation competition medium sufficiently discussion engineering insight towards practical indeed markovian good primary traffic towards derive scheme extension arm bandit markovian result markovian channel complex exploitation bandit multi armed bandit medium access extensive cognitive medium armed bandit employ selection establish availability compete consider channel partially feedback information spectrum investigate difference availability probability secondary consider work centralized access access access exchange user channel theoretic cognitive medium learning game weakly equilibria nash equilibrium assume equivalently random observe recently consider combinatorial bandit channel assume secondary channel respect requirement decentralize propose still user contrast remove requirement albeit decentralized access secondary scenario secondary pre allocate rank analyze detail logarithmic regret addition secondary bound scenario liu rate uniformly decentralized work another schedule user channel achieve discriminate user policy extend incorporate sense organization read deal system deal secondary centralized solve classical result multi armed bandit distribute access provably section consider section low simulation scheme conclude since deal multi jump main high vector rest bad channel alternatively ease abuse kullback channel slot width slot slot availability vector mean channel initially secondary learn decision exchange primary user user sense variable slot sensing obtain value indicate user record channel channel none receive whether general policy user feedback result design successful subject interference primary user policy secondary channel high entry access sn regret distinct incorporate expect throughput policy time nk centralized access appeal classical multi round base index entry channel sense free secondary user policy minimum process armed bandit regret henceforth arm arm high index slot summarize transmission mean policy n jt I select channel jt channel tradeoff channel predict availability throughput sense channel exploitation exploration term channel exploration statistic channel exploitation regret optimal mean jt policie statistic logarithmic regret good channel satisfy spend channel statistic secondary centralized central policy avoid centralized sense centralize good spend availability hence achieve algorithm availability high sense channel channel arm learn allocation policy first bound satisfy channel channel availability section access policy provide guarantee uniformly lose transmission due selection bad performance due good armed bandit distribution variable analyze technique learn access observation target multiple user communication user access avoid channel avoid contribute avoid channel rank slot slot converge free adaptive randomization feedback otherwise generate retain channel rank statistic input round high entry loop select channel c ensures allocate good channel transmission go infinity regret policy every implement number spend spend channel result expect ideal user availability attempt reach free stochastic process markov markov number channel chain self uniform channel exactly consist channel transition chain assume appendix knowledge allocate rank lack communication channel occur reach configuration define rank logarithmic appendix incorporate number spend channel perfect expect good u appendix channel spend bad channel logarithmic access logarithmic u access secondary user regret explicit communication imply lose successful logarithmic design scheme distinguish sense equal good channel logarithmic simulation demonstrate phenomenon secondary implementation policy entail truly exchange user unknown duration channel learn availability designing channel feedback policy bad scenario regret linearly bad channel slot execution algorithm update sample high entry slot current transmission loop stop jk ji mn
sequencing framework short inaccurate unique identification present generation sequence large read constraint present enable reconstruction specie approach simple comparative discussion read support order experimentally preprocesse convert measure represent see preprocesse measure sequencing reaction resolution approximately nucleotide sequence reaction higher later sequence reaction amplitude position division bp position normalization size bin sum bin column result input preprocesse describe preprocesse nucleotide frequency amplitude constant sized square transformation sequence mixture division total amplitude sized bin iv obtain preprocesse align predict effect position sequence mixture nucleotide database peak peak nucleotide th peak trace peak model center peak position height nucleotide peak peak peak width sequence evaluate space thus trace basis trace transform square preprocessing sequence sequence display basis highly bin position reconstruction initial bin offset reconstruction distance root predict verify know composition offset circle peak peak peak position red correction peak red correction local employ ht root measure position align predict use specie position database cm cm conjecture assertion equal contribution unseen million comprise enable small number tool composition single sequencing reaction compressive deal many comprise specie simulation sequence base may reconstruction mixture thousand realistic measurement promise reconstruction toy mixture may availability everywhere population environment create complex system body cell order cell typically specie sample specie characterize human high species cavity community composition physical disease broad understand interaction composition technical limitation scale survey conventional profile relatively inaccurate incorrect identification rely availability laboratory identification much attention standard direct sequencing extract gene use sequence identification sequencing therefore require hundred sequence depth mixture sequence result reconstruct human dna microarray identification review microarray platform aim sensitive sequencing still microarray scale microarray array specie detect recent universal account single sequencing obtain read region read sequence composition reconstruct method dna enable throughput community detect length basis limit obtaining long read sequence sequencing reaction mixture sequence constitute sense sequence equation database frequency frequency experimental compressed wide variety fact natural certain compressed information small thought need cost simplify various single camera computational biology design sequence drug cs combination specific mix microarray specie probe cs solving determined great uniquely accord cs condition notably rip uniquely logarithmic instead furthermore exist size thousand intuitively rip similar close orthogonal detail requirement matrix representation cs application pool sequence community note numerous characterize level enable relatively certain compressed use database represent dna coming step reconstruction reconstruction simulate mixture specie thousand biological addition applicability sequence species community reconstruction composition hand gene assume specie gene purpose specie mixture frequency vast sequence gene sequence region well serve universal sequencing gene region ability distinguish dna abundance mixture constraint composition try characterize frequency specie specie vector hundred specie sequence matrix nucleotide sequence length specific position four mixed sequence composition certain frequency th position nucleotide similarly nucleotide sequence therefore eq hence sequence reaction typical sequencing reaction considerably free hundred thousand hence system solution crucial cope reflect formulate cs seek sparse solution small set species cs sense concatenation matrix concatenation notation general np hence optimal remarkable theory replace lead require measurement frequency linearity mixture fortunately cs paradigm cope enable reconstruction account measurement utilize importantly problem vs sparsity lead fit possibly require vanish enable error tolerance experimental achieve rather sparse specie frequency fine accuracy measure frequency threshold reconstruct design desirable order enable unique reconstruction isometry rip uncertainty principle briefly rip perfectly property invertible sparsity furthermore achievable general system thousand suggest reconstruction mixing represent database specie database sense far advance version check reverse align match difference unique sequence input enable pc sequence unique sequence database mixture universal r dna gene mix dna obtain sequence reaction describe position specie sequence peak correspond nucleotide identify peak complicated sequence previously perform sequence multiple dna peak sequence nearly impossible locate preprocessing identify nucleotide sized bin see apply sequence position score base database predict matrix step measure performance propose specie select frequency random normalize result frequency later input figure mixed figure reconstruction specie well original large positive measure rmse precision rmse root rmse I measure account frequency rmse score present vector reconstruct sensitivity define fraction present reconstruct predict continuous recall rely minimal present reconstruct mixture precision mixture specie uniform nucleotide mixture mixture use circle mixture successful reconstruction need incoherent accordance rip determine though thus aid bring example even base reduce previously feasible information mutual coherence coherence column coherence pair specie correlation center exist correlated pair show high mutual coherence place sequence highly distinguish difficult sequence relate specie distinguish acceptable still small measurement translate read sequence sequence dot product expect exhibit coherence around even typical algorithm length line plot rmse specie enable run subset sequence number sequence large lead rmse cumulative position typical basis basis sequence species reconstruction reconstruct distribute mixture specie database black randomly generate sequence bar derive sequence vary species mixture blue specie show incorrect identify black reconstruct frequency present effect sequence coherence mix mixture sequence compose value coherence sequence rmse sequencing species present species reconstruction rmse positive specie frequency sequence successful reconstruction mixture percent frequency specie distribute specie test reconstruction mixture figure rmse law b specie large tail frequency sort specie frequency reconstruction mixture specie simulate distribute minimal reconstruct present measurement practice turn effect minimization determine solution nucleotide figure slowly real black green approximate sequencing performance specie noise reconstruct rmse present predict show sensitivity precision attain realistic degradation performance one reconstruction realistic level rmse specie frequency uniform sequence length green line represent vs curve sequence minimal inclusion experimentally mixture introduce mainly sequence exhibit variability stem activity standard sequencing qualitative require maxima may combination variability peak overcome utilize peak local accurately predict sequence database sized bins single sequence peak peak predict effect height sequence perform position peak local correction deviation height prediction feasibility reconstruct sequence dna universal result gene proportion preprocesse figure peak measure dependency bin root error stem peak reduce accuracy reconstruction successfully three remain identify basis sequence sequence manually add addition identify presence another identify space highly sequence differ basis test sequence mixture solid line five proportion correct nucleotide position range reconstruction result runtime frequent specie frequency quantify specie sequence reaction study amount database ability information
rule edge everything sense helpful distance distinct graph different define thought graph remark geometric detail chapter develop toward algorithmic application geometry everything need careful four nest check familiar edge dx city q city inside diameter segment city graph figure illustrate http com edge construction become intersection open boundary iii open invariant axis ax ax transformation translation rotation scaling take template associate relative neighborhood origin instance proximity subset extra origin radius edge e vertex corner proximity monotone family template one analogous geometric graph neighborhood graph intersection two radius pass open center think one fix configuration vertex say cost one subgraph graph probability spirit study use design vertex exactly north east se conceptual previous edge create imagine infinitely long thin able log position space instant define position process page successive jump remain place axis axis part follow west replace single west straight procedure successive collection path city random edge finally realization poisson process need external randomization initial deduce external randomization near boundary square directions ne se immediate adjacent trajectory figure intuition confirm say road network network familiar impose deterministic efficient general phenomenon infinite sense consist pattern area network study random loose remainder city unit exact limit convention per normalization natural regard write q straight number possibility context however statistic descriptor world inefficient subtle drawback length collection line connect exclude discuss characteristic shape slowly characteristic hold calculate quite insensitive discretization characteristic shape common imply alternate prevent grow typical city straight network difficulty city road thus minor insight city normalize enforce constraint visible world quite arise structure city one lattice nearby pair paper exact analytic formula calculation resort carlo obtain explain value minimum span version interval suitable configuration deterministic eq carefully I prove background strong attain network infinite low boundary believe skeleton attempt well ad construction close skeleton look something value tractable say know exist clearly rigorously unable rigorously continuous toy road intuitively place road city nearby city city distant proportional context consider optimal network benefit functional rigorous viewpoint assertion nontrivial sufficient neighborhood prove fact quantify normalized return around correspond square decrease increase decrease increase economic prediction normalization close substantial literature relative graph deviation give extend rather focus different instance focus critical certain maximum little seminal various statistic closely detail proximity seem object attractive alternative connect remarkable result find road network line mathematical question spread contact process random interesting usual lattice critical q contact asymptotic make loose particular particular deterministic equality view realistic real world road mathematically progress discrete imagine finite consistent mathematically natural structural process ii translation rotation invariance assume constant analog kind arise scale invariance acknowledgment nsf grant dms thank anonymous helpful comment remark lemma sep motivate one proximity come possible question attract attention year example www edge geometry concrete visualize city road dimension would graph purpose reader review connectivity attractive review geometry potentially interesting purpose imagine specifically point road primarily interested short length turn subtle innovation section theory quantify network efficiency amenable calculation tractable carlo simulation particular
expect provide leibler kullback q easily turn majority vote binary decision term mixture discrete continuous consider factor relate threshold consider attribute priori choose last pose learner identify select stochastically learner classifier choose interval offer another kl divergence posterior limit kl continuous kl small whenever kl divergence suggest small guarantee risk minimize group direction refine reflect hence close sg ad expression except replace risk gibbs majority decision consequently difficulty main previous framework obtain various optimization detail ideally conjunction bound use cover classify example misclassifie penalty misclassifie positive remove repeat reach heuristic compression cover machine sc however approach utility aspect strategy subset let cover choose maximize u p use maximize decision decision reach greedy learning determine cross attribute small keeping utilize greedy cover gibbs low gibbs measure piece indicator use utility instead need utility partly fall partly versa follow far first jx assign jx new vector attribute ix I jx cover decrease decrease decision suggest remain example decision cover cover cover contribution decision divide example cover amount example soft add gibbs add reach totally cover greedy totally cover utility number increase analyze attribute number cover take attribute attribute microarray remove cover training kind guarantee cover example algorithm run news prefer threshold oppose fix alternate threshold choose attribute however previous c test real microarray expression tumor gene identify high intensity across level gene patient set microarray sample gene set contain failure gene expression value breast tumor various level example ex sc pac framework attribute first name obtain point contain svm elimination present svm state mining validation cv five set gene training perform nest fold selection criterion fold nested deviation side confidence fold classifier interval adaboost cv gs adaboost frequently respective choosing boost inconsistent boost round attribute experiment frequently gene far exceed dataset stop follow boost run algorithm use fix table pac respective microarray bind provide uniform fold refer classifier testing fold respective illustrate current c c note quite relevance observe limit factor microarray large give tight currently dataset percent current percent hence limitation come availability bound tight sc find classifier able acceptable classification two try use separate hence classifier compression minimum encouraging domain alternate explanation suboptimal extremely offset pac perform combination approach pac competitive add importantly bayes bayes reflect utility try report observation adaboost notably marker cancer factor commonly give insight worth investigate experimentally identify pac include prominent marker disease prominent marker gene identify disease discover gene cancer breast b gene biological regard finding study follow protein complement nuclear protein classifier breast er interact breast cell also discover element second gene gene identify microarray md express er cancer confirm rank eight gain ig sm statistic tt sum one cancer identify rank four criterion ig sm similarly rank sm ig tr tt provide strong cancer dataset marker division perturbation one finally discover gene regard relevance system dna aim attribute characterize three formulation different principle sparsity generalization small size trading addition seem extent allow yield competitive classification utilize significantly traditional feature selection approach approach need basically furthermore generalization practical potentially guide dna gene propose find validate various justification approach throughput provide approach yield marker utilize gene microarray validate rt costly impractical full gene finally mention wide relevance significant implication design justify approach combine generalization guarantee result classifier property assume significance limited microarray limit approach domain array within acknowledgment science engineering research research grant research medical objective performance approach successfully goal analysis size limit give bound far expression learn conjunction sample bayes identify reliable identification dna much give tight guarantee unlike propose approach design application microarray feature accurate depend attribute far associate huge obtain guarantee acquire biological microarray investigation genetic parallel front result two type normal base gene dna focus give insight microarray quite variety reason easier oppose gene microarray deduce interaction number facilitate make technology gene indicator disease subsequent lead well disease attempt yield instance involve subsequently combination gene another centroid e appear traditional filter attribute perform conjunction acceptable empirical approach theoretical justification work come formulation algorithm combine feature selection consequently selection tight combine algorithm classify microarray correspond measurement part motivated variety strategy extend immediate classify framework lead class guarantee exist error absence sample three optimal code string compression attempt classifier separate subsequently empirical microarray task string predefine bit one respective minimum attribute definition attribute take way need fall bit identify value risk depend receiver priori possible give priori string length preference threshold string decrease bind rapidly dominant contribution definition finally attribute
close hyper k rp exponential less used investigation reveal occurrence substantially small reduction limit singular recommend approach acknowledgment thank anonymous useful suggestion support natural engineering research r krige j genetic convergence w circuit buffer factorial hypercube f compare capable frank rigorous surrogate convergent finding solution simultaneous equation grid van md bayesian process numerical investigation phenomena krige fidelity evaluation distance optimization expensive function blind assessment evaluation highly computer simulation simulator comparative traffic architecture bayesian ill system regularization computer code deterministic approach contour estimation complex mit j w constrain computer stein simulation use hypercube stein l ny engineering pattern search solution incorrectly formulate van ed electrical theorem example mathematics ns department expensive deterministic computer desire statistical spatial commonly simulator determinant computationally due close overcome introduce along inclusion cause unnecessary smoothing interpolation gp inverse use model physical engineering process expensive consume simulator say replicate decade deterministic widely computer al design deterministic deterministic circuit simulator analyze reference prefer demonstrate preference stochastic counterpart paper simulator deterministic confident deterministic simulator realization gp simulator desire deterministic computer simulator numerical solver differential accept simulator representation paper gp simulator gp maximum technique approach inverse several positive definite also ill study conditioning quality fit experimental design ill conditioning krige use singular near use sum gps overcome near stage krige model lee predictor second develop new approach reach tolerance effect iteration require desirable popular overcome unnecessary section iterative several illustrate remark recommendation practitioner portion usa high portion difference water low period energy extract hereafter notion propose electrical energy consider infeasible variety economic recently rapid wind place diameter ideally find potential extract greatly portion power optimally place numerical examine power simulate water grid see triangle differ model set triangular center triangular david institute chen power location simulator average location give objective turn gp fit simulator detail undesirable interested simulator maximizer function put location prototype roughly smoothed power helpful lead computer simulator denote response respectively simulation hold simulator gp distribution al like square sense popularity radial krige discussion stein exponential discuss vary slightly structure gaussian gaussian develop structure replace et al detail fit numerous often use determinant ill condition large norm correlation unstable ill precise computation likelihood ill often pair close close k neighboring design size setup design et al pre contour quantile popular ill conditioning condition condition white vary z z gp model produce gp viewpoint interpolation desire tolerance achievable section along major fails overcome ill conditioning fix close introduce unnecessary unnecessary smoothing ill singular condition threshold objective threshold behave van diagonal shift small function point close expression hence arbitrary unknown like often pre process design prefer goal follow near expression infeasible obtain eigenvalue course compute numerically matlab build compute low threshold getting behave near sufficient follow stein choose consequently simulate compute proportion near figure matlab find simulator suggest ill condition system recursively iteration regularization final iteration one cholesky decomposition follow forward propose interpolation accuracy lemma iterative generalization von series version taylor series th predictor popular first von log correlation behave define converge also result near singular behave even near iterative practice propose choice estimate change portion surrogate increase secondly different value numerical change unstable recommend optimize profile regularization outline depend key implementation computation lower design replace optimize profile compute use compute iteration depend interpolation one build stop specify use predictor measure predictor von lemma tend converge appropriate achieve matrix original whereas application illustrate even good popular example simulator eq illustration hypercube ill condition turn condition contour simulator outline section implementation close fitted significantly reality gp fit mle fit iterative show summarize term von accuracy summarize fit hypercube design matlab build design interpolation surrogate fit behave small fitting interpolation forced turn computation propose lead improvement interpolation summarize result build matlab design iterative improvement approach denote denote applied simulator give variable scale fit use propose section generate evaluate median tail fit gp set hypercube design matlab specify estimating approach c popular value process row table indicate behave size correlation matrix behave interpolation increase design propose require fix ill much small get singular become less likely dimensionality accuracy point hypercube example h popular denote realistic simulator flow surface head head variable compare median obtain gp model surrogate fit hypercube simulator candidate keep c expect popular approach optimization last near need improve table summarize value choose accurate approach certainly popular apply c simulator coordinate present hour processor computational cluster cluster sided increase
rs selection match snp rs report though criterion turn strongly snps report namely interestingly gene snps lie many snps share clear must gene genetic variability testing correlate snps represent snps perhaps remarkable detect snp approach use modification bic statistical argument preferable comprehensive confirm individual project important trust marker influence correlation causal snps positive power might widely discuss phenomenon discussion miss snps effect indicate really still study include snps rest mainly sample complex would study individual difference testing deal huge potential rather strategy simulation manuscript strategy present mapping might useful discuss certainly remark theorem conjecture proof association publish marker elementary statistical consideration clearly trait bayesian criterion deal comprehensive simulation snp substantial proper tend false link causal complex aggregated snp snps believe explain power advantage datum publicly project year wide study genetic review involve g major detect trait quantitative categorical genetic marker snps snp million snps within marker lead paper report single marker test snps review recommend significance family wise error significance understand likely correlation include permutation like control recently interest trait age snp individually account fairly design marker marker generalize model due notice classical criterion criterion even bic marker bic suited situation marker marker signal informative prefer relate multiple version correction series base recently property section ease presentation consideration demonstrate single marker stress marker test incorporate causal section particular selection criterion dimensional multiple deal closely range great apply huge search particularly huge marker strongly associate trait perform strategy simulation study rather surprising insight procedure snps detect causal marker procedure publicly quantitative project detect snps lie region snps snps report able several case motivate discussion context regression principal conclusion might study quantitative trait snps l k index snps person snp q sake covariate complexity intercept deal snp additive model index order marker large marker model elementary tell square x usual residual square ie f nan none causal snps statistic calculation rather straight figure model model snps sum residual chi test ratio much considerably incorporate effect power test effect fix orthogonality power situation snps marker consider kk testing expect causal orthogonal complex trait influence concern marker variance marker orthogonality thing slightly correlation might joint causal snps problem snps false loss marker complex justify favorable marker assume model denote statistical like aic function form aic bic linear coincide minimize parsimonious aic go infinity consistent explain assign model dimension much likely choose bic formulate criterion model interpret causal snps snps chance incorporate distribution minus logarithm causal snps orthogonal closely correction rule control family recently new extended criterion prior dimension coincide bic model prove assumption maximal dimension fix trait sparse undesirable encourage pick large article consider equivalent regressor context work thorough relate similar suggest criterion extra select asymptotic snps question selection interesting search multiple advanced strategy simulation whose step modification fact snps snps initial single marker test snps take snps consist search bic marker p proceed snps marker decide bic enhance snp consider snps snps practical selection lot causal principal bic square section subset snps population manuscript p gap file txt subsample study comprise homogeneous population snp snps snp neighborhood snps snp predict two trend large yield huge association use relatively explain procedure order snps respect entirely order snp substantially detect influence snps square detect causal snps give numerous positive snps quantitative trait simulation threshold first snp runs causal explain snp really positive detect snp positive c snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp snp first observe snps simulation clearly distinguish snps believe instead report snp snps report value fairly small decide consider suitable exception detect classified relatively snps detect twice time might classify causal snp expect nature frequency table frequent snps several practically snps detect positive correlate prominent snp time false correlate snp explanation effect centrality rewrite square q proportional individual centrality become number signal pairwise snps chance fp first show snps centrality regular behavior causal power snp crucial question report detect multiple snps p might effect false snps trait fluctuation sample correlation different false positive figure give snps occur positive correlate snp turn positive snps relatively sigmoid plot detect causal snps vanish association influence missing gene chance detect false snps functional testing procedure trait et analyze expression population european background china major objective find association snps snps mb probe permutation obtain additive exclude individual consider population datum four population base permutation originally col number tag association col snps match account structure col value col rr rr snps cat e hmm hmm hmm association snps pooling gene detailed supplementary result comparable additive believe would interesting population snps link list snps decide snps snps snps project accordance snps snps snps bring information marker marker replace regressor modification snps snps pairwise choose accordance snps selection search procedure local step strategy perform selection combine snps detect snps perform backward naturally snps strongly correlate comparable apply snp accordance
constant low fix integer conditionally q case indeed realization pair belong I pair act proof modification assume q take conditionally see guarantee kullback leibl expression analogous property completion close establish frobenius upper uniformly let inequality addition note get thus prove denote rr x cf p assumption satisfy least large entry constant side extend replication determine reformulate gaussian eq give denote restrict canonical basis q ab oracle necessarily dictionary value inequality replace modification improve inequality brevity follow immediate bernstein matrix define least norm let dimension constant easy hermitian self cf give statistical surely eq apply distribution follow recall n x xx separately lemma c deduce proposition absolute eq condition exponential union variable corollary scale dms part linear corrupt general sharp isometry apply estimator admit simple form satisfie fast logarithmic factor low coincide constant recovery rank also statistical find approximate restrict estimator sharp inequality pair observation though obtain motivation matrix convenient model independent mean scalar product bilinear denote uniformly canonical form orthonormal matrix particular interest clearly isometry however isometry usual cf since space operator general give orthonormal subject type reference design replication nonzero zero column identity r generally learn interested consider identically model reformulate longitudinal design matrix I replication subgaussian root compress sense either I rademacher noisy setting study analyze treat dimensional dirac design become usual accordingly become us deduce consequence general inequality improve sharp successfully emphasis paper example suboptimal matrix active rapidly penalize nuclear von penalization also worth point application exploit procedure involve regularization norm mainly coincide usual emphasis paper set thresholding factor simple matrix absolute intuitive bind error suboptimal obtain suboptimal slow hermitian nuclear motivated density optimality derive prediction error frobenius finally discuss frobenius class prior main section derive sharp matrix rank low bound rate briefly implication devote appear fact rectangular decomposition orthonormal orthonormal quasi trace duality property subdifferential exist q discuss introduction satisfied equality role q follow brevity matrix belong normal cf easy represent belong normal cone arbitrary representation arbitrary monotonicity duality identity fact deduce mp p mp p mp q l l l frobenius matrix satisfy condition recall support denote cone linear equip frobenius intermediate linear cone dominant low selector vector eigenvalue restrict design design let yield l also characterize minimize penalize risk minimization coincide random become lasso I sum bound bernstein union implication imply matrix completion explicitly singular singular v form thresholde subdifferential minimizer characterization matrix consider representation understand property always preferable computational standard become view oracle inequality probability almost surely inequality next theorem denote absolute constant possibly uniformly distribute pair satisfying almost follow lemma theorem improve concentration choice maxima indeed maxima respectively remark form q equal useful minimax depend view suffice normalize frobenius error large
physics stanford university keyword lie code transformation scene lie fit eigen basis reduce allow inference encourage discovery sparse minimal distance affine operator video sequence show video description frame difference standard decade research learn image sound representation efficiency wavelet representation observe pattern video video code content change approach largely translation encode motion result occur quite motion change onto plane translation large prediction bit seem change transformation dynamic lie transformation may smoothly large visual transformation affine transformation intensity spatially localize version transformation capture affine temporal despite simplicity lie part evaluate gradient previous full video change frame full transformation coefficient technique perform operator lie describe inference local extensively transformation translation show overcomplete arbitrary initialize inference coefficient initial pyramid solution resolution seed piecewise coarse minima searching proceeding part constitute indirect smooth robust estimating transformation directly operator demonstrate infer learn reduce tractable smoothing transformation inference test contain operator movie implementation transformation frame generator occur seek ensemble video infer minimize rule naive operation computationally reduce approximation eigen perform term matrix consist complex hold eigenvalue matrix facilitate periodic transformation rotation orthonormal benefit eq therefore replace multiplication non many white transformation generator translation perform problematic overcome map transformation replace minimize coefficient along transformation direction u effectively along transformation direction u along translation operator rotation single inference match coarse way dot show value local minima translate transformation change visual transformation index transformation order maintain due constitute lie group transformation generators structure affine transformation capture though encourage learn operator learn patch direct move sum distance act code encourage occur long transformation pool patch video transform range pixel degree scale horizontal inference transformation fraction recover recover reconstruct db transform evident fraction recover coefficient image code simultaneously recovery patch reconstruct majority patch reconstruct ability transformation translation rotation skew patch transform used affine operator motion horizontal rotation compute learn operator exhibit property correspond pixel location array influence instantaneous intensity pixel basis show image patch block pixel location therefore location contribute instantaneous change intensity pixel note block spatial motion translation patch b c intensity interpret pixel image consecutive frames video move patch apply central patch wide buffer transformation pixel wide buffer act pixel horizontal vertical translation expect focus transformation contain full field translation translation provide useful basis exist motion base translation variety transformation learn intensity contrast spatially transformation learn capture video patch reconstruction transformation frame pixel pixel pixel bilinear central compare pixel translation smoothing horizontal allow vertical horizontal translation smoothing translation transformation operator unsupervise fashion learn fashion hard code translation figure increase become operator frame also translation substantial pixel suggest useful employ standard describe image movie build previous operator transformation make key eigen operator allow tractable operator reduce minima infer video translation motion attempt image translation
proof convergence euclidean nonsmooth apply explicit regularization simultaneously regularization statistical comparative empirical error vector similarly define dd excess comparative theorem exist lemma require moment nonempty x therefore desire assumption q conclusion write right hand bound unlike set expect mean due thus still term depend decomposition control let sample denote entry replace verify role desire denote easily rademacher variable complexity es desire q constant q direct application rough follow bind I directly rr apply sharp desire assumption conjunction conclusion lemma p convergence analysis analysis manifold behind gradient controlling term regularization excess exist directly decompose excess define structure true set lemma proposition rr sharp estimation omit argument prove derive problem overall splitting algorithm positive n rkh completion span lie dimensional span convert finite functional span function function k transfer infinite optimization coefficient c j pi nc nj nf nc column non optimization descent newton method etc develop backward convert change semidefinite k identifying follow set sparse gradient simply similarly note forward commonly process split backward splitting estimate proximity operator q frobenius need term relatively therefore q problem subdifferential calculus equation q subdifferential subdifferential element set see optimal iteratively convergence minimize operator correspondingly derivative small threshold variable keep unchanged norm norm furthermore convergence theory split converge regularization sparsity sparsity large solution correspondingly select practice gradient solution none obviously minimizer large iteration minimizer eq desire initial initial focus vector derivative wise thresholding entry wise involve weight could small introduce transformation main note rank high use singular decomposition unitary notation involve number calculate nr use greatly involve inefficient sample computation kn n kk decomposition v nr k ki pd perform I singular decomposition desire sparse briefly introduce py output value also binary define fitting ny I odd posterior mild regression taylor pf jj j consider expansion extra term formulate minimizer nj set minimization reformulate p split backward splitting become unitary dimension omit artificial expression method nonlinear setting equally detailed mention simulation study minute gradient lasso point lasso assume view linearity equip simulate five uniform form contribute ten uncorrelated lasso symmetric respect select norm limitation bandwidth median pairwise point regression select regularization vary able select five summarize repeat choose return five variable select lasso select fail treat contrast much great frequency advantage lasso c lasso h k partial derivative regularization respect dataset sample lie dimension half point spherical dimension noisy draw sphere space implement distance htp level vary correctly capture emphasize generate two report return dimension reduction call capture original dataset compare derive preserve explain plot norm derivative method sparsity small empirical ef test surprisingly fail omit variable dimension expression gene expression thousand sample become important biological widely accord tumor type difficult datum build datum code response represent variable unit normalize variance apply extract reduction compare loo datum implement choose distance point leave lasso leave one loo table implement although method perform build addition lasso lasso illustrate pathway distinguish type pathway selection many selection simultaneous automatic reduction optimization algorithm generalization regularize regularize integrate advantage previous refine one point lie rather implement calculate occur semi supervise several direction often efficiency use implementation term j mainly motivate paper introduce I regression know laplacian approximate support marginal intuitively smoothness penalty supervise view generalization remark proposition approach traditionally treat integrated call selection gradient impose gradient select derive covariance error gradient one develop forward backward practically scalable medium extraction method loading efficient biological science biology common million snps modification million site increasingly deal response many variable selection fall metric combination strong selection aim drawback evaluate group problem function term accounting prediction term control elastic net widely implementation perform try overcome base smoothing spline make impossible high dimension commonly approach deal belief real concentrated manifold accuracy visualize therefore likely suboptimal finding onto capture go sir explore subspace sir impose distribution regression easy limitation sir yield classification include variance save contour limitation direction quite direction include method function methodology derive term worker estimate gradient use reduction offer tool dimensional available variable notable exception principle analysis produce principle sparse loading mainly use linear dimension reduction variable nonlinear motivate combined microarray expression normal tumor microarray likely responsible expression cell dimension feature subset gene improve remove noisy focus extract biological extend worker selection supervise optimization gradient prediction irrelevant partial expect variation p hilbert associate optimization use norm sparsity component make norm widely impose possible value gradient fidelity term term propose optimization refer q key ridge appear significant many component potentially derive comprise primary depend impose help eliminate improve infer relevant et case learn view space directly since impose regularization remove limitation gradient special lasso invariant regression learn make linearity thus extension nonlinear framework
randomization sequence increase square finite write general form unbiased standard unbiased provide integrable n e eq alone positive lagrangian determine constraint finite entirely believe least tail sequence deterministic convergent complex shift geometric minimize wish constraint occur variance stop mse approximately plot interpret bad scenario indicate rate little relative increase mse eps optimisation computation generate integral since minimization budget minimum occur otherwise somewhat fast achieve large budget produce choice unbiased toy wish require initial iterate eq course step estimator use estimate start close adaptive choice permit large sequence quadrature rule integral particular variance therefore variance estimator compare easy l interval efficiency monte integral advantage replacement exist create obvious neutral price asset volatility brownian drive asset run volatility black price option volatility rate option yield price option neutral conditionally usual bss even exact latter unbiased raise choose functional page euler order sufficiently continuously drift case eq suggests shift reasonable shift price standard error evidence simulation agreement option price condition positivity fail volatility may question value minute run matlab intel quadrature equation bias bias simple problem finance provide extension allow theorem axiom conclusion exercise theorem process sequence deterministic numerical integration obtain carlo involve approximation unbiased value include integral root find pricing volatility finance
like anonymous suggestion nsf fellowship nsf theorem goodness fit division bin goodness division bins powerful availability computer efficient box variant problematic many circumstance feasible confidence level chi goodness fit test norm identically distribution model consider value accordance terminology class common bin use root construct empirical page draw fact root statistic confident draw square draw root square draw come confidence specify significance namely unfortunately mean statistic distribution avoid average root probability associate various availability computer direct box calculate monte carlo statistic would standard statistic easy circumstance extensively root mean asymptotically statistic example present article detail level goodness computation discuss involve part confidence sum independent briefly illustrate power square direction research detail section come specified confidence draw mean draw unknown specify root statistic draw begin notation statistic analyze associate bin obtain bin perform number obviously expect statistic statistic focus goodness fit replace contrast use multivariate central joint limit dirac concentrated origin restriction origin multivariate define sum variance axis variance square particular draw restrict along principal project onto complement consist clearly adjoint construction eigenvalue axis diagonal computing require usually efficient unless hard accommodate homogeneous accounting analogous impose extra describe cdf center tool theorem cdf evaluation gaussian variable number addition random variable cumulative root take characteristic principal branch function define cdf number principal almost cdf identically calculate yield cdf value branch cut though contour obtain eq contour thus decay fast cdf gaussian obtain goodness level draw arise cdf draw figure versus six formulae summarize attain display tail describe cell probability draw specify quadrature require evaluation figure total second display figure bin constant positive real great involve run example ghz intel mb l cache less digit take precision yield digit exploit precision digit suffice computer high fast example section fig vertical axis horizontal axis e k statistic classic statistic comparison complete much comprehensive treatment constitute root observe bin bin whereas draw take actual root statistic square associate classic pearson statistic label ratio statistic convention plot hellinger line limit number draw example however differ level simulation rely say draw draw model mean least simulation simulation simulation draw I generate successfully arise significance level simulation confidence great significance define show root statistic fail plot draw specifie bin draw increasingly increase example success example statistical plot remark distinguish bin root opposite fig draw plot define remark specify example specify q via draw consider estimate likelihood specify
correspond krige analytical formula give local indicator distribution new sensitivity maximize condition case moreover particularly deal computer select experimental informative reach test also sample independently well hypercube variable xx mean mean obtain scheme reach variance form poorly enhance fill dimensional one powerful distance etc criterion lead design conceptual popularity application uniformity initial volume interval interval exist kind form functional measure norm easy compute measure remarkable center discrepancy good term amongst exchange genetic find simulated anneal temperature slight result criterion point ht space design robustness dimension indeed ensure proportion contrary create dimensional project onto design reveal non step sample solely include retain input compare initial reference consider lead criterion behave two keep criterion value different size three conclusion computer ht numerical determination percentage xx involve dimensional provide result time initial output toy evaluate mean efficiency robustness design discrepancy optimize optimize increase show conclusion design property much variability important lead width width htb dimensional analytical several comparison design concern repeat toy evaluate coefficient design discrepancy optimize summary furthermore conclusion distinction quantitative sensitivity study gp even dimension guarantee point projection type result systematically less design fit course design one step instance simulation fit estimating issue safe sensitivity precise capability input fit computer compare prediction residual analyze measure coefficient call validation require calculation computer face point cpu sufficient tool validation answer test approach monte localize example near point leave large domain fine strategy could avoid proximity proximity optimistic bias validate cross propose divide sample learning sample residual obtain sample residual use global leave validation become geometric cause new adequate might quality many prediction optimistic therefore solve main drawback minimize test discrepancy step away point perform process choose prediction capture additional idea computational efficiency method mean error bias vs xx sequence pattern low discrepancy discrepancy center discrepancy additional sequentially design point discrepancy difference initial new low discrepancy require especially design advantage size add point compare design validation analytical toy represent ht gp sample range wide variety initial increase sequentially point keep optimality maximize distance design choose design contradiction objective validation reference coefficient leave point leave process curve paragraph minimal repetition greatly satisfactory case maximal value interval obtain increase contract fit coefficient design add close compare give confidence rather large choose carlo evolution basis range dot blue sequentially carlo illustrate poor monte contrary sequential validation size precise monte blue dotted blue line nuclear water break peak scenario part model operation light water propose economic operation development implement illustrate give q exercise peak temperature scalar cpu minute iv pc lie consider law essentially material either normal variable fit dimensionality make thank devote include center measure range design give estimation begin result case estimation clearly validate confidence extremely large variation dimensionality sample lead optimize red relevant expensive I resolve uncertainty sensitivity test concentrate year widely use objective discrepancy measure adaptive design gp design adapt availability predictor secondly leave use design put required test real application analytical minimal necessary validation effective useful sequential acknowledgment support p france fr computer instance often expensive directly use sensitivity robustness
prevent incorrectly outlier correctly outlier carry variant real situation scheme primary period failure gradually teacher curve student topic machine understand algorithm make flexible function favor outcome although ignore simply maximum outcome prediction discuss boost source computation naive classification major limitation fail feature take take represent similarly replace independent gate pair form overall product interestingly come criterion replace bin arbitrarily sufficient represent number choice bin increase feature become code bin construct memory bin fall bin link datum bin bin always train read express bin intersection count bin unity distribution function theorem initially uniform fail increment incorrectly bin incorrectly show iteration increment fraction increment factor learn paper name network central rule compute equally outcome problem range hereafter research high uncertainty observe default optimal entire appear straightforward outlier issue represent illustrate distribution prediction call clean example green distribution sample classify without high confidence estimate prior cause cause figure region red red incorrectly observation alone red dominate red example overcome estimation difference become classification class may use training remove difference classification might straightforward effect become nontrivial optimisation argue school method go round describe section balanced subtle difference feature classification pattern equally likely prediction generally flat pick one training close take training test comparable fraction sample criterion fail wrong fail test data failure compete incorrectly round failure round cause fail sample add round bound fluctuation likelihood converge unseen test sample remove difficult boundary outlier test add give datum start train classifier training file test add train classifier classifier save pick add datum fail look process collect test incorrect label incorrectly cause subsequent fail similar good example propose teacher student correct come example important note optimally select rest datum unseen training estimate difference identify would good produce boundary fuzzy classify example deep project unlike laboratory infer object basis namely pseudo curve learn curve test example pseudo level gradually fail boundary example label practical detail increase rapid prediction green accuracy sample time boundary class account spike curve prediction accuracy fall mostly cause incorrectly label object fail maximum confidence turn incorrectly pick boundary get however long network accuracy represent pseudo curve continue remain low incorrect get rather assign dataset training continue note spike degradation examine time mostly update continue eventually optimal training less realistic accuracy entire uci repository consist neighborhood pixel predict code respectively grey grey type grey dataset available uci training test uniformity evaluation well report name training original select optimally datum sample optimal selection maximum optimal selection ie dataset provide uci repository optimally datum result accuracy comparable unseen training obtain slight test mix training uci datum maximum entire uci repository result show attain uci repository good accuracy good heart cancer classification accuracy interesting fail subject tool get level give confidence level use take like reality center galaxy although band many believe massive emission integration large device survey one extensive survey discover high simple filter call colour five colour colour input correctly predict spectra appear star object also spectral universe introduce drift efficiency region shift identical add sample region overall classification reduce assign confidence belong overlap patch identification observer overcome improve observational figure histogram release methodology know concentration blue correctly red incorrectly star since galaxy remove remove fail example fail confidence cut important point note object resolve colour outlier totally object identify object object exist colour another predict verification light surface address light evaluation trivial verify ray many precise measurement either observer thousand star different magnitude across star broadly class name basis nontrivial measurement star galaxy assume ignore case spectra impractical method straightforward state predict type spectra belong class sample correctly network test table table accept illustrated spectra good entire
graph topology propose average minimize local efficient sharp network walk update include stochastic communication protocol protocol communication provide update dual body agent maintain gradient update q jt proposition arbitrary xt combine distribute composite dual finally lipschitz composite projection ex ccc department electrical sciences berkeley berkeley berkeley berkeley decentralized network form nonsmooth convex localization sensor analyze average sharp topology allow clear separation effect arise structure gap network prediction confirm low well optimization development problem network structure arise variety domain science agent tracking sensor naturally distribute common necessity decentralize locally light network periodic failure example quickly processor challenge canonical arise minimize average e vector desirable small subset datum different processor minimize loss entire dual paradigm know significant seminal analyze process important network fast convergence general processor must agree recover sharp consensus study topology path argument underlie e reference allow stochastic gradient average minimize equality rate shift focus processor potentially whereas recent al project subgradient distribute minimization non contribution algorithm distribute maintain average essentially facilitate issue mean topology careful demonstrate close spectral analysis term optimization technique nesterov splitting issue constrain optimization stochastic elegant averaging scale chain protocol deviation comparison previous characterization scale term often give paper network tight polynomially topology require paper guarantee optimally obtain optimization essentially topology connect graph network tree protocol attain dependent structure current maintain optimum online expect comparison give network scaling scale spectral graph factor obtain solution cycle iteration two bound simulation excellent cover begin study communication protocol cluster fix dependent randomized communication protocol well failure communication tradeoff transition theorems structure regularize objective often remainder section formal statement whereas main consequence main section depth basic derive depend network treat communication present collect throughout one vector means write formal statement distribute averaging optimization specifically undirected vertex edge subject belong convex sub need assume generality translate maintain impose directly immediate neighbor ni nature arise domain motivation follow sensor equip device environmental application sensor temperature computation minimize scalar function variance quantile estimator motivate learner empirical processor suitably processor cluster processor directly small case computational dual averaging design minimization nonsmooth discuss extension set dual strongly satisfie make generality proximal quadratic convex il cost satisfy lipschitz lipschitz denote generate step step receive eq type iterate first approximation proximal stepsize seem originally nesterov relate appropriate novel dual distribute maintain pair associate node subdifferential receive node neighborhood base weighting doubly stochastic notation update projection compute compute projection stepsize sequel convergence local optimum locally manner definition sequence weight see average equation version investigation corollary follow dual averaging provide define basic sequence generate access locally sum term common subgradient third estimate deviation extra significant concrete convergence roughly deviation tt v asymptotically approach statement optimization subgradient incremental subgradient distribute algorithm advantage per rather highlight subgradient machine statistic minimize randomize incremental access every lead disk seek algorithm avoid gradient albeit turn investigation communication occur doubly singular summarize projection pp connection method quite since know rate walk tie walk graph probability explicit interesting four topology first placing connect analysis connect near neighbor panel grid topology network panel random geometric place random connect node separate pattern device wireless node extensively generalization position distribution finally show belong sparse good property attractive option distribute computation spectral gap graph typical probability satisfie see construction design construction interest specify matrix choice adjacency ni invertible doubly stochastic graph need order doubly illustration protocol dimensional connectivity let symmetric construction moreover doubly technical relate convergence summarize network topology cycle grid graph high probability bound ratio factor optimization remain term topology instead state size topology corollary define actually sharp mean centralized optimization iteration address issue spectral follow establishe network function graph give hard require conjunction implies predict quadratic match varying matrix still obey impose communication want decrease network refine communication incur hardware failure provide doubly stochastic theorem upper bind consist last shrink stepsize tradeoff compete stepsize minimize yield boost hold modify penalty communication establish convergence bound whenever consequence contrast stochastic communication fix dependence scale fast finally none gradient correct straightforwardly result case corrupt wireless observe field information satisfy special subgradient add gradient give discussion note average theorem stochastic update agent receive addition l uncorrelated communication cover essentially replace result result introduction researcher design maintain update minimize stepsize give clear slow quite proximal address euclidean geometry example dimensional simplex e become technical difficulty update precisely work essential fast ease extend stochastic incremental maintain token determine token neighbor let update q optimal ip pi bind never weaker tight dimensional grid well define quantity use via sequence show average evolve q matrix consequently evolve almost like subgradient subgradient subgradient average avoid linearity challenge early work proceed regard average give dual averaging non restrict easier analyze centralize convexity break piece remain control lipschitz continuity projection divide side convexity give concrete dual averaging walk doubly crucial cluster location sensor sensor corollary adopt notational simplex frequent eq brief relevant frobenius refer control namely sequel via write evolve notational clutter break cutoff term second consist step summation concavity sum basic see appealing allow note thus ft statement proof bound spectral matrix upper spectral devoted bounding note control walk imply random walk equip address graph cover recall place neighbor imply th q perform taylor expansion node neighbor right kn thus regular connect substitute case grid specifically horizontal vertical analyze grid eigenvalue product small precede cycle boundary nk grid corollary b immediately see von small particular say obtain geometric graph factor gap remove corollary give dependence tight objective convergence slow eigenvector matrix eigenvector eigenvalue equal walk generality eigenvector loss generality otherwise flip sign indexing need define lipschitz see evolution q one guarantee eq little appropriate matrix theorem failure sum tt show sum communication agent occur underlie network relax vary doubly update usual namely analysis make use however still evolution unchanged eq communication control uniform claim modify simplex arbitrary recursion symmetric chebyshev replacing similar random index high probability break separate throughout structure ease q make remain sum doubly stochastic ns q master outline sampling giving procedure consensus quickly edge drastically communication network yet fast topology asynchronous protocol computation proceed round communication clear adjacency diagonal matrix identical send round edge graph cluster centralize select round vertex still achieve factor graph relax function agent stochastic simply combine type suit completely decentralize environment communication protocol edge edge edge failure protocol psd psd hermitian upper average communication protocol node pick neighbor pick double matrix random adjacency pick dt adjacency underlie stochastic define definition dt slow factor proportional maximum degree amount factor algorithm edge fail independently use communication dt underlie apply factor rate generalizes receive subgradient receive easy virtue simplicity dual network prior care analysis pass nonlinear thereby obtain derivation nothing assume h old imply recall proceed put around norm couple argument q complete statement show statement x
independent orthogonal another distribute course claim origin qx independent generate transformation arise jx overcome choice sign transformation expensive number multiplication per generate take special use plane rotation vary compute angle convenience compute matrix multiplication one division detail loop implementation similar inner generalise generator observe desirable value pool pass transformation experience aspect implementation generator several appear plausible produce acceptable fail regard value additional hoc odd mod permutation odd appear satisfactory seem ad hoc little theory sum number would chi return user sample scaling ensure permutation phenomenon become less increase entirely pool rare variate occur I pool thus event adjacent devise correlation significant reduce easy discard generator every pool generator generator conventional generator although generator entropy random number uniform orthogonal etc uniform per normally pool rejection choose generator period least normal orthogonal random generator great uniform random generator contrary spirit long period certainly avoid generator could guarantee period period extremely unlikely care need normal generator end discuss generator conventional generator ignore speed acknowledgement comment david version unfortunately pass version computer journal outline contribution recent generating normally without rely number generate random number idea uniform generator discuss avoid many graphic anneal carlo substantial pseudo number dedicated contribution several aspect hardware hardware generator mention software generator normal uniform idea appeal transform consume give number pass construct product make public exchange hardware device stream device connect map appear content technology idea depend availability volume stream obtain generator close word require stream bit require popular normally random number rejection von recent reference elegant generator rejection distribute pseudo average normally generator slow involve slow uniform pseudo method five machine generation discovery depend generator idea pseudo fx distribution statement calculus bayesian entropy annotated idea speedup percent slow generalise random generator report probably well restrict comment one arithmetic point
fouri insight recent thin tree proxy optimally thin bethe entropy rather sample derive develop thin tree bootstrap method bootstrapping analyze tree experiment validate implement theoretic test machine ghz processor analyze begin experiment quality issue understand approximately vary relative able approximation despite comparison run split recover perhaps surprisingly split unbiased factor return effectively ranking far sample fourier capable handling order run fourier running set structure simulate draw jointly thin vary able possibly require recover full underlie correct trial indeed also note recovery correct model log likelihood draw distribution figure structure thin know generate chain know eventually enough also note jump jump success rate leaf long correctly run simulation datum generate structure mean recursively partition sized figure also interestingly discover discover nearly balance less chain analyze dataset full ranking ten compare twice many item ccc sake roll roll study unlike set divide directly sample use split evenly group distribution biased plot help significantly lower bias useful behavior show approximate first ranking bias marginal people people provide distribution sample somewhat say interpret certainly related type cluster typically choice away remain item understand small bootstrap size partition recover leaf correctly leaf leaf recover sake leaf apply large house use candidate five candidate fine detail well candidate party fine english fine p party show party candidate party candidate notably party low rank portion may necessarily candidate independent minor party candidate order independent visually marginal marginal significant matrix belong candidate approximate plot principled see visually exhaustive running second include mutual fine leaf leaf small stable smaller learn recover leaf major agree original tree insufficient candidate belong group party voting hierarchy thin chain likelihood achieve hold training improve one overfitte practice learn correlation suggest rank correlation crucial run similar support north west show consistently leaf dataset consistently group dataset potentially exploit independence complexity idea throughout show bayesian problematic indicate main immediately pt pt retain advantage real item rank generalized assumption sample answer explore recursively real currently method depend ranking however compose easy ranking rating extend parameter algorithm handle valuable mutual already rank interesting estimating ranking learn understand careful small place structure considerably able simply identify independence independence analyze rank potential give new insight believe crucial develop procedure form acknowledgement support n provide dataset idea upon item since number absolute necessary abuse count datum equivalence generated rewrite composite evaluating likelihood first item item absolute rank item count along input optimize optimize thus datum objective equivalence rank independent rank natural objective certainly necessary simply generate ranking ranking recurrence write recurrence obeys recurrence delta permutation take support size recurrence know recurrence binomial array structure p computing binomial indicate collection fourier transform write recursion uniform particular give fourier fourier coefficient branching see detail linearity convolution fact domain arrive fourier recurrence irreducible detail irreducible implement recurrence careful dynamic sure thing triangle binomial graphical adapt variable estimate triplet probability amount triplet use triplet h union bind mutual triplet within define complement let subset strongly n kn triplet internal internal define triplets eq goal connectivity strong triplet inside lie internal e b k eq bind formal argument lemma plug connectivity triplet edge ba establish desire pn root complement strongly ab uniformly k almost require simplifie behave I bind hold probability conclude theorem empty theorem fact permutation recent storage argue full strong structure expressive family distribution reduce complexity draw permutation game form rank ranking formal fourier theoretic framework automate discover ranking dataset decomposable class machine ranking reasoning preference survey retrieval certain ranking many problem ranking way ranking yet preferable tractable overfitte achieve exploit independence structure naive rank base constrain two item rank rank novel relaxed rank item rank contain ranking political vote voting typically constraint histogram recursively ranking unlike estimate structure outline contribution section ahead study summarize relation distribution introduce contribution intuitive novel permutation base ranking appropriate notion independence rank evidence relation approximately rank independence relation complexity novel independence transform biased perform scalable use theoretic factor factor empirical evidence interpretable item iteratively large recursive stagewise section partition item partition objective subset partitioning exponentially space structure dataset method assumption effective voting paper ranking association item rank assign convention think low rank rank item place mean prefer also mean rank notation certain concept express notation rank rank rank rank order notation mapping difference ranking notation permutation ranking permutation order set ranking hand permutation item item rank item item order composition permutation collection permutation object simultaneously rank run throughout analyze known rank american association candidate name five candidate year figure proportion vote ranking vote occur vote also visualize represent matrix level overall high far winner notice candidate vote place throughout via vote distribution percentage vote ranking marginal reflect rank ranking pose challenge ranking store array second marginal probability item example computation issue nontrivial impractical ranking realize dataset aware problem lead exponential algebraic fouri probabilistic independence briefly probabilistic ranking rich day expand upon body detail think analogy paper ranking allow expressive conceptual bridge popular rely family independence old ad hoc maintain good hypothesis update sense discuss permutation either distribution successfully less object detail among object mass vote figure rank vote recent research around maintain pair store number example store fig rank last follow vote divide one store order tuple perhaps encode joint rank require storage fourier transform permutation primarily marginal correspond sense frequency fouri low fourier matrix frequency fouri fourier theoretic reasonable view principled frequency contrast sparse scalable making exactly reconstruct order marginal moderately scale demonstrate exploit dramatically track confusion association define say store keep size general typically recursively without independence constrain rearrange row impose thick gray factored permutation order despite argue permutation restrictive independence mutual allow rank subset permutation complement ranking rank preferred refer restrictive block structure marginal permutation identity position field reasonably potential identity confusion track red team track team rank condition force third place seem quite approximate vote factorize solid factored true capture candidate poor assigning permutation receive vote support infinite next section condition flexible notion cut successively drop inspire novel independent subset object ranking form final object intuitively complex relationship within allow correlation generate decide prefer within decide fig fig fig prefer stage ranking preference item result offset left denote offset way independence first random walk rank drawing mapping formula h mh commonly probability distribution convolution answer question property cutting preserve relative rank say preferred assign permutation preserve rank turn permutation preserve description distribution assign nonzero permutation preserve four notation everything else formally independence fully independently fully independent drawing stack prefer position might cut independently q convolution define relative ranking denote rank fig rank notation write possible distinct map follow algebraic uniquely decompose compose stack ranking mean number show think coordinate uniquely item perhaps relative ranking every factor definition assume definition direction q ranking use element q satisfied convolution view also analyze concept remarkably ranking require convolution fact definition reader also theoretic description fully factor along ranking would capable place ranking symmetric extension draw independent sense item special independence distribution ranking delta assign delta interesting delta never delta respect thus independent fact complement later reflect even preference set relative item amongst delta ordinary independence strict figure marginal independence regime rank delta think ranking useful incomplete ranking candidate candidate first approximate distribution like independence commonly would rarely independence small sample size indicate almost always ranking ever independence dot accurate kl factor distribution visually interpretation result approximate candidate winner independent inferior partitioning however candidate examine order marginal approximation show factored candidate cc case storage full require storage family distribution simple assign relation within simple correlation across amongst amongst completely subset recursive drawing pick drop pick pick setting first marginal figure region recover preferred generalization drop write pointwise fourier theoretic call join simply sample memory bias one low employ recurrence linearity fourier transform proposition detail appendix transform would apart show relative perform deconvolution algorithm show independently split ranking set q ranking respectively rank consistent ranking eq summation exactly marginalization compute function fouri step establish first fourier transform apply convolution dual establishe notice compute mle factor know normalize split fortunately normalizing divide complete intractable computation instead receive parameter distribution quality result fourier fourier reconstruct marginal fourier reconstruct marginal likewise fourier marginal return marginal know theorem state give join reconstruct marginal joint distribution since joint operation pointwise fourier domain enough reconstruct marginal reconstruct fourier coefficient frequency run join split bad cubic fouri directly appendix must compute transform uniform biased computed plot experimental running understand binary partitioning explore natural simplification subset run example one imagine consist consist fig draw secondly set rank express hierarchical decomposition visualize tree item leave hierarchy leaf example decompose tree impose assumption suit possible structure encode common consistent call level partition partition ranking together consequently hierarchical way establish write independent leaf decomposition decomposition line since know desirable way require decomposition hierarchical thin chain thin refer hierarchical express thin thin chain analogous thin clique never scale polynomially ranking thin sequentially marginal thin describe clinical opposite political somewhat within verify conjecture use independence remove perform search nearly kl divergence identify candidate belong group hierarchical datum kl divergence group set biased mixture biased parameter indicate since lie impose independence available rank obvious particularly really next structure rank relative rank subject constraint graphical optimal distribution base address alternatively want partition tree base ranking subset complement automatically determine independently sense formally solve kullback reasonable training infinite truly independent actually single evaluating globally relative already hierarchical propose locally cluster show tractable compute prove fig six absolute whether prefer formally subset absolute assign denote interpretation equation ranking ranking ranking ranking guarantee detect independence necessary absolute ranking argument establish reverse equation evaluate converse rank determine thus equation optimize intractable however mutual triplet item mutual mutual value triplet internal triplets triplet set aa mutual triplet show graphical finding term invariant index triangle bar instead mutual view involve computation triplet time instead tuple reflect rank tell know prefer fig absolute make commonly whether objective sum problem triplet low weight internal triplet sum triplet symmetric result poorly dataset candidate easy visualization set form since highlight correspond mutual mutual conclusion show candidate mutual information surprising candidate align know rank preferred like cut graph objective tendency prefer partition balanced partition unbalanced cross avoid optimize optimize useful thin alternatively objective encourage balanced variation intuitively denominator equation exist encourage experiment proposition accounting dependency involve pairwise independence look marginal factor upon examine practical variation objective quantity triplet q measure prefer tell preferred sum arise detect insufficient ranking yet moreover would consist measure measure nonetheless subset use almost partially rank argue function access triplet approximate remain reasonable denote use regularize adapt triplet mutual estimate accuracy
several two network thing size roughly go achieve gradually indicate get gradually bad community large substantial beyond traditionally size scale become phenomenon large network network consist moderately large community notion maximal removing associated rest average importantly though remove core indicate interest identification common largely way confident domain specific conclusion responsible low partition optimization network participant recent machine draw area combinatorial scientific implicit study often lead good boost fit additive objective computation parameter somewhat fundamental diagnostic crucial development perform regularization well prediction emphasize neither algorithmic consideration computational black closely couple statistical understanding seem problem design constraint implicit statistical consequence decision discussion dna discussion co numerous participant form recent idea scientific interact increasingly development scientific dna nucleotide idea area serve exploit complementary order solve apply recent advance internet broadly term chapter I trend increasingly area trend appropriate way body diverse distinction former science adopt issue algorithmic roughly application strong specific concern include phenomenon reliable datum effort union depend increasingly etc business perspective perspective national security perspective find force union thus look versus substantial shift away issue broadly term explicit involve participant vice versa substantial shift generally great detail computational extent statistic relatively learn ph mechanic molecular water protein water science lot later scale thing I transition conceptual remarkable class computer software engineering remarkable lack come understand datum confident conclusion output fast chapter like thought work take complementary versus statistical understand make claim justify choose problem represent dna microarray dna nucleotide historical trend identify wide range micro market query internet apply previously specific internet idea internet issue would algorithmic hope two statistical algorithmic consideration apply black coupling seem scale design understand statistical would like detail modern although perspective statistical account everything customer transaction course month consist site perspective goal patterns associations association also quality fraction database frequent hard much algorithmic devote exact solution unobserved pattern achieve variability around proceed noise well former necessary event yet assign give give particular event ask question common whether undirected unweighted edge interaction model value encoding describe two novel section column exactly scientific linear motivated large computation deal computer science sort often describe genome sequence base code protein microarray device measure genome protein individual environmental variation numerous amenable nucleotide human genome nucleotide negligible snps occur genomic marker track gene population use infer population human either encode people snp encode individual computation attention application matrix svd dna microarray dna snps gene snp perform eigen capture eigenvector heuristic eigenvector certain lead heuristic may justify happen tend ellipsoid along justification domain mathematic reason easily process linear snps isolate one interested one reason task input find good dna snps pathway reconstruct etc common algorithmic include look large variance maximally uncorrelated intractable consider call subset choose exactly eq spectral frobenius span deal focus several include algorithm currently spectrum decision hope sophisticated rule practice call qr emphasis question backward issue whether run multiply result essentially involve enumeration top singular onto problem general focus typically make algorithm column fail desire typically constant big notation order exact also demonstrate column importance column fast small additive could disadvantage immediately applicable scale even uninformative heavy graph adjacency network quantity decay heavy tailed power manner frobenius bound matrix compute top proportional span accord detailed prove bottleneck subspace compute span rapidly motivate probability recall look relative directional singular hidden reason since euclidean encode encode suggest choose sampling structure send subsection sampling algorithm scientific analysis describe previous subsection describe intuitively reconstruct quantify well look l arise moreover interpretation natural interpretation viewpoint arise measure spectral amount sense algorithmic perspective relevant depending number cholesky qr full question right thing answer outcome predictor adequate rely assumption perfectly sure violate response column nice sensitive influential determine good interpretation coherence gain insight consider visually ten intuitively stick particularly leverage magnitude leverage suggest diagnostic regression investigate course turn might point statistical score square fit leverage ten point mark leverage visual inspection call color code leverage red leverage document construct mail leverage metric popular gene axis star gene red dash unsupervised leverage supervise algorithmic consider leverage singular vector span leverage equal euclidean norm column uniform hadamard extremely column identity row element use solver black subproblem vector solve opt ax opt opt highlight essential bad case l see statistical diagnostic regression analyst tend bias toward ls input suffice nontrivial many subsequent subsection singular acceptable generalization case hadamard transform hadamard tend score reason localize orthogonal projection lead development l problem run implementation deterministic thousand hundred describe leverage come perform note combine look collect heart hybrid algorithm deterministic qr perform hybrid phase let span top importance accord rescale qr return qr matrix non uniformity speed algorithm complicated sampling bound p ok ok interestingly make importance probability concept describe subsection worst critical qr matrix bad case respect bottleneck matrix subspace qr apply million column implementation traditional fail quality bound select comparable constant good previously goal dna microarray snp snp diverse non trivial goal select column generally diagnostic unsupervised selection problem algorithm behavior score typical call edge laplacian notion related tend edge community stick leverage span present code gain uniformity term document publicly mail choose plot score high leverage nearly order large size high score suggest surprising email corpus though successfully plausible generative associate reason expect nice phenomena concentration occur microarray snp present plot normalize leverage describe different cancer remarkable dna snp present evidence two phenomenon responsible term domain believe property like occur history stick express axis snps back etc snps elsewhere mutual metric particularly prominent datum maximum axis datum course strong bias model quantify condition dimensional unsupervised like supervise issue observation inner usefulness qr algorithm decision version qr qr phase one algorithm behave low tend surprising practitioner observation implementation scale preprocesse randomized phase perform room qr tend make sophisticated less randomized direction select qr keep column qr preprocesse get run qr norm directly column tend make thing thereby deterministic setting rule randomization result practice choose behavior deterministic qr choose somewhat column improvement column randomize statistical statistical leverage concept bad traditional answer seem datum score informative worst reliable bad effort informative application answer seem intuitively scale application assume base consideration surprising stick relative model suggest score statistical reason diagnostic application perspective extensively scientific interested balancing vision computer interested primitive divide network interested meaningful specific search contextual ad top engine page search typically text important construct bipartite graph discretization discretization keyword bid bid phrase perform mining graph quantity click problem example identify worth analyst well coherent thought useful testing new query market place match phrase also occur specify bid ignore game theoretic issue imagine original middle homogeneous large similar phrase nearby topology advantageous engine phrase phrase nearby original bid perform expansion application applying analyst constitute community intractable exactly algorithm heuristic problem set thereby look otherwise modify iterate might singular nice hierarchical intuition nice noise adversarial less intuitive problematic reasonable readily wrong whether improper correct intuitive insufficient resource noise etc visualization apply large lead largely illustrate visualization problematic situation network principled algorithmic statistical understand explore ccc toy nice realistic network illustration network low responsible typical community method flow find flow ask look piece equally sized opposed moderately like leave way formalize refer family objective approximation involve cut partition quality weight cut balance two lot chapter cut balance pay give possibly denote end complement cardinality alternatively substantial normalize great case eq could replace min denominator replace product formulation slightly big big preference equivalent function achieve cut formulation importantly partition intractable learning cut version partition general approach include laplacian cut cut degree expansion meaningful application robustness divide graph far deal include focus cut return node graph time analysis achieve observe basically imply task eqn piece objective note improve application million arise locality nearby seed piece kind nearby cut say initially two method low cut whole view spectral complementary strength find multiplicative well relate amenable implementation community detection begin observation experience network precisely community connection amongst rest
every every super set rank set learn submodular difficulty submodular discuss proof sketch discuss intuition prove jj prove interesting maximal basic could set case disjoint constraint third necessary similar must additional small nearly disjoint theorem prove eq think non special construction family generalize partition unfortunately achieve reason collection polynomially super polynomially case relevant goal super polynomially reason weight well code super unfortunately small plan combine observation mean intersection expansion turn perfect strong ratio nearly nearly idea family suffice theorem insufficient prove modify precede introduce sort truncation truncation ordinary ordinary rank away keep large get integer call function define large family family family cover know truncation size broad appendix partition correspond section general suffice ti property commonly note property sufficient unable construction empty satisfie axiom maximal disjoint replace I every use empty show define apply priori hand side function modular submodular applie issue constraint tight submodular cf ic ig minimizer intersection minimizer j second satisfy consider two lemma I gd construction let index denote construct instead take common parameter I start bb achieve g set bipartite graph additionally construct family u v satisfy last allow concrete probabilistic construction follow al match requirement state match proof require slight edge independently nod achieve parallel edge edge decrease family define additionally claim b whenever since definition immediate b da trivially remain prof desire submodular submodular distribution compare chernoff bind arbitrary roughly additionally view submodular prove result prove paper survey naturally median expect submodular let product distribution application corollary function construct column corollary imply submatrix concentrated expectation row technical provide theorem submodular theorem supervise paradigm draw I unknown may perform goal real small stand probably function error pac boolean special submodular note pac submodular natural problem distribution building learn class begin useful submodular lipschitz distribution provide monotone function value sufficiently approximation return example product label training begin expect output corollary constant carefully zero monotone submodular union close boolean zero implication approximate factor assume l together approximate fraction mention recall monotonicity furthermore imply inequality goal definition statement zero event hold probability least idea proof set suppose example time formally argue measure amongst none hand n n contain rank remark case minimum simply modify output modify show new show monotone formally draw draw approximate low point ratio learn sized even make formal probabilistic guarantee note function possible hypothesis approximate within underlie distribution query word augment let draw underlying query probability approximate within least theoretic hardness slight modification way exist learn submodular polynomial algorithm support family hard though prove algorithm open question construction submodular submodular normalize non negative w submodular symmetric q let fair coin flip label constraint output sn monotone submodular approximation use multiplicative learning instance I passive supervise slight passive u xu md repeatedly fair coin label coin label label linearly say claim every variable independently condition belong hypothesis fs sn first program feasible discussion fact draw label approximate correctly fs sn proof fact similar prove n reproduce probability least inconsistent incorrectly produce approximate within simple et bad robust handle extension clear algorithm general assume submodular assumption factor negative learn use describe fs label extend agnostic case agree target fraction arbitrarily inefficient submodular q learn approximate multiplicative proceed error mistake agnostic learn see appendix np hard mistake result procedure inefficient statement surprising rank proof let concave surprisingly partial converse follow sufficiently small idea value value uniform concentrate concavity say probably sufficiently get henceforth variable set cardinality finally identically distribute appeal great recent work fy variable obtain calculus non concave fix scalar concave implication follow pick independently remove element call easily monotone approximation q median define hx k argue complete immediately imply since whenever l complete original section hardness illustrate minimize formally eq submodular running compute construct minimizer lattice survey far minimizer encodes minimizer lattice lattice minimizer combinatorial much hard impose submodular cardinality variant submodular submodular analog tractable main minimizer perform query output query factor size minimizer large return negative monotone function approximate factor jensen minimizer strong construction sufficiently slowly set u u define query attempt cardinality bit correctly represent error determine whether whereas suppose query I apply choose neighbor remain vertex random neighbor randomly distinguish vertex call cut min cut modify graph vertex monotone perform structure represent minimizer moreover query almost proof require disjoint vertex neighbor pick independently analyze repeat neighbor v n du follow remainder theorem let path minimal cut word let apply neighbor perform cut whose correctly prove perform query determine whether second every vertex cover return f et submodular restriction seem unnecessary et cover ratio many query modify construction theorem bipartite monotone submodular minimizer within factor compute minimum form matching cover contain independently probability suppose query set first perform query multiplicative theorem consequence show satisfy property substitute open economic briefly economic function economic combinatorial intuitively increase price certain demand change formally price gs price old contain preferred price structural implication extensively many efficient condition necessary economic
independence diagonal however correlation thousand reasonable generality fisher discriminant depend increase sign turn rule naive depend put visual inspection equal discriminant discriminant perform away rate tend analytically accuracy impossible select apply classification depict well bayes restrict discriminant outperform restrict fisher powerful enough variable rise feature represent closely discriminant whole implement impossible empirically among reason focus ideally possibilitie counterpart consider naive method penalty way scad mcp primary procedure constraint classifier solution fisher portfolio reflect exposure preference drive accommodate application coordinate implement road use spirit diagonal regularize eq road fair study independence road road fisher marginal also incorporate road track performance variant road road road along theoretical property correlate feature discriminant alone two feature employ though standardized mean difference word mind large absolute objective standardized mean difference st rd hand covariance simple calculation lead st rd lagrangian problem propose constrain descent tailor minimization constraint optimization solve piecewise common affine replace pool enforce affine serve normalize fisher discriminant regardless confirm solve descent popular descent direction denote element vector search cycle meet coordinate particularly attractive explicit enter analogous use warm solve suppose need optimize become calculation form soft thresholding operator convergence coordinate strictly decompose therefore coordinate wise strict convexity differentiable theorem descent converge derivative computational cost operation cycle warm start use denote cycle enjoy road similarly replace version require third road sample classification misclassification question address assume pa pa nd misclassification rate say misclassification road view objective fix discriminant version eq prominent technical challenge multipli method reduce utility much complicated projection oracle discriminant constraint fisher discriminant pre feature road advantage two rule selection specific screening sure screening demonstrate road discriminant road road drive motivated fdr choosing screening permutation nan carry calculate statistic feature intuitively index let quantile make whose alternatively know road track sub fisher space include road achieve road loss screening block next verify theorem kn simply accurate scope pp word piecewise path reduce include differ affine purely spirit particular stress property trivial believe dimensional constraint replace involve composition lipschitz compare road road road road screen road version centroid discriminant fair independence naive generalize covariance oracle study testing repeat stability generality mean set dd size sensitivity due subsection path road realization clear index cutoff road dramatically road road road road road fair nb result range would mention subsequent around guess screening version road road select road road pre set median oracle decrease phenomenon dimensional go contribute classification road reasonably oracle method fair nb fail discrepancy employ fail road road rate road emphasize screen mainly lie pre road perform even lot table road almost road fair independence road substantial road recommend quite microarray well substantial road similarly thresholde standardized difference fair select marginal road penalize summarize feature road fisher coordinate number select road road road road road setting subsection road median broad range choose almost primary simplicity subsequent l classification road road road road road road road setup except block correlated matrix block examine varie percentage select road road road road road road road road perform road look contribution road show table express advantage highly correlate road pick take road recommend road road setup size equal correlate pairwise performance estimator vary percentage road road road road nb nonzero evaluate stability road structure give integer matrix definite unity signal nonzero element summary road compete fair road road road road error nonzero road broad expression cancer first come set contain vector cancer datum vector microarray phase project gene expression profile gene trial nb analyze event year year diagnosis year information positive negative select subject positive negative one total reader find set standardized road road road fair nb road desirable contrast road competitive select close performance number vary three robust tool dimensional road road gene road road fair nb error select gene road road road fair nb simple dimensional employ un regularize difference vector curse accumulation choose evident simulation discover pattern resolve part introduce variant road control worth explore selection easily extend nonlinear order polynomial spline kernel learn community transform road challenge root stochastic role completely preliminary proposal extend road section outline road suppose mean fisher approach classification dimensional centroid project centroid onto observation centroid close onto necessary project population centroid spread multi class scenario select coordinate th maximizer coordinate determine b binary regularize coordinate regularize discriminant coordinate additional discriminant find coordinate base centroid span road topic associate comment greatly scope financial support grant dms gm greatly theorem w w w c w set consequently lipschitz pa nc
replace modify likely tail idea behind heavy could possibly help speed work develop use approach approximation unnecessary base way statistic way would decade k perfect simulation coupling perfect simulation process dominate context develop purely purely process multiscale area capable modelling point pattern vary topic regression suitable coefficient coefficient perfect within promise compare method independently couple monte birth chain issue markov reach problem past perfect simulation therefore need rigorously burn appropriate error carlo identically distribute reduce simple price long difficult code many give present dominate locally point potentially literature go context modelling point pattern wavelet particular area relate zero coefficient wavelet simulation begin coupling dominate spatial justify perfect standard introduce describe infer turn attention problem review index perfect appropriately modify approach feasibility address investigate example section offer brief intuitive introduction principle behind formal description see suppose irreducible intuitively go infinite chain run chain couple chain I minus infinity chain thus state enough coupling two coupling stochastically order whenever stochastically ingredient attempt notably discuss truly method process couple allow simulation interact soon type general locally subset define locally stable exist point obtain poisson evolve markov fix death mark process recursively low process process evolve suppose death happen accept birth happen event remove define identical underlie process mark keep calculation calculation costly version calculate algorithm general dominate partial ordering induce preserve markov whose wish locally value function monotonic respect partial induce intensity interaction step write computationally deal clearly simulate dominate expensive least linearly practice burden follow alternative evolve way process death occur happen accept happen remove event obey coupling suppose wish simulate calculation dominate coupling past involve calculation write identical attractive multiscale theorem construct refine clearly necessary several class cox stationary area interaction process produce cluster moderately order clustering fill gap produce neighbourhood area configuration regular poisson configuration constant compact neighbourhood point addition process configuration case reduce flexible yield regular spatial randomness cluster unfortunately small distance display sort behaviour distribution tree patch physical physical law behaviour behaviour class q equation ball scale measurable integrable extension standard area perfect multiscale process already uniformly substitute extend multiscale might indicate scale existence process perfect datum multiscale appropriate circumstance fit systematic adjustment pseudo parameter sample spaced interval assume independent normally problem transform large behave distribute identically combine wavelet discard small since wavelet spread evenly discard noise cross discovery bayesian prior population wavelet wavelet thresholding capture may dependency area interaction section idea appropriate approach coefficient subject noise value wavelet noiseless wavelet level coefficient level transform independent equation extension consider discrete index wavelet wavelet nonzero specify binomial lattice negative integer assume point concentrate pure variance extend consider reasonable produce make natural structure clearly discrete wavelet represent tree configuration mind intensity expect reasonable exactly theorem tell couple discrete complete specification area prior need interpretation neighbourhood location index possibility decide adjacent coefficient child coefficient adjacent make nine illustrate capture time periodic show discard part unless neighbourhood function include child immediate coefficient variation wavelet boundary modification practical close posterior coupling past advantage normal lattice equation convolution density simulate ignore mark amenable abuse third refer monotone subset condition process process constant intensity dominate lattice maxima minima maxima minima global maxima low consequence feasible dominate maxima modification essential chapter give dominate location intensity dominate lattice start point section probability upper remainder carry way birth dominate although reason rule give ease assume jk jk jk jk jk algorithm feasible value birth necessary issue live point location whose large purpose nearby assume strictly location large problem q monotonically monotonically dominate location good gain taking dominate
inferential ratio popular program user pair drawback comparison tends favor rich model aic criterion commonly aic compare nest nest favor criterion select ratio penalize increase accounting uncertainty aspect factor bayes incorporate uncertainty intuitive comparative evidence probable probability compare popularity publicly available likelihood complexity computational burden high compute marginal straightforward integrate subspace branch substitution eventually sum extensively condition nest bic firstly sometimes recent appeal laplace neither option implementation require tuning suffer extra point sampling harmonic integration review context large advantage applicability protein need highlight integration reversible accurate interesting alternative mean estimator estimator share simple make easily substitution tool complex introduce formula estimator nothing normalize unnormalized negative integrable define unnormalized instrumental integrable contain q sample introduce consider hence case harmonic posterior sampling probably explain original importance arithmetic q generally reduce integration harmonic harmonic estimator tool exception argue theoretical explain yield chance region large standard end yield approximation fact raise sometimes whether reliable generalization improve alternative review likelihood estimator harmonic approach basically share density formulation particular choice instrumental originally instrumental perturbation total target eq arbitrarily perturbation multidimensional density perturb length origin visualize act perturb instrumental density density respect instance yield condition full moreover perturbation extra computational mass perturb depend asymptotic via delta method square ultimately estimate key role infinity kk act address minimize estimation consume one obtain perturb original purpose computation easily http site google com site home shown recommend simulated covariance density origin frequency transform entire real call jacobian density successful simulate output simulation software specifically upon request author publicly software benchmark harmonic indeed estimate marginal though infinite unstable sample posterior must aware quantity surrogate rigorous bayes factor example simulate comparative evidence alternative benchmark synthetic panel implement couple simulated evidence whole order improve rate equal marginal know true scale relative mean interval autocorrelation correction since correct perturbation logarithmic scale harmonic arithmetic relative similarly estimate sample quantity monte carlo estimate although condition error sufficient guarantee accurate q denote bootstrap replicate formulae account autocorrelation correct relative error small arithmetic mean simulate point really marginal nonetheless distant corresponding value strength compare method estimate mc serious reliability magnitude robustness range specie show topology model couple evidence approach repeat time table somewhat relative carlo relative correspond factor logarithmic consistently model bayes possible extend deal select compete advance compete evidence bayes evidence favor topology continuous evolutionary nuisance belief comparative summarize compete substitution subsection density fix indeed hadamard topology aim compare topology marginal likelihood also exhibit performance evidence possibility simple evaluate evidence compete bayesian eventually compete probably date model context applicability computationally light burden appeal explain option routine simplicity match estimator biological provide effectiveness marginal class harmonic estimator share simplicity feasibility unlike enjoy theoretical moreover simulation appeal evaluation like stream burden reduce also search optimize verify substitution comparative respect posterior estimator circumstance outperform precision candidate evidence estimator even fine appropriately cm cm cm em molecular bayes marginal marginal solution term simplicity burden researcher far simplicity arithmetic result estimate bias inaccurate reliability conclusion generalize harmonic simulation share infinite alternative fully satisfactory estimator currently publicly sample marginal model state related connect specie infer specie study genomic chapter substitution address fully propose organized briefly review concept focus substitution tool popular tool harmonic estimator harmonic section name model numerical comparative discussion dna specie consist specie site relate replacement nucleotide specie describe replacement tree among call leave external represent day species internal genomic branching branch period substitution dna probabilistic modeling change evolve site
dirichlet regret kullback propose optimistic aim observe kl reduce probability vector maximize resp easily lie p neighborhood neighborhood case ball vector dimensional represent represent direction simplex maximize maximizer vary continuously p consequence optimistic assign compatible optimistic mdp transition really exist small never equal therefore optimistic ball assign positive even true accumulate optimistic favor transition experimental indeed assign strictly positive transition eventually unobserve transition potentially work except difference neighborhood together illustrate represent evolution example take bottom cm section explain solve arise maximize lagrangian maximizer follow np imply sum iv denominator satisfied way determined v define function play maximize decrease mapping jensen easily function denote z solve initialization series iv iterations paris fr consider reinforcement decision optimistic mdps favor kullback purpose studying kl term solve optimistic extend kl benchmark significantly improve mdp reduce observation element keyword reinforcement decision approach kullback leibler long model decision mdp assume agent need face fundamental trade experimental consequence action act order reward consider state maintain running estimate expect approach exploration exploitation armed context extend several instead act optimally accord surrogate collect policy armed prove optimistic logarithmic mdps optimistic use optimistic extended transition highest remove less transition elementary interpretable extended lead undesirable may optimistic importantly optimistic assigning probability connectivity persistent transition accumulate impossible optimistic kl kullback leibler kl pseudo indeed smoothness metric first issue thank relationship simplex kl optimistic trade promise accumulate search kl consequence build logarithmic kl traditionally number practice environment property kl paper brief description discuss kl mdp rx action agent receive action reward consider mdps mdps states reach mdps policy respectively reward fix optimality policy focus reinforcement problem know e probability consider jx optimistic receive reward count proceed episode start episode length th episode episode episode soon action policy follow episode optimistic take optimistic solve equation extend maximization action maximization rx c dot product probability radius confidence maximization appendix algorithm lagrangian depend current value probability ir f newton newton argument accumulate reward play adapt kl neighborhood start constant appear theorem eq dependent logarithmic upper quantifie depends inspire every proof rely concentration sum pair probability regret write episode matrix optimistic episode otherwise summation denote resp resp transition matrix resp third written th x addition combine complete compare kl randomly environment model transition six
optimality noting give word side constraint substitute optimality already solve explicitly calculate solve cubic follow maximization solve part keep maximization cubic equation repeatedly alternate alternate specify care implement overall thing nd step lagrangian multiplier evaluate nd admissible overall satisfied part check objective zero invertible candidate acceptable acceptable guarantee satisfy suppose candidate part number side admissible side write basic cause instance update initialize newton give pt dirichlet find optimize objective find root newton quasi newton initialize step repeatedly instead iterate solve question problem connection bss differential version bss mean without mean second
action team behaviour simulate general confirm recognize set represent team b front forward b front forward forward front forward leave front forward b front front leave forward b b forward front forward front front forward forward forward front b forward front c forward b front forward b forward front b front b forward front b front forward front b forward front forward forward front b front c front b c forward front forward forward front front c forward front front b front forward b front front forward b front team behind forward
r p functional line propose monitoring perform use simulation q distribute term parameter root innovation uncorrelated monitoring unit setting numerical corresponding monitoring rule limit take define nuisance parameter weight past nuisance west point carefully yield investigate rate hypothesis delay signal analogous conditional delay quick method table average delay result monitor curve simulate law overall seem explore figure trajectory immediate detection negligible yield hard detect
locally first state optimally strategy directly train latter concept consist simplify estimation extension classical normality problem know normality quantum learning problem proof depth sufficiently likelihood side rao inequality estimator example coin q statistics concentrate rao unbiased find mechanic situation essentially dimensional version er rao quadratic overcome adopt modern perspective provide instead problem idea statistical sense know restrict adaptive rough estimate step sample rest estimation parameter normality smoothly converge model observe single mathematical two statistical quantifie dominate distribution densitie analogue quantum define infimum le
agglomerative algorithm deduce agglomerative hierarchical linkage subset application cluster result search function quality function diameter number call problem strategy agglomerative agglomerative go see drive force classify discuss agglomerative agglomerative bottom cluster cluster cluster create hierarchy two possibly useful property cluster bioinformatic cluster priori agglomerative specify distance object equivalent use distance object cluster agglomerative complete linkage hierarchical polynomial length input diameter approximation time although agglomerative widely consider factor analysis guarantee linkage agglomerative diameter closely problem search center minimize distance center center come cost optimal solution know hierarchy center provably center know metric diameter problem euclidean also metric space approximation discrete center diameter clustering
hypothesis probability transition eq contain one ask control whether control law pa tt operation pa obvious rest essentially stream htbp probability want converge rewrite denominator obtain reference pm illustrate simultaneous divergence process controller measure unit htbp characterize depend apply hence growth depend
mathematical arguably theorem integral real value distortion wang call distortion distortion parameter fairly suggest distortion start transform new assume eq therefore call parametrization necessary loading assume net index therefore become natural notation infinite depend cumulative intuitive value always later shall hold load basic loading satisfied whenever non vx vx
illustrate calculate mm x x imply apply third second reciprocal x surrogate g x x x x x x mm obvious check mm iterate accord second iterate whenever converge infimum identity imply hope much surrogate reduce separate mm x limit function complicate unbounded component coincide
posterior involve know distribution wise model easy implement except gibbs draw variable color need component combine produce feature simply relate follow dimension equal dimension
project proof contraction continuous almost proof rademacher lie point draw dotted geometrically onto crucial examine figure clear point still project seem necessarily lie corner right corner believe almost map affine exist hausdorff lebesgue zero simply sense invariant different unique intuition degree freedom matrix dimension boundary boundary sign neighborhood trace zero stein conclude pt sort detail statement involve involve argument indeed continuous see lasso boundary quantity invariant everywhere apply stein formula result x divergence x space interpretation fuse also freedom correspond fused graph degree connect component extension connect component slightly modify degree lasso fuse fuse problem freedom generalize correspond two connect interpret dual correspondence decompose contain coordinate last define coordinate yield keep correspond coordinate yield furthermore coordinate give conjecture degree fuse equal fuse cover much general result nan one corollary omit sake brevity list along fuse sake completeness degree
component compute simplify logarithmic term construction distance kernel rely dimension size nd error come place feature probability invoke chernoff hoeffding x q kp time sample k approximate subset np integer determine subset reduction transform size q ct I qx x q valid distribution deep measure embed possibly infinite paper algorithmic fast computing point near provide technique reduce convenient set set
subset refine try belong neighbourhood solution neighbourhood build neighbourhood obtain modification start generate solution improve otherwise end optimum important biology since protein fold logical secondary domain predict five fold alpha beta protein b bind protein protein beta alpha beta class class round treat separate classification example fold value experimental fold validate conduct impose applying indeed validity
cauchy finite use induction almost singular bound vanish almost consider proceed check theorem condition hypothesis empirical bound variable converge finite almost surely indeed q hence induction hypothesis variable z show combination pseudo fy side pseudo function jointly surely transform due symmetry difference cr ts td limit hypothesis form definition q form h almost write proceed easy check variance induction variable note show variance combination pseudo c g tb ty e side equal z surely z z surely statement equivalent jj distinct check immediately I let z theorem trick avoid repetition algebra generate coefficient form column converge lipschitz limit exist
replace surrogate instead optimize optimize mm denote last discuss numerically optimization simplicity objective mm read science besides specify explain plain way discuss monotonically function prove decrease instead step argument compute global minimum last equality expand omit assume appendix subsequence converge I meanwhile convergent subsequence k x k proof besides obvious contradict get base two lemma model sequence generate x upper surrogate tangent taylor accordingly evaluate know point x model could impossible claim converge find always satisfactory result numerically solve optimization constant convex objective mn ij tr equivalent rewrite
variance ever recommend matter provide remarkable mcmc importance oppose importance density must support like hull simulation completely probit ml obtain replication compare harmonic significance covariate simulation compare approximation harmonic diabetes simulation source explore rhs select approximation harmonic solution preliminary dimension model attractive natural rao estimate simulated estimate unbiased approximation conditional probit obtain replication approximation usually reliable dominate normal imply harmonic estimate compare bayes harmonic importance diabetes rank order worth marginal good rarely generic solution whenever available approximation generic model compare model jump reversible constructing method try limitation simulate markov existence stationary immediately produce chain include besides target enough
wind drop summary exponential smoothing write truncate normal easily ahead forecast function expression one iterate ahead forecast apply approach wind simple benchmark benchmark persistence random forecast forecast truncate persistence ahead location hour forecast location set variance consider persistence constant forecast describe benchmark forecast third benchmark forecast unconditional forecast exponential smoothing move empirical computational estimation density day density appropriate conditional forecast origin ahead forecast parameter forecast keep show decrease j maximize conditional forecast summary benchmark approach generate forecast compare minute hour ahead forecast tn forecast tn forecast lt lt tn tn lt stand tn stand truncate generate forecast forecast forecast density particular forecast form
call cluster denote contain output k asymptotic target consistent pf sample work first find empirical distributional maximize already cluster assign point cluster assign notice iteration initialization initialize I jt jj k two complexity distributional tuple tree efficient furthermore therefore double weight zero way joint ergodic
fire present three strength unity strength two interpretation graph connection direct thus specify specify second regardless inconsistent evidence specify neural remain value typical solution neural mix source represent traditional network sum
distinct pair sample distinct propose recursive way express except therefore allow straightforward marginal marginal observation point closed likelihood compute irrelevant computation bayes factor practice derivation recursively I component update update j update nu jj nu nu
e sparsity pattern interested computationally characterize network structure degree chemical encode various chemical involve hundred stochastic task consider proximity fluctuation specie count network would chemical area tackle state stress graphical sample infinitely continuous course subsample space raise depend sufficiently approximately stationary latter large intuitively limited information confirm efficient support discuss relation exist literature
number exploitation epoch player play spend go step epoch arm sample increase epoch go show achieve reversible eigenvalue sr I arm reward policy condition end epoch appendix policy nontrivial bind knowledge choose logarithmic state basic th exploitation epoch length play length arm irreducible sr arm reward condition detail order similar replace regret choose nontrivial available achieve regret close formally irreducible reward b decentralized player play markovian play notation adopt transition passive arm former evolve arbitrary even play arm due know player play adopt receive former give play reward player otherwise ideal perfect player arm effect set strict achievable ease global exploration exploitation player offset share eliminate
vi direct proposition prop argument exist far prop p continuous identical sequence equivalently consequence relate somewhat proposition imply first p sup norm subsequence p prove see ex measurable real nf element satisfy l lf continuous value sup norm compact non value every additionally satisfied use convention supremum empty claim hypothesis claim additionally extend logarithm interval establish b prove analogously b continuity continuity extend logarithm element converge logarithm already establish claim follow argument already establish b every part theorem law large number separable banach fx establish conjunction sup clearly variable every evaluation borel see e van compact borel mapping hypothesis prove analogously similar separable banach continuous already conjunction take function x away show compact establishe give part essentially sup claim intend dx inspection analogous fx f ff f uniform proposition appendix application hence probability theorem space unique maximizer order sup choose proposition prop constant cover fix r mb f j note
agent unless around estimator change step converge assume span definition unbiased observe estimator estimator recalling agent happen step long converge stop unbiased iteration least unbiased limit simple private measurement social rational raise complete structure future relax common network pick agent proceed calculate expectation
module error time solver minimize influence factor affect intel processor gb graph node enyi clique edge triangle create complex create random represent accuracy efficiency test clear create part part pick entry compute harmonic compute square solution harmonic another rely residual pick entry form stack matrix square harmonic definite magnitude semidefinite magnitude nonzero number conjugate substituting satisfie require gradient use laplacian spectrum include graph free involve property type connectivity edge remove single conjugate graph formula easy upper degree connectivity path graph acyclic vertice vertex degree regular star degree connect center comparison actual graph star enyi graph fewer exist graph cell complex embed boundary place vertex dual connect edge boundary surface sufficiently complex possible complete euler characteristic complex boundary lead solver gradient cg residual solve rectangular form solver mathematically equivalent mention early semidefinite nice nontrivial effective quadrature differential solver formulation number triangle complete rectangular solver list indicate tucker formulation exact identify homology lead harmonic component report harmonic part require create hierarchy linear decrease hierarchy call coarse
universe function submodular lipschitz building bound self bounding lipschitz submodular expectation expectation structural theorem together concentration inequality structure say decompose submodular one domain prove begin simple monotone block bound submodular even follow submodular submodular lemma submodular ordering b vs fs respect define submodular submodular satisfie vb bs submodular well release strong vb gs submodular always terminate size vb bs shift g mapping complicated want able choose give former define deterministic carefully procedure gradually child procedure differ construct single influence ps fs intermediate state fact property completeness ps b sx fs sg item reject element reject
variable large dimensionality toy series free widely different field economic finance effect crucial activity vector autoregressive
envelope affect let lebesgue subset vc corollary cover collection loss generality represent define essential supremum analogously routine b cover thus argument unbounde integrable envelope major truncation g precede f covering let ergodic every average converge
actor method treat super accounting interaction nature community membership determine number method agglomerative cluster analyze citation network examine recover agree know network simulation ensure feasibility structure identification great hierarchical formation start enable inference role actor link possible link prediction process actor position infinite feature goal al annotate discovery nonparametric issue annotate infinite blockmodel recover branching among grid offer dependent membership link network mechanism latent actor interact crucial call membership coefficient actor actor generalize
eq lipschitz solve programming however moderate size cost newton linear application attract interest relatively implement challenge huge achieve convergence descent successively nuclear must solve solution descent formulate include fused projection fortunately usually compact nesterov construct objective descent fuse numerical utilize instead work people gradient sometimes one typical
sum cardinality namely configuration ks thus obtain instance result development degenerate finish case structure come write l prove rely state permutation notation permutation equality constant permutation expectation back may namely q j l l j non argue negligible negligible theorem write central limit triplet chapter differentiable classical almost sure parameter compact criterion jensen hand side attention lead inequality far far usually taylor expansion derivative quantity almost asymptotic function omit almost fisher invertible obtain fact convergence proportion know theorem describe first consistency normalize composite value preliminary necessary uniformly consequence almost sure consequence assume f eq leibler entail positivity establish permutation label continuity theorem
theory small enough follow induction distribution obtain randomly generate application interpret basis extraction phase generate large result observe increase close expect framework extraction universal testing cast rank easily effective constrained optimization
analogue te rank scalar te ty conditional two sum use derive vector consider continue component ty account last ignore case entropy probability te position vector rank joint possible possible represent pmf pmf white find double distinct fig probability white bit bit assume time follow future sample possible possible instead neither joint example form single rank ahead white augment vector
time service job job enter job time vector similarly general useful treat complex consider later independently arrival observe arrival equivalent service regime framework type transformation job job service job job queue minus interpret job regime come employ service processor queue single within queue request remove queue manner queue simultaneously request queue request arrive call processor density arrival service service processor auxiliary job indicate job server clear time solve system equation jacobian arrival jacobian triangular depend arrival queue processor job job process queue service arrival queue queue service time equation set notice job immediately queue service latter number service joint q reasoning processor sharing design computer simultaneously share way understand queue imagine system slice consume job queue service slice remain service time service time drop ps queue limit job precisely queue independently arrival compute equation solved iteratively hold hold procedure converge
discussion hoeffding deviation independence bivariate hoeffding hoeffding page hoeffding interpretation f px x f f x axis analogously four argument two drawback firstly five secondly certain continuous formulation characteristic replace square straightforward version correlation give preference due simplicity preference ordinal literature note case family formulate appropriately empirical coefficient ordinal utilize ordinal nature
spread thing equal limit experimental covariance characteristic length hypercube pseudo input adopt standard prior parametric gamma space similarly random remain directly walk proposal parameter measurement model identical say scheme stage gaussian simple one analytically function keep explicit advantageous factor need since parent pseudo function issue jointly everything define vector sample univariate conditional v dd dd k dm plain metropolis entry parent walk proposal parent proposal justify follow row submatrix standard
energy would fail near material atom may geometry another examine atom sometimes atom eq near neighbors cubic lattice atom edge material neighbor contribute equally neighbor atom atom state atom behave solve hamiltonian
aspect measure symmetry large sign relate convention horizontal handle describe band aspect angle htb b band side arbitrary sided co show five sided figure htb side aspect angle band incorporate need model ideally train started scenario versus evenly space verify increase aspect object set start evenly distribute point type final random angle rotation figure object aspect angle versus final datum upper moving sphere
note could two additional notice setting g supplementary great satisfactory weighting cm induce dedicated incorporate predictor result simulation investigate follow enhance induce capability extend concept norm factorization la project european project characterize value minimization use order order simply twice conjugate definition property consider compute mt j fa sa j fa indicator fa fa maximum fa fw fw sa minimize hypercube convex envelope conjugate
question similar exponential lasso jeffreys normal gamma prior observe light natural I hope well conditionally local greatly simplify write write preferred error scale prior prior marginal double obvious px draw version version enter level expand local
estimator root consistent pilot estimation n bn constant tx vx ex vx tx vx vx tx e tx xx vx nx recall nh ni place result vx tx accordingly whole square estimate although exist quasi importantly square directly less nonparametric involve quasi limit vx pp dimension limit quasi fisher information fisher help estimator small theorem adopt tv proof indicate estimator plug suggest estimation classical quasi also bad position element correspond uncorrelated classical quasi likelihood asymptotically equally
mle lead limit scale corresponding average however subsampling show correspond consistent effort identify optimal issue write diffusion dimensional estimation limit equation multiscale theorem drift I system appropriate asymmetric hausdorff hausdorff distance drift estimation depend maximizer discretized mle slow averaging describe diffusion average minimize clearly system detail
small genomic recently system analyse variety factor size disease variant complex disease marker identify replicate phenomenon attribute underlie expression biological drive phenotype review describe additionally inherent rather genomic analysis biological interaction genomic cause disease crucial mechanism single amongst attain association identical may see additionally necessary activity interaction share subset note take advantage multi snp literature article snp technique statistic sum use frequently amongst snp marker statistic systematic yet subtle purely snps reach issue detect association search space multi effect remove snps significance novel identify association status degree reveal would type scheme drawback snp put limit snps examine arbitrary marginal notion control motivate interesting detecting dimensionality reduction snps combination examine cell minor find interaction expert limit must limit combinatorial recent powerful taking processor make feasible genome pairwise still remain computationally snp rather restrict pairwise interaction combinatorial exhaustive search narrow combinatorial discover
pt rectangle abundance rectangle rgb rgb rgb rgb circle rgb rgb cycle rgb cycle rgb rgb rgb circle cycle cycle rgb rgb cycle rgb cycle rgb rgb cycle circle rgb cycle rgb cycle rgb cycle rgb rgb rgb rgb cycle rgb rgb
behave probability lemma cause part nb nd suffice n argue f finish splitting n slight yu van behavior satisfy assumption q leave unchanged estimator b low come eq bind multiple virtue c inequality deduce q nz n n iw I n great equality chebyshev inequality concave maximize derivative fix real w pt notice tm g
norm otherwise weight mkl mkl norm mkl mkl net mkl allow interpretation posteriori eq regularization hyper regularization correspond marginal kernel bayesian mkl rewrite weight regularization model mean term quadratic analytically integrate marginal use bounding rewrite mx mx base regularization regularizer mkl unfortunately block regularizer however hyperparameter maximum update minimization respect weight section respect
auxiliary asymptotically auxiliary result combination variable central limit normal cm lin optimal require spectra priori used inference within framework field concrete application reconstruct unknown five separate spectrum measurement reconstruction maximum posteriori power spectrum maximize joint iv spectrum wiener flow filter operation exhibit jeffreys spectrum spectra exhibit statistical filter superior wiener correction smoothness noise like vision enhance stream artificial since signal permit filter focus mode sufficient available permit signal straightforward available exclude purpose reconstruction simultaneously due trivial general original scale well homogeneity knowledge spectrum permit construct filter linear galaxy distribution signature matter non scenario put capable would property avoid fidelity tuning structure key matter spectrum field verification reconstruct thus spectrum map make immediately optimization soon critical signal filter wiener linear wiener filter knowledge response differ covariance spectrum wiener spectra filter effective wiener effective differ wiener exhibit operation account critical implement behind frequently kl jeffreys threshold
eq bipartite graph satisfy eq constant observe factor see follow design set expansion tuning construct unbalanced denote adjacency degree observation satisfie satisfie restrict invoke finish observation compress sense derive result polynomial exist unbalanced approximation
bandwidth assume discuss rather assumption encounter setup I asymptotic put determine approximation observation kernel smoother study allow monitoring process asymptotically process ts satisfy functional limit limit special turn limit asymptotic depend question bandwidth respectively sequential calculate past criterion sequential achieve square prediction integrate distance avoid omit good fit sequential estimate eq statistic regard sequential leave define put asymptotically put distance want response q array tn
solve optimization submodular show monotonicity property monotonicity fa fa fa sum equivalent unique solution proximal submodular unique proximal u jj property prop fu fw optimum proposition solution single solution proximal unique minimizer minimal minimizer proposition set know large may problem restrict separable base many minimization prop describe detail certain say denote give base say set tight tight everywhere desire tight exchangeable tangent prop exchangeable dependence function non tight pair prop tight tight exchangeable conjunction property pair exchangeable pair let exchangeable tight proposition check support maximizer tight prop tight decrease true tight tb ta fa w prop prop deduce base optimality cone exchangeable give proof prop vector tangent cone cone contain tangent cone weight
rest paper integration system deal kb motivation investigate develop preserve crucial e integrate relational database dl cl paper implement solution semantic degree integration say kb dl combining cl indeed allow describe two case integrate dl hybrid former safe hybrid body rule allow resolution calculus deal run constrain sound answering query atom language devote dl I answer answer logic extend rule concept occur answer rule answer iv combination require dl log answer constrain resolution modify version calculus logic logic programming concept logic inferential mechanism induction prominent concerned automatically induce inductive clause form induction presence classify accord orthogonal discrimination characterization ground clause
nm usa many vertex topology attribute choose query learn much vertex stochastic block choose query two attribute heuristic consistently network represent vertex represent hide take way topology simple assignment likelihood q number tu tv direct loop self loop connect may reverse kind community specie interested
component distribution similar component decision make unconstrained project decision reach eight demand decision observe state decision weight equally available underlie hour ahead wind decide energy amount wind energy energy expensive price make excess lost day wind history two hour contract price c l current price hour current wind variable energy wind receive variable know hour wind data north american assimilation system location tx wind pattern contract price contract price price generate varie day year analyze separately year train wind wind weight base rule package function hierarchical day von exponential family unit sphere
mle perturbation turn fmri study movement intensity easily likewise foreground surveillance across represent another sparse high intensity magnitude
artificial investigate mixture problem include protein start cut iii depend task achieve theoretical analysis mkl novel base perform sparse scenario appear additionally offer complete derivation additional experimental extend previous publication novel illustrative improve publication mkl subsequently researcher project optimization mkl view unweighted svm six bioinformatic generalization bound analytical optimization also mkl composite kernel learn small medium scale structure sparse mkl discuss introduce conclude view cast multiple unified framework present norm mixing show comprise mkl include dual make regularizer loss latter cover easily structural prior incorporate regularizer begin sample unseen h form hilbert constrain regularization allow mapping consider minimize optimal avoid overfitte regularize note approach regularizer non arise inherent parameter adjust favorable convention regularizer arrive problem primary investigation regularization contribution show exist optimal multiplicative switching refer proposition proof prop prop satisfy regularizer contrary yield infimum regardless contradict yield show mkl implication ray class addition couple trade versa optimize implicitly search problem preferable model begin expand slack norm lagrange incorporate introduce lagrangian multiplier incorporate lagrangian saddle denote derivative reveal yield attain arrive ct conjugate maximize subject let ignore moment
band scale band transformation identify log band locate structure ht illustrate loss band first band simplicity select band first widely see frequency business line type year horizon select extreme heavy interest infinite suitable claim finance paper particular close analytic expression poisson compound property form es derive analytic es ii gaussian stable location restrict stable process include infinite skewness tail loss year stable denote univariate determine degree location term characteristic large imply heavy tail respectively member special admit close typically proceed characteristic discussion intractable setting efficient estimation datum random characteristic analytic mixture bank model positive stable expression es result linear proposition lose truncation implication general
modification demonstrate divide four modification nonconvex convenient two nonconvex lie partially outside justify overlap boundary even nonconvex correspond mention sec body body overlap intersection nonzero volume outline uniformly intersection bin change histogram mark index consideration tackle normalization probability could justify normalization unit convex eq disjoint integral normalize include four density component quasi common principle decomposition straight determine appropriately refined index additional produce combine symmetry distinguish histogram symmetric directly generalize
observable discuss misspecification estimation stand number another chen examine suppose observed contamination another independent measurement univariate provide measurement expectation measurement error panel
program check correct practitioner ad ad module modelling mostly due difficulty multidimensional difficult treat model noise package measurement extended kalman approximations order task multidimensional conditional work rigorous time require specification effect restriction theory point view variability precise dynamic follow diffusion triplet th expression log expansion symbolic calculation value dy dy du k du iy iy iy I j I j I I I p j j ik expansion j iy j iy
etc decomposition case vector energy rotation linear find neighbor question happen want concentrate conclusion empirical motivation code
phase perform letter kolmogorov exploit different reduce need low method sort cdf edu novel method
number predictor pairwise predictor indice fit spaced rule figure number percent explain total tend fairly numerical looking take fine grid stay solve notational convenience rule discard predictor discard predictor package safe rule elastic net discard actual extremely excess assume logistic letting likelihood discard inner kind investigate analogue regression max use sequential broken curve classification binary indicate particular http stanford global
approximation imply novel smoothing straightforwardly method determine distribution smooth filter smooth proceed filter high sufficient smoothing result derivation smooth gibbs sec provide concept numerical evaluation conclude dynamic measurement tb otherwise accord x purpose filtering smoothing approximation bc observe assume p soon measurement extend compute state given infer posterior map
input group note reduce close applicable however conventional sparse sparse consider treat case wide regularization parameter stochastic process mutually make remark derive general specialized valid input result weak deterministic great section demonstrate evaluate performance evaluation five
scheme value alternate complete linkage suffer inherent concept category construction concept section conclude remark section let denote metric point distance equal follow definition relative correctness encode multiscale persistent pair rr tr tx associate persistent underlie partition consist agglomerative hierarchical distance relation block easily notion discuss linkage usual definition agglomerative definition merge useful mathematical construct encode nature object together formalism extremely class share structure consist g vector x corresponding transformation etc composition assume ff f category side indistinguishable except category consider go refer category must identity two object three identity exactly exactly three identity must set equipped element composition
define include unit arguably implicit modification see external specify however partition novel coherent realization weight beta lead well random call generalize regard distinct sequence natural specification characterize application dirichlet prior beta outline basic compare beta hmm suggest beta exchangeability heterogeneity virtue beta hmm unknown structure section analyze genomic array investigate among al hmm partition tumor dna copy et extend latent markov dependence heterogeneous markov single adjacent du et dp state impose persistence dp mixture respect proposal assume persistence beta weight specie mechanism adapt intensity link breast conclude proof theorem modification rule characterize sampling beta characterize distribution define randomly
could seem global adjacent chain swap chain suitable confirm good ess temperature overlap prior size acceptance replicate delay rejection exchange range operator range surprising automatic acceptance increase prior ess discuss detail ess report simulate marginal ease drop subscript index precise marginal model replicate order ess order record large burn vector compute lead large large replicate ability deviation search strategy least replicate stability report time across ess matlab gb ram marginal inclusion average replicate triangle vertical blue triangle perfectly covariate contribute covariate blue surprising less situation hand small small effect evident second respectively posterior replicate small blue solid line replicate ess small posterior replicate triangle vertical two evidence ess able ex account detect ex surprisingly assigning replicate mark ess ess model addition receive marginal red triangle replicate vertical red dash line ess capture coefficient contaminate effect ess present variability red line solid vertical receive contrast posterior agreement respect see surprisingly main difference ess reach well stability expect towards drawback far apart compete ex ess
laplacian denote vertex inequality loose well cut expense runtime bregman much guarantee associate non eigenvector laplacian laplacian subdifferential value moreover subdifferential denominator eq constant eigenvector laplacian q sum anti symmetry positive component zero median odd even median eigenvalue laplacian constant constant direction subsequent result constant median sequence terminate find zero f contradiction terminate iterate second direct compatible eigenvector eigenvector nonlinear modification
merely behave define rate admit entry rearrange obtain hadamard e entry wise nonnegative definite eigenvalue interval equivalence theorem component eq formula definite lemma wise hand nonnegative sum frobenius chapter
algorithm datum arrive systematic trading trading intervention trading trade relate portfolio optimization idea make algebraic classic variance ordinary least construct mind certain regard trade financial attribute adapt stationary dynamically incorporate portfolio financial counter estimation mechanism technique consideration idea allocation devise trading technique dataset asset allocation multi period portfolio exhaustive investigate trading strategy computer machine standard finance article technique portfolio finance financial improve appear finance exposure typically et correspond instability encounter mean asset mention adaptive adapt environment finance two algorithm mean variance idea process
unique maximizer expect log contradict inspection fix maximum section e mention exchange exchange log near conventional em concentrate record system program request start uniform mle em galaxy fit model mean grid equally spaced deviation mle em dramatically reduce computing add conventional em appear increase per decrease appeal implement strategy censor review problem proportion numerical collect unit accord censor inspection failure subject govern
finish schwarz claim start adversary tf tf argument vanish regret guarantee vanish rate adversary one solve lipschitz optimization optimally function subset lipschitz conclude game set set game game play eq corresponding minimax player game minimax playing theorem corollary second conclude note get conclude cover appropriately extra application add break inside nest three arrive mention theorem loose supremum draw assume simplicity supremum equation follow bound pass bind repeat arrive bind next quantity conclude proof cover member close sub use triangle quantity last bound supremum value tree depth corollary element sense thus proceed assume arbitrary enumeration crucial non indeed member eq remainder shorthand hence union hence since n appropriately statement sequential rademacher proof payoff class moreover class cover eq define fix depth assume enumeration pair zero path cover
region train layer variable code learn mrf show state art denoise generally clear variable regard circular symmetry coefficient conjecture model cosine phase structure complex pyramid phase always corpu image however concentrated variable dependency difference I phase distribution identity express cosine bivariate term pairwise statistic angular
computable computable differ computable computable computable computable computable measure set computable let computable metric space let computable computable computable computable measure measure computable abuse computable restriction continuous topology almost continuous density measure almost computable almost computable measure version describe pair computable computable conditional description conditional even almost rule space enumeration variable variable random assign equal fall variable computable coin enumeration version calculation eq side consider function suppose continuous exist interval contain image fundamental obstacle admit able pair computable version continuous even distribution computable existence denote machine standard enumeration map index computable computable computable coin computable mutually greatest uniformly distribute expansion computable variable x lebesgue probability integer admit
net interpretation graphical section following grow xt xt case affect point indicate direction child conditional parent separation configuration child xt net clear reasoning separate conditional irrelevant generally irrelevant model generally speak case inference remaining obtain update low gamble guarantee inference coherent positivity hold also right smallest large calculate sign recursively root calculate follow equal eq counterpart first possibility message gamble tuple number ms belong eqs course reasoning form replace lead course take strong turn recursion tuple possible single equal node local message tree transform node local simplification sign greatest close continue eqs reach eventually ab moreover eqs find cx ce conclude separation draw anchor west node unobserved root x x root use extension message pass calculate towards root
analogous alternate section sparsity number informative would enforce entire well dna strength signal sparsity desirable microarray gene goal require exclude different gene active factor finish enforce pca explore x u v x discuss datum similar first signal simulation
acceptable recommend method slightly high nominal partly dependence partly get true believe power uniform event apply ks repeat show figure ks generally test somewhat
exploration standardize let mode eq treat marginal integration demand accurate select approximated th omit simplified expectation identity follow quantity suppose exist denominator let approximate q eq combine sum weight include take account convolution component several model medical proximity study belong implement goodness
sparsity case strong sometimes moderately suggest base powerful optimality variety balanced way modern design arise effect similar effect normally class testing relationship response fisher trial tool statistical clinical microarray important consider simple th th treatment assumed goal formally test analysis software package test nan identification nonzero reduce independent effect testing variance reduce chi square power maximize alternative invariance versus apply implement minimum compare adjust variance nan exceed max might reader margin small max substantial minimum alternative class kind likelihood close alternative distinguish entry absolute single particular biology
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order coefficient snr snr make pattern sort condition significantly boost section practically motivated generative arbitrary varied tend degree transform study capability attain detection threshold introduce generative model hierarchical multi ise denote internal characterize dependency vector characterize parent probability parent tree govern contact infection disease latent agglomerative hierarchical covariance leaf proportional denote root subtree contain cluster contain thus covariance easy verify covariance agglomerative perfectly recover graph covariance unbalanced haar lead binary pattern draw
join empty intersection join maximal next elementary full vc collection subset full join dim establish c disjoint element measure let finite follow cover
equation equivalence market strictly convex regular scoring proper exist convex function immediately imply regular strictly convex see imply n define equation equation suppose equation score particular meaning equivalence exactly realize market price occur know optimal I function kkt require market scoring market probability cost base finite score maximize regular achieve market price prediction market represent n recall l maximize expression n equation trade regularizer market regularize cost yield stable bind bind playing role previous imply market scoring regularizer conversely strictly regularizer view strictly rule perhaps scoring rule score scoring rule rule market market distribution treat market correspond understand stable
carlo events pdf belong carry histogram carlo reconstruct experiment simulation value pdf initial value event accord event th bin value estimator setup adjust variable physics consume simulation relate medium complex calculate histogram pdf shall
homogeneous state simplify value space case since henceforth chain random identically similarly matrix depend current affect state c pt traditionally define since observation suffice adjust triple finite observation definition model state observation make since permit time wish determine use past sense define sequence discard potential leave represent actually make random write hand hide process actual process situation make hide underlie actual mean practical physical underlie need take deterministic need way underlie observation precise recall probability letter letter underlie definition q naturally eq distribution probability banach measure variable maximal observation independent symbol symbol take place underlie observation independent union definition conditional represent information make hidden prescribed policy fix mean induction see function write aim
possibility dense analytic turn reduce chebyshev determined study stable deal take generate pz result say normal iff rational satisfying condition deal clearly indeed generate normal strictly stable even power zero within let generating represent sort
function conjugate smooth smooth derive proof therefore ball large summation v technique obtain project accord construction value bound therefore bind tight lemma arbitrary lipschitz lipschitz constant th iteration bind eq bound f simple require acknowledgment like thank verification breast cancer pe related order associate constructive comment improve quality remark grant nsf gm fa high structured sparsity induce structural either widely adopt kind example graph fuse efficient general structured spectrum induce combine proximal method subgradient scalable scalability simulation genetic feature arise science engineering typical response lie interested truly fitness nonsmooth lasso coefficient limit capture structural advantage encourage closely related idea explore structure response mapping output structure relate relevant development remain address develop structure impose order e group
derive multiplier bregman iteration mild irrelevant choice split bregman explain method popularity get equation inversion check explicit constraint square symmetric eigenvector root eigenvector replace correspond approach slow fast separable separable easy solution update negligible third equation straightforward run mostly update update definite symmetric possible demand decomposition newton
example gene investigate sparsity latent formulation small loading latent loading hierarchical incorporate sparsity prior independently result marginally priori loading develop context type dimensionality factor selection choose significant manual reversible jump mcmc many dimension approximate latent could mcmc define appropriate ibp allow specify number dimension nonparametric development equation th dimension include model inverse factor delta distribution infinite infinitely nonzero entry
trade relatively potentially formally kullback leibler third illustrate population desirable information discrimination framework nuisance density framework comparison parametric density value comparison realization component density outcome g g comparison common assigning usually correspond biological feature conversely thereby normalize since integration main arbitrary leave order issue arise sample bayes popular take arithmetic average minimal iid would likewise
high throughput flexible experimental protocol demonstrate feasibility towards automate throughput functional biology early wang extraction automate software preprocesse highly dependent report successful novel amenable root parallelization seminal et variation direct set typically show vary preprocessing depend acquisition vary growth kind hardware standardize remove issue ensure optimal standardized outline gray basic follow image outcome desirable
perform infer signal variance plus minus observation gaussian random centering effective much ideal input uniformly hypercube signal ten method coincide six process hyperparameter two day square mining run mean show group bar rescale always bar height bar cm problem work baseline involve derivation expect method realize
mapping extend create generalize document concentrate discrete relate document spirit influential differ fully local parametric attempt make visualize track word oppose collection important external resource wikipedia google wave framework tool present representation control space simplex tf vector applicability predict predict operation highlight significant structural change benefit collaborative edge discover
discussion united ep conjecture example case study solution abstract generalization variational theory physics problem relative entropy leibler reason measure family absolute show property characterize optimal dual absolute separate transition play decision deterministic transition strictly information resource dual illustration deterministic infinite unbounded infinite measure solution information theory physics mutually discuss simplest index measure normalize element correspond exponential course set banach compact later family multiplication exponential compact compact hermitian operator hilbert characterize absolutely absolutely imply mutual continuous property important measure direct normalize uniquely define transition absolutely non within deterministic transition another perhaps sense mathematical variance attain physics distribution maximize entropy maximize capacity geometry banach construction quantum optimality family measure one seem time criterion deterministic transition strictly sub importance communication algorithmic complexity transition input mutual
maximum md ir n cg inspire framework restrict learn cg form cg machine technique often fast solver solution stress gradient approach learn via approach mention abstract partial least square euclidean norm partial competitive benchmark approach usefulness examine particular rule optimal rate main next
pl proof denote problem compute success argue proof directly transform compute formulae disjoint therefore various tackle pd engine inclusion principle propose symbolic report diagram thousand differ avoid disjoint require proof depend bind proof drop proof become expensive ultimately infeasible path algorithm slight propose use formulae obtain probability work context method remark describe set mean proof never formula formulae subset query success probability shorter always probability information decrease carry include branch motivate rely threshold stop formulae tight level proof current formula derivation stop due reach goal threshold probability shrink algorithm proceed iterative
sdp draw define concentrated sdp generally way start procedure effectively gaussian sdp provide sdp natural function distribution diffusion walk sdp want parameter relative convex infinitely invariant semidefinite think convex von concentrate vector possibility main implicit via computation matrix sdp play role relate structural previously
weak symbol follow real symbol exist symbol self adjoint lead give word depend fouri sense h unitary boundedness mean hausdorff uniform globally operator boundedness localize function construction modification prove set open mean increase flow neighbourhood neighbourhood necessarily unbounde modify follow modify compactly way estimate specifically state construct notation scale angle depend enough contain angle access complement formula decompose block provide square elimination invertible triangular formula effective operator auxiliary operator matrix operator invertible say pose successful reduce nonlinear spectral problem zero valid left verify denote integral center effective review formalism later banach check property immediate recall extend flow consider near operator replace scale confusion likely occur denote quantization letter focus subscript aim theorem canonical neighbourhood kt mutually allow energy layer use arrival notice different equivalently end dynamic energy map fully
project g state classifier sample feature extraction step integral report several publish biased prediction scheme randomization design sophisticated subsampling etc gene vast amount gene datum last decade analysis apply biology g nucleotide snp array analysis generalize sciences health financial etc like relevance gene microarray experimental result appear dimension gene expression level show dimension reduction binary extended multiclass prediction g regression straightforward currently extend latent g etc author support application e increasingly handle
choose correspond root rest sufficient ensure perhaps obvious solution cover lemma incomplete intercept rational last design polynomial system parameter estimate guarantee write consequently polynomial constraint simplify design satisfie ensure argument design polynomial brevity omit scaling choose corollary also upper numerator numerator optimal count hand root nonnegative zero distinct zero interested specific parameter high degree generality information pseudo compatible develop compatible equivalent note optimum complement characterization semidefinite zhang eliminate finally simplify essentially linear criterion semidefinite come whose coefficient main appearance show rational rational design union find
iii tail fy fy short medium tail long generality short tail tailed satisfie von satisfy von monotone satisfy short short tailed satisfie failure failure hypothesis medium hypothesis test es medium tailed alternative hypothesis es es significance favor reject test long tail tailed sub classify asymptotic hazard super moderately short slowly vary theorem provide consistency test weakly short mild need theorem short tailed assumption theorem statistic short tailed condition mild distribution function denote log long tail short tailed condition satisfy vary long e seems demonstrate many unable long tailed eventually test test vary result
pseudo derivative jx jx j jx yy jx jx risk argument prove nf right nx f equality w display ny p n definition integrate p well duality argument expectation
clearly advantage list heuristic fit fit neighborhood neighborhood bind cluster choose cluster identify cluster outlier case report sensitive removal naturally respectively htbp misclassifie three traffic ms ssc r c pt median median ms ms standard deviation misclassifie database r pt mean median median ssc l c ms ms ms ssc data dimension fix follow mixed fixing generate hybrid linear code http edu subspace uniform ball multivariate generate outlier cube maximal former origin ssc mixed support specify dimension subspace mix unnecessary instance misclassification record algorithm e various artificial hybrid modeling affine obvious outlier g ms case unlike affine particular non ssc fit heuristic effectively estimate noise running time algorithms ssc algorithms function kf
performance unseen signal dictionary suit representation bind depend number problem minimization erm prove convergence admissible bounding sample rely consist naturally represent similarity inner product diverse text represent extend usefulness set representation practice diverse field etc dictionary motivation coefficient typically compression commonly store representation condition unique dictionary chen atomic sense sparse representation retrieve need implication dictionary simultaneously choose prior motivate dictionary use compression denoise dictionary extensively
rescale forward backward normalization approach recall recursive ji sx k rescale backward jx manner sx limit label align mt mt decoder base base transformation break first claim concern write upon require eq original recursion handle straightforward consequence continuity respect claim fourth claim maximize arrive characterization viterbi practice thus decoder side decoder transform beyond decode let v break rule exercise calculus positivity make note decoder unnormalized limit decoder normalize transform let cm align leave I I decoder transformation decoder use break reduced factor numerator denominator expression induction variable proposition jx jx recall numerator observe kf sx sx give I example last hmm emission alphabet initial distribution emission f suppose show original bottom transform decoder power hybrid indeed satisfy also realistic hmm indeed convergence hybrid decoder viterbi naturally generally summarize purely algorithmic align fail viterbi viterbi since understand interpolation choice seem practice except trivially underlie normalize well aspect mainly rescale indeed work analyze clear general member regardless explicit make interpret complex cycle g viterbi cf apply decade aware idea inference plausible reason see mostly lack illustrate viterbi task predict structure purpose base entirely first hmm particular allow significantly elaborate include attain
scope current argue statistical challenge ask classical approach dual type investigate encode character alternatively genetic fix geometric multivariate character specie variation closely challenge selection address statistically discussion suitable bayesian
anneal long history trace maximum twice free strong aic number minimum description principle mutual closely source approximation quite space cluster dissimilarity weight vanish dissimilarity clustering rely ultimately fail need identify theoretic perspective uncertainty measurement resolution hypothesis program set
tx belong class henceforth integer stand convex continuous non negative ball relax subgradient projection mapping employ scale path carry view study project minimizer subgradient subgradient class quasi mapping large strongly mapping give save reader metric onto weighted ball j component follow sequel regard definition assumption subgradient projection mapping c n lx k jx j j stand convex hull hold true hold relaxed mapping save calculation standard easily choose w l establish subsequence example section function sequence already online learn class strongly construct wide applicability even large usage algorithm sequence nonempty close user index sequence slide
decrease gold proxy air likely take proxy identifiability include proxy somewhat average coverage fig discrepancy diagnostic reasonable diagnostic expect scale scenario discrepancy large scale scenario scale discrepancy diagnostic estimate cause identifiability suggest diagnostic rather discrepancy diagnostic diagnostic distance drop diagnostic scenario fit separate month mid proxy four km grid spatially overlap km assign cell fall take cell purpose hoc retain essential character computation miss due cloud smooth additive cover month location cloud operational model nearby road smooth spline road three road four km spline point source five per year source km consider calibrate height scale comparison co relate proxy month fig find relationship pm ignore proxy pm explain discrepancy fig consistent exclude prediction correspondingly assessment proxy predictive worse exclude likely assessment quantify qualitatively hold suggest area informative strength gold able exploit proxy pm bottom month proxy mid panel u panel prediction proxy domain analysis p inclusion mid without proxy isolated proxy proxy proxy proxy separate output proxy km area east give
adaptation account positive corollary adapting double laplace scale gibbs sampling augment p relevant posterior section treat marginal mle describe facilitate bayesian cdf pdf representation follow likelihood parameter posterior conditional truncate normal complicated infinite prior I combination highly inaccurate even evaluation derive conditional rejection prefer integration helpful mh obtain alternate cdf normal proposal accept mh good proposal improvement acceptance rate high posterior therefore scheme preferable acceptance increase mh rule thin draw fast fold sampler speed evaluation arithmetic square root
indicator regular design coincide design design regular design non fractional factorial design fractional factorial design design regular proper design q indicator coincide indicator indicator indicator complete constant small give fractional factorial design full factorial desirable identifiability add real factor relation purpose basis factor formal include let fractional factorial suppose additional among new q conversely
dependency correspond evolutionary breaking markov slice sampling perform method flexible expressive family unobserve end task address arise latent nonparametric constructing provide posterior explore variety density nest mixture depth tree branch also useful multi task vision motion discover develop unbounded depth live node se unlike exchangeable
specific simulator contour gp sufficient run simulator must complete ei yield good improvement multimodal ei estimation minimum branch outperform sequential step dramatically computer experiment estimation employ simulate realistic consuming evaluate desirable whereby initial surrogate spatial surrogate combination along suitable subsequent set trial great improvement interest popular trial expect criterion varie point exhibit local optimality greedy trial interest high search away previously design small simulator point change ei require another ei find focus branch contour simultaneous estimation maximum minimum bound branch
construct equation step could construction strategy construction third say game scheme finite half payoff base might game resp choose generate player receive mapping extend si iy si distinguish player stress realization law realization observation nature choose outcome pick choose right label payoff payoff law receive probability cc cc payoff signal action response generate generate action player response say behavioral resp player nj external framework player average payoff uniformly player might pure mixed pure bad
ks chains simulation split mcmc batch quasi limit ks experiment statistic result beta truncate modify autoregressive order besides precision influence parameter order c precision rmse c rmse c square rmse acc effective ess average ks diagnostic batch precision precision parameter behave differently autoregressive truncate rmse value get third column list parameter acc acceptance rate improvement acc autoregressive ess draw markov order ks last acceptance hypothesis convergence display chain get outcome prior well truncate autoregressive rmse calculate modify prior acc slightly ks
simple method diag lastly diag replace every give may tune validation cv suggest graphical advantageous simulation decide select cv pay attention choose cv bic un replace alternative bic select sparsity k binary standard coefficient consistent odd preferable section well previous likelihood small conclusion confirm sparsity namely positive correctly identify false positive incorrectly association well identical positive harmonic oracle regard conduct penalty neighborhood coincide alternative would approach worse select bic recommend line package adapt allow un standard logistic un method value small
size especially cccc linear setup lin whether code totally response generate plus minus mu mu minus mu mu minus mu mu mu minus mu minus glasso adaptive wang frequency correct replication wang glasso tune aic degree freedom lin equally spaced result glasso model contrast produce work cccc glasso composite ii lin create previous code true totally group composite sometimes wrong interaction correspond fitting always order theoretically yu cccc glasso novel aspect use shrinkage give consistent ensemble produce averaging inference unify group constrain penalty yu implement mu plus adaptive lasso
empirical image image address separately frequency factorize q explicit law law robust paper gaussian consequently observe q naturally embed classical choice practical algorithmic update law law conjugate law parameter pdf law improper informative jeffreys insufficient reason nevertheless choice limited law possible easy point law build multiplying law explicitly rule law normalization describe however preserve frequency addition prior eliminate integrate accord appendix law parameter remove penalization law denominator remove dirac equation integration respect denominator
similarly detail classification sim via author sim classification use gp sim integrate improvement combine statistic derive illustrate constrained arise health policy combination include sim function classification context design support sim package sim contain use hierarchical difference similarly orthogonality identifiable orthogonal matrix care rw proposal reversible mcmc finally gp sim gaussian acknowledgment grateful foundation education whose constructive comment lead three sign sample gp sim solely identify label aspect direct true gp discuss employ possibly unit involve collect reference find sample obtain collection sign wrong index plot average index easy much one rely predictive perhaps reliable automatic involve look directly
allow step hence possible evolve fix learnable element essentially see set choosing reduce function rather process always high unless distribution efficiently learnable every access set polynomial circuit circuit access construct output access f imply learnable concept efficiently learnable exist inverse evolution monotonically tn pn sn circuit representation theorem require point algorithm explain uniformly nee sure high present mutation denote representation high candidate generation pn nr mutation sn rule tn fr fr fr empirical mutation empty fr tn sn establish slightly representation increase candidate influence empty pn sn improve decrease substantial transformation transformation
satisfied optimum optimum identify solely henceforth though convex feasibility incur computationally problem establish rs problem linear equation np instance integer rs remain rs rs establish search optimum computationally adopt square alternatively consider error handle every per reliably identify sensor envelope large g use hard efficiently implicit norm albeit unconstraine norm residual circle sum euclidean norm optimal location constraint either interior relaxed rs reflect link interpretation section versus ls rewritten mention sparse establish connection assume non reconstruct determined equality minimizer pseudo block sparse degenerate single reduce regression cf cs pursuit thorough instead substitute close namely convex yield tight norm apart logarithm determinant propose box kt sufficiently strictly tend
confident drop precision baseline core cascade exponential power law generate cascade edge cascade choose cascade point three much break particularly especially careful cascade cascade cascade edge cascade performance regardless work reliably regardless refer similarly network power law dramatically drop heavy exponential problem much hard break drop performance break axis transmission reliably recover achieve forest bit important note network free core remarkably seem depend try type cascade sharp drop greedy start low marginal gain edge probability make mistake cascade spread easier never identify transmission edge identify cascade examine cascade cascade break cascade transmission event cascade transmission event cascade amount cascade transmission recover edge big produce cascade interestingly cascade cascade large infer parent parent break find opt right therefore opt computation per algorithm localize hierarchical exponential localize improvement order solution practice use infer node matter hour gaussian experiment far true draw break add cascade even function add time reach increment interestingly break real cascade able cascade jump technique miss
generalize suitable demonstrate finitely ordinary differential regard change transmission behavior process gray rgb analysis action potential cell detector ensemble possible number follow delay enable process generic response well phase jump transfer periodic class process applicable framework process event basic enter device typically discriminate arbitrarily detector nature upon arrival action might release component many network
inexact line search require one variant search iteration dimensional acceleration dynamic depict difference three since em implementation small iteration obviously cost run mainly relative next set compare time line search optimize detailed line unconstrained appendix nine algorithm set simulation effect performance accelerate em need need implementation fast fast term cpu vs flat quickly flat clearly let covariance matrix
may rescale actually without generality keep rescale therefore observe property fast asymptotic behavior function present result fix assume differentiable taylor take g odd z eventually yield achieve prescribe theorem depend vanish super polynomially observation distinguish ridge function b ga neighborhood f r u fx ga u ga point evaluation respectively hand notice condition impose give function class generalize approach ridge obviously ridge appearing hold motivate algorithm access direction respectively collect directional similarly
quite complicated quantity asymptotic exist give equation solution mean table stock market usefulness share package bank portfolio package monitoring share quality realistic statistically desirable market share package portfolio choice package characterize bank bank invariant share portfolio share order package priori qualitative characteristic stock simplify real assume possess financial capital portfolio share resource package bank analysis modify package quite portfolio market investigate theorem exercise compete portfolio statistically strategie markov
median self exposition omit literature logic discuss idea search database content content discuss database structure learn computational dedicated structural updating phenomenon language formation exposition evolve already list human thought memory last way read numerous replace space field exist grateful advance comment acknowledge due abstract set partially endow implie say borel natural sub easy hold say say subset say element resp mean relate possibility observer sub object graph state embed structure memory long implication question structure unchanged property favor contain big interpretation concrete return arc formal tb vice versa relation choose consequence sure n w would change relation hold complementary mean relation hold see hold graph paragraph originally terminology fit intend application coherence coherent coherent call selection mean relation iff symmetric back endow structure w visualize simple cube answer every turn incoherent adjacent cube symbol generate proper coherent skeleton cube exercise exercise finite nest element cut exercise converse know exercise disjoint immediately kind vertex vertex pn union set shortest connect
insight ps bounded rhs fy fy exhibit ps mcmc score ps show continuity sum finitely point rate mix gold apparent extent practically big prior particular remain normally start finite moment posterior likely moment make function integrable suggest near converge may slow posterior degree rapidly contribution large provide insight mcmc converge mixing support figure maximize remain make transformation parameter go high maximize nest parsimonious inherent showing prior stand issue maximize likelihood nearly concentration around mode namely range space range lot fluctuation mcmc mcmc output tell proportional see remark lot explain score decay spread bf reflect figure ps factor detail check point accuracy well test remark two nest modification ps new work quite differ log discussion would effect theoretical path ps notational
involve cancer reference ratio intensitie sample plot position segment various particularly level possibly encode assign segmentation copy several provide apply popular deal resolution assign discretized unit clearly array platform interest copy protein nature dna piece piece relevant unit association clinical simplify terminology array although approach enable consider resolution region region neighbor discretized profile contiguous direction clustering consider spatial pattern perform simultaneous cluster wise idea multiple testing
much stability properly select learn big rate demonstrate usefulness five microarray gene expression commonly scientific classification gene varies implement effective reduce classifier case issue associate factorization issue scope issue specify cluster origin knowledge one stability varied guide whether cluster consider recently al interpretability latter correlation original variable microarray currently go record
selection band jointly kernel spectrum decay spectrum mkl model nine toy problem goal spectra goal subset kernel well kernel
term unlikely fraction step additional step bring fit threshold variant amenable accordance role likely fraction adjust likely correctly allocation acceptable allow correctly section mistake role formula mistake extent positivity formula favorable reliability target code moreover evaluate mistake arise threshold standardized product
cardinality gap introduce slightly form allow replace gap suggest david establish elementary relating gap weak name sensitive result x let family borel stationary ergodic asymptotic dividing minor modification positive equivalent stationary process measurable importantly use proof key nevertheless family condition nice measurable countable sub immediately ergodic extension drop exclude example process illustrate rotation measure composition xt stationary ergodic easy see f give sufficient case show ball df process give
variable follow next martingale w k martingale q variable sign follow hoeffding conclude case modify old inequality word argument finish q real choose bfgs bfgs hessian gradient goal function mkl model apart investigate mixture coefficient weight single highest sparse cover entire setup mkl elastic
two nature consume interested surface shape site common protein efficient search scan surface two candidate bind site mcmc method small confirm match difficulty use efficiently htbp htbp htbp htbp sample last sample variance calculate hold constant conjecture school sciences uk recently shape markov monte improvement exist convergence jump configuration match bind connection gibbs configuration point challenge application bioinformatics vision matching set matching may relevant object
integrable partial thus optimal smoothness randomization digital nets rmse integrable partial derivative result necessary net describe digital net underlie randomize stem function vector ds sd f monte sample aim convergence write bad q c star choose z integer b
determine min I unlike particle perturbation ergodic readily parameter achieve rate present employ langevin mala ir qr density perform step retain average acceptance ratio ergodicity mala perform equation computational expense exhibit mix physics ratio successive density obtain update importance identity estimator multiply normalization require section basic propose inner subsection compute loop cardinality expansion gain exceed tolerance desire temperature break equation p describe calculate temperature often interested landscape temperature goal fact firstly landscape nearby
friend event neighbor low friend event group high overlap group friend determine diagnostic diagnostic across different diagnostic friend group neighbor diagnostic iteration hand type reach group event group five friend event indicate customer observe k correspondingly converge show isolated reflect convergence approximate equilibrium reliably equilibrium bias e connectivity discard number run collect verify remain discard reach stationary contain user keep remain formally qualitatively burn determination also inspection run fig reveal walk note fm method stem status network rich base connect isolated individual user tie call walk base particular reach graph might typical belong friend friend neighbor friend friend friend
intra level shift particularly affect trading consider tailed inter day case shift ignore occur real abc observe run sampler perform model innovation analysis synthetic study setting table estimation mmse mmse mmse mmse mmse variance simulation markov away parameter ignore shift significant exact non abc mmse intra shift propose consider identical stable tolerance abc table versus basic mmse mmse mmse mmse acceptance mmse set start away model incorporate summarize mmse inter day section accuracy important algorithmic trading price series deviation trading basic mixture abc cd sample min interval market perform contract worth produce length raw circle represent open inter boundary h transform deviation perform batch day batch average sampler table c c var var mmse var batch demonstrate account
large plug code word modify plug sequence redundancy code unless since must case must member code almost lebesgue I rough sketch detailed theorem special say pp arbitrary constructing source rewrite redundancy convenient family g take divergence trick
way prediction treat possible art machine use prediction sized pair short imbalance predict link anomaly dendrogram connection prediction numerical al co occurrence evolve compute centrality evolve aggregate adjacency similar netflix rating predict focus rating change preference time et filter tensor web link email communication exploratory several link author conference service context application internet traffic temporal combine slice single matrix summation also bipartite graph analyze matrix fully temporal signature cp temporal cp gain computing use matrix see datum auc tractable however behind new link tensor tensor illustrate predict link underlie process tensor incur
significant amount expect bregman iterative usage wide require differentiable treat explicitly see proof detail theorem subproblem involve optimality solution order p v formulate compare denote subtract side subtract fourth side q substituting sum subgradient involve therefore lead eq prove whenever contradiction subsequence height bias hinge complicated observation l ns subtract k k ty k inner product second third fourth sum subtracting
strategy select include intrinsic another default informative change comparison comparison performance oracle assume tuning selector selector relaxation normal two group limitation net ridge design six dataset uncorrelated component iid simulated ii response multivariate simulate example multivariate gaussian gaussian correlation simulate assess tuning selector elastic minimize one simulated measure performance error mse
relationship bi graph special direct marginally bi like dag conditioning vertex graph page domain vertex acyclic remain variable dag bi suffer example graph parameterization introduce bi vertex contain probability parameterization operation parameterization form complete sense however price exponentially conditional table bayesian cumulative convenient direct additional term factor graph purpose bi direct cdf parametrize sufficient cdf direct graph
modify memory angular momentum evolution alternatively correlation term take physical contain problem autocorrelation beyond program author upon support scientific fellowship cluster maintain national job grant grateful ann discussion anonymous paper institute university nature system measure angular body reliably relaxation rate angular find angular momentum walk slow
bound follow strictly continuously continuous weak topology weakly continuous attention continuous functional weakly definition topology directly weakly continuously banach convergent bind dominate g whether weak concern hilbert weak continuous set weak bound weakly compact moreover subsequence weakly necessarily vanish uniqueness convergent note reproduce kernel space pointwise thm thm thm example thm predictor present banach intensity parametrize reproduce result representation log latter additive specification estimation functional linear enter specification inspire multivariate
variance extent residual another correlation numerically row total give mean generate connectivity zero equilibrium correspond exclude deviation divide generally stability divide figure across outlier confirm simulated generalize genetic could unstable ga ga computed fitness select removal exist follow sufficient fitness e still highest obtain across property unstable respect examine sum row heterogeneous system stability computation rather together confirm stability ratio system numerically term review causality simplicity extend describe satisfactory decomposition stationary lag eq transform x lag characteristic lie outside circle condition residual split eq spectral decompose part causal address transformation leave causality uncorrelate whereby split causality frequency ref establish fundamental motivate relationship g
measurement affect traditionally affinity show define measurement prove unitary replace compressed dimensionality problem dimensionality datum costly dimensionality tractable apply transform feasible support domain transform detail compress random pose sparsity recovery result compressed prove exact recovery ambient compressed isometry q bernoulli fourier random sensing provide measurement uniform let x eq take root give q divide subtract satisfy c ci measurement rip close gaussian find bernoulli row fourier
dp dp paper define prior choice prior encourage natural dp normal induce estimation interpret regularize covariance regularization arise introduce arise introduction encoding cluster independence unstable similar although specific generating translate pair conditionally conditional independence derive regular arise analytically tractable graph resort mcmc extensively year divide gibbs sampler exchangeability explicitly mix jump sampler paper marginal sampler option indeed integrate precision sampler dramatically burden proceed joint indicator conditional j l th exclude cluster predictive cluster
user interference whereby secondary channel channel optimal channel match user sum throughput maximize scenario two user e link assume show throughput secondary user optimal secondary allocate channel user allocate channel aware sequential bandit matching channel throughput channel allocation random convention variable reward permutation channel slot slot apply storage update slot polynomial matching problem bipartite channel tight short source transmission delay minimum sum I maximize maximization problem clarity
example k deduce give range reduce estimate large almost entirely close hand lemma unconditional distance order since mean begin mean eq poisson formula albeit rather reach contribution area great st research uk limited national nr scale section evolution molecular different sampling
hamming set eq proposition attain exponential particular exponential sufficient q appendix ham ball relation coding theory convex hull code mm let strictly convexity mm bit elaborate part boundary dc c cc c induce homotopy sphere interior mp mixture must number p family without row hausdorff
review validate formal presentation count traditional argument reproduce sketch completeness qr factorization rearrange arrive imply lead deal inequality partition adaptation pp sort singular resp resp follow unless require amount lemma result show proceed qr transform complementary dual rank continue towards constraint always would inequality increase unless uniqueness conclude factorization affect uniqueness within explicit distortion version respectively general provable change much solution assume theorem towards confirm leave norm specific practice however assumption uniformly chance
short memory fractional artificial sample ii estimates ar spike lag pattern fractional lag inspection indicate period seem may decay period strong structure I apart estimation different table parameter calculate frequency display empirical square ols small ht c mse short accordance may reduce bandwidth however task practical behavior value may indicate short part model fractional affect whereas fractional affect estimate small regression
uniquely copula function dependence translate function variable transformation preserve novel generate multivariate specify distribution nonetheless marginal correlation frequently strong generating marginal equal organized idea simulation accommodate plausible range detailed example bivariate family uniform beta minimal correlation correlation call al never summary
rarely practice algorithm concern rest illustrate et fisher choice algorithm start uniform respectively advantage count potential algorithm display
claim two induction yy theorem diagonal sufficient yy xx dependent standard vc origin necessarily lr yy yy yy right side sphere require lemma norm variable psd yx u ds v eq matrix fourth moment psd exist depend lemma extend n column far psd matrix ij center ij ik
regression aic bic summarize bic outperform second explain cc ccccc ccccc size bic aic behave candidate dataset nj keep dataset bic table summarize comparison uncorrelated ccccc ccccc size aic bic aic bic single e set keep working certainly true nonlinearity aic model summarize conclusion aic cc ccccc ccccc ccccc c aic aic bic new criterion add variance model assume keep rest assume model model aic bic summarize result frequency aic criterion indicate usefulness capture
let need convenience continue moment trivial counter conceptually length entry vector random skew stable I analysis stream matrix entry demand stream prove estimator unbiased see theoretical property moment skew generate maximally skew random variable avoid q simplification explain show accurately represent since straightforward stable define cc sample harmonic moment simplify unbiased asymptotic adequate unbiased mean
neuron former fire fire post stimulus almost spike even stimulus per presentation entropy spike paper determine train priori take optimally markov spike train length confidence check demonstrate quantify algorithmic content quantify spike quantify spike simulated train train record layer demonstrate increase structure practical technique quantify structure intuitively neither highly order strictly periodic spike train think since possess generate order spike organization neural activity would imply calculate stand assign high complexity train reproduce high algorithmic rate sort entropy desire state define condition event entropy differently use symbol string train string appear spike symbol string markov bernoulli exhibit possess extremely future process note train hmms state macro macro graphical hmms different linear kalman encoding decode recursively constitute minimal drive formalism generating influence implement spike train neuron locate reflect neuron internal nonetheless neuron neuron state appear stimulus detect stimulus
suboptimal study use digital communication orthogonal multiple channel estimator digital quantization scheme analog transmission unknown symbol symbol mle exploit channel complexity rapidly mle performance low communication snr result suboptimal digital communication outperform transmission channel quantization decentralize multiple quantization superior strict constraint quantization quantization relative high com suboptimal decentralize wireless sensor access digital transmission training mle implicitly reduce propose estimator signal datum level redundant traditional fusion diversity communication observation perfect message analog digital transmission simulation digital communication estimator analog transmission channel energy quantization superior quantization medium observation noise ratio level wireless sensor physical processing decentralize decentralize fusion sensor process observation fc inter sensor
mean assume ergodic generate cluster consistent sample variant require regime length formulation literature model hide model example bayesian cluster formulation generalize one homogeneity two test generate distribution cluster process I markov show ergodic binary ergodic demonstrate asymptotically joint stationary mix consistent distributional distributional
function cardinality variable independently frequentist freedom factorial lead submodular submodular function marginal likelihood associate highlight factorial exactly function extensive factorial provide trace statistic w large lead norm however analysis apply algorithmic sparsity norm norm unit ball potentially number vertex face cm get descent compare differentiable particularly consider accelerate variant apply efficiently p equivalent minimization namely fast submodular show solution moreover minimize directly unique submodular also
jacobian density differential privacy remainder excess minimize convexity objective regularize follow excess minimize classifier objective risk minimize perturb optimality q use cauchy schwarz combine probability least prove tight trade interested classifier perform new source private perturb function risk classifier expect value
pearson assess result produce network entropy reject pearson slightly mutual deal contingency table shrinkage score criterion grow hill combine test figure hill algorithm date large display variability min hill confirm make instability h statistics monte carlo along fundamental
note condition write event trivially assumption f consequently condition turn trivially union coherence modern processing arguably become nevertheless still think million entry design seem reconstruction restrictive variant thresholde regard carry thereby fail optimally worth point past variant perform illustration regard key model asymptotic deterministic vector nonzero result use multiplication solve regularization trial amount average take selection order second apart thresholding selection analyze agnostic understand true selection point conservative nature small scale still room reduce threshold mainly loose conference reduce experiment regard specifically kf f respectively set carry prove somewhat conservative nature constant contribution low complexity matching pursuit succeed generic deterministic average used establish carry recovery irrespective phase nonzero collect concentration
overlap consequence strength weight learn imply learn variance govern induce furthermore pattern duration obey time basis capacity draw replica cluster probability threshold pattern compact shape typical accord classification yield spike representation ir implement ir entire solution
encourage induce penalty cost option unnormalize low unfortunately
meaning integrate expression relation aim human like ap user share different language different color position play describe game object user try select figure contrary image user guess describe root language constraint mean computing learn semantic description step need collect description shape description feature generate new description learn web user sort semantic share constraint
induce rate drop significantly specify bandwidth direct connection link represent start store forward switch delay know remain duration traffic path traffic link notation denote interested determine large traffic flow almost seek define complete transmission specified period denote rate available available relie develop bandwidth fact moment link relationship rate p r establish satisfy determine satisfy pair tight imply l link formalize employ train measurement form network available bandwidth model available train evaluate outcome whether train instant path outcome specify probe link sequence form credible interval tight
code length locality image image constrain agglomerative merge region adjacent texture effectively deal majority texture window intersect boundary region degenerate window neighborhood region reliably estimate window size degenerate marginally deal propose size start window ever degenerate reduce please hierarchical htb window nevertheless new much accurate image much segmentation result summarize texture encoding distortion max window stacking construct maximal maximal r r w mr l j require determine segmentation measure segmentation image scale optimal solution distortion parameter segmentation assume user ground truth
parameter analytically natural equation employ lead quantitative us quantity case np definition rewrite analytic account prediction translate prediction matrix linearly column freedom remove degree fitted remain necessarily linear independence linearly linear compose arbitrary discussion conclusion give practice freedom cm fit motivated argument fit prior must negative figure demonstrate two prior flexibility
examine daily return gaussian dp concentration use rely iw sec hyperparameter process noise return fall start agreement give package fall bank joint economic fed cut rate exchange bank posterior display variant towards self rapidly redundant align bar bar roc bar bar roc hdp towards hdp hdp hdp table raw daily overall point hdp hdp hdp ar non roc maximum window probability use advantage hdp hmm sequence hdp return observation process datum center around roughly take outlined hdp regime case dynamical physics motion complicate dynamical description force appropriate herein target tracking application dynamical know target define transition dynamic equivalently acceleration explore along experiment multiple demonstrate flexibility al variant et structure switch dynamical describe phenomena utility hdp ar hmm one able volatility stock exchange ard sparsity induce flexible variable
markov spin dynamical nearby apart dynamical follow term dynamic nearby initial indistinguishable algorithm closely term coupling lose initial configuration coupling chain sampling coupling applicable carlo coupling element entire coupling avoid couple large large space avoid ise heat spin patch inspire rigorously generate
look algorithm carefully exploitation exploration evy well commonly good cast exploitation solution use walk obeys adjustment search evy potentially efficient step large far fortunately solution neighbourhood effectively diverse evy evy good lead simulation insensitive dependent tune specific
efficiency boost class boost boost additive tree multi logistic I adopt flexible weak learner tree parameter log otherwise typically adopt build gradient newton first information loss construct derivative respective split
interested beyond enhance develop proposition see paper steady queue length g scale version use proposition mean close p shall stationary say steady queue length see give subset guide construction efficient splitting develop behind splitting shall divide position target put j nc nc induce discuss intuitive becomes place splitting proceed follow weight initial particle say reach integer child parent child particle start particle particle either level apply weakly splitting algorithm unlikely particle motivation construct balanced convenient formal mechanism indicate sa
draw distribution draw uniform implementation function evaluate double library double diagram figure quantile double double apart due taylor expansion zero peak due peak error
solve optimization q relate together ie covariate model version graphical correspond prior incorporate place laplace ij b zero definite central obtain iterative construct lead procedure modification benefit generalization group hyperparameter negative belief focus keep tendency observation optimization particular distribution map estimate motivate primarily deal bayes estimator loss issue address perhaps
average degree degree degree predefine link turn characterize frequency give sub increase heart possible etc assign related sub proportional sub graph parameter study maximal model approach study frequency object simple order sort scenario stop network higher computationally hierarchy self describe nature turn theory construction adjacency randomize locally matrix
situation homotopy smoothed parameter solve solution next prefer homotopy large smooth warm start often allow homotopy constant update weight vector typical svm svm programming l unified form lp svm calculate lipschitz loss therefore smoothed hinge solve algorithm regularizer hinge loss hinge calculated smooth parameter eq initial homotopy l svm regularizer eq sum quadratic hinge loss lipschitz ls solve demonstrate efficiency
usefulness deal dimensionality rank absolute marginal rank covariate result rank covariate name screen sis marginally iterative sis begin response covariate sis sis meet sis sure independence sis linear robust several important sis cox proportional index true cox sure procedure satisfy property select model tend partial single covariate x large marginal utility correspond survival rank covariate covariate index especially formally show noise covariate expand sure property lot
valid asymptotically valid later er rao tend introduce concentration inequality depict one find independently arbitrary vector tends exponentially tend straightforward show eq want concentration want simplify construct index exponentially approximation asymptotically approximation orthonormal index probability tend exponentially orthonormal eigenvector eigenvector approximate substitute q say enough exponentially concentration similar state substitute
relative error probable make unnecessary set batch instead possibility iterative split batch statistical reduce entropy lead linear positive spectral eps default focus optical typical easily transform transform axis modification work start recover exactly deviation different approach fig optical default model optical peak transition exact default contain default output calculation similar result broad spectra organize introduce formalism
available domain coincide coincide consequence normalization concern without assume universe important include incorporate interest crucial independent identical whether term factor affect rank summarize rule make inference information piece know relation likelihood update process emphasize relate contain completely insight smoothly incorporate
alternative sampler vector time efficiency markov utilize sampler large become computationally impractical involve variate full sampler difficulty utilize reflect variate variable ensure update acceptance mcmc methodology line mcmc several mcmc basically mcmc online way though markov simulation preserve result variate gibbs typically bayesian automate alternative adaptive sampler additionally bayesian perspective tackle bf analysis rank e lag lag work lebesgue denote unit vector markov chain transition realize adaptive sequence index variate model formulate discuss integration rank include discuss justification select respect explicit
high document plug naive classifier accuracy would common identification address compete application letter unknown also three review across letter approximation numerator factor quantification procedure application writing collect measure document kullback kl chi square verification difficulty create write accurately observe score potential resample subsampling writing contribution equally stage development writing acknowledgment manuscript give support part contract investigation laboratory name inclusion necessarily support ic fellowship study document sample document categorical classifier large writing
depend concentrate universal code derive mixture index specify weight thus scope mix sequentially well state state long weight positive unimodal great impact sample derive model purpose side counterpart absolute back sign computed close conjugate prior side respectively plug full abuse prior two instead know code theory model laplacian distribution tune row unknown apply model see central coding induce center differentiable regularizer become robust alternative regularizer regression consistent identify context compressive well recover signal compressive sense regularizer arise approximate pseudo element report partially confirm regularizer show regularizer use non regularizer recovery coefficient patch well jeffreys determinant jeffreys hyper parametrization lack reason jeffreys regret code jeffreys grow jeffreys attain
increment draw u draw increment increment draw kk jt illustrate group course box group customer come customer join either upon group contribute popularity box box whole factor indice pool global moreover induce index exist take choose increment mechanism group generate member sharing note global relationship category main university various popularity share manner frequency popularity although describe bring distinction viewpoint viewpoint inferential identifiability finite view limit finite model weakly bound hold counterpart shall distribution due randomness place density call weakly finite group normal mix q place rectangle place arbitrarily sufficiently spread dirichlet omit thank relevant viewpoint interested recover local identifiability mixture union mixture mixture component marginal density group component distribution give consequence
transform characteristic know note though test particular optimistic however require purpose explore property compare base replication show fix unique either solution right column error rr c variance joint remarkable extremely heavy tailed typically lie particular standard consider realistic consider tail interesting gaussian fourth heavy tailed estimate row bottom rr mle w c r rr c na r gaussian mle ratio r rr median w na ratio tolerance replication proportion middle deviation around truth mle mle implicitly see fair include median impose critical affect solely directly result deviation fourth moment exist grow contrast heavy tail median outperform median finite location inferior estimator unbiased mle continue unbiased extremely heavy moreover last show na
constant interval since seek hierarchical successively approximate example follow subsection framework symmetry operator symmetry also role transform encode terminal root encoding contribution coefficient set imply code x x non concern generality lose lowest lose operation object effect one level product bottom dendrogram either cluster result apply operator singleton give terminal end element preserve implication let take term inductive inductive define subset lead
profile infer plot biological attribute relevant grateful literature break random distribution begin result dirichlet close identity exchangeable form model computational readily adapt partially gene profile gene profile partition h four close probabilistic might encourage statistical worth connection methodology important volume dirichlet satisfying mixture broadly characteristic mixture cluster identity shall call identity character unit call item unknown parameter subset conditionally exchangeable covariate simplicity write
separately let addition meet suffice monotonically tend monotonically small stem second character establish iv side expression reduce value yield see factorial q solution lie condition let monotone monotone decrease interval monotonicity follow analogous function definition kk uniform proposition use determined hand particular establish
mr auxiliary report simulate example replicate p respectively likelihood factor due believe base delay copula approximation similar burden increase give h base later reduce evaluate complete monte carlo auxiliary language ghz job assign worker believe code factor contain stock major uk france continuously return nominal obtain capital index delay rejection one estimate sub gaussian copula sub marginal suggest provide evidence list due consideration level discuss powerful hasting either single block multiple slow delay rejection discuss delay generalized block factor gaussian factor survey inference
bivariate decision significance measure couple ordering significance give sided value surrogate level correct statistic graph rejection coupling vs though driving detect great series rejection confidence achieve agreement apply embed procedure patient generalize eeg transfer anti brain channel channel improve resolution potential reduce activity neighboring channel correspond p record length patient patient take channel one segment second duration scheme mixed vanish delay mi lag right channel channel number vector pair direction patient patient htb conclusion patient seem exchange indicate almost always channel channel channel patient indicate observation exchange embed panel record agreement brain patient diversity regard channel dimension c substantially component indicate area seem embed channel explain top interval minute dimension respective profile agreement profile
threshold ray detect subsection able ray dr ray identify cluster consist cluster ray cover limited band compose band confirm optical cluster area span dr combine ray algorithm effectively completeness cluster separation project ray match cluster ray cluster ray separation ray match reliably ray separation ever optical go wide range identify galaxy cluster detect galaxy feature galaxy line contamination universal preserve exclude feature particular form even blue use band exhibit tight galaxy cluster spectra regular break color show large use use red color galaxy getting detect match precision algorithm within serious disadvantage use computation produce dr hour core long detect plus red note good weighted optical mass count member galaxy direct galaxy member determine formation cluster work room deep work dr reliably guarantee implementation color red serious preferable color span color additional improvement deeply co income dark tm acknowledge nsf grant er
change decision social terminate false alarm penalty incur alarm event happen alarm penalty alarm course false alarm incur expect false alarm alarm penalty obviously ii delay learn protocol continue incur event change delay remark public belief depend determine decision terminate link view modification decision define include incur measurable correspond sigma algebra denote determine economic discount guarantee kolmogorov criterion geometrically alarm vector discount objective assume unlike classical belief give public belief cost maker minimal policy summary distribute time specify eq transition observation cost alarm discount factor decision social extension standard incur detection large classical policy programming belief characterization analyzing stop bellman eq maker belief define convenient rewrite bellman define bellman programming equation transformation algorithm captures involve unchanged coordinate maker public change stop iteration use previously time result confusion iteration bellman proceed bound real generate continuous bellman yield computing since bellman value computational view section devise stochastic illustrate detection trivial belief belief update mathematical induction bellman positively denominator equality clear concave necessarily concave concave map distinct explicit function stop read denote incur optimal detection function preserve concavity easily piecewise linear value
point condition curvature surface quantify curve notion curvature second fundamental informally restrict region manifold less hyperplane implicit riemannian seem fraction tangent plane apply manifold turn manifold let q x close property consequence imply x nr nr x r rx combination hull point x q rs means fx consider random provide improved number however
throughput reasonable resource section theory graphical describe procedure inference place expression model propose present conclude remark gene graphical expression gene point gene characterize v vertex edge vertex adjacent connect conditional vertex contain edge connecting carry carry unweighted introduce range iv iv connect term directly indicate specific vertex cycle graph compose tree vertex adjacent tree cycle call span tree maximal forest span span span tree clique maximally separate path vertex vertex contain ks ks ks k clique intersection clique
affect statistic first statistic change numerator statistic give denominator statistic datum certain n statistic define ij follow covariance column correlation degree originally independent statistic row scale proposition however penalize major observe diagonal calculate value statistic central portion outline denote quantile indicator central central portion test central matching since follow variance estimate conservative directly test common evaluate performance many simulate example pre process center method normal observe dependency microarray robustness compare dependency select five portion note refer column center first row test test fdr without subset
file separately assume posterior posterior constrained matching value panel summary match display panel probable record quadratic give quite produce match compare whose frequency marginally distribute simulated update matrix metropolis hasting step report summary probability match display panel case record obtain loss constrain provide latter seem uncertainty c solid line latter guarantee estimate figure step consider ratio simple plug also display probability great high potential match almost certainly due assumption absence match information retrieve provide datum multiple match maximize produce figure likelihood use constrain illustrate exercise enumeration census enumeration include person census survey record people census parallel enumeration survey block individual note block survey census enumeration
design particular correct respectively similarly upper via nan invert fix p behave numerically describe detail set repeatedly dataset interval confidence maintain coverage probability become value low probability agree c importance currently confidence inference dataset size exact unable often fail hour unconditional interval poorly electrical brain neuron electrical activity various spatio process challenge pair neuron much complicated demonstrate accelerate hereafter value mark mark value k fire near ms time increase recorded period minute perform ms conditional firing window use test power alternative minor speaking fast test type temporal motivation nan carlo unfortunately test challenging size need successfully conjunction
arrive pp n omit nr pp n similar pi op op see complete theorem omit involve p axiom case conclusion condition theorem exercise remark dimensional loading carry loading loading result original series consistent independent curse latter case procedure prefer asymptotic propose together far report title factor model classification primary h h phrase curse dimensionality precision modern age practical series finance economics environmental understand dynamics key portfolio pricing risk management economic phenomenon environmental
calculate relation rare parameterize previous measure asymptotically entropy probability iterative rare first markov property retain ii associate rare program iteratively path chain simulate bad state
duality require evaluate sum moreover brief map vector strong solution kl k k w f parametric max show flow reduce efficient parametric max version basis cosine transform dct compose pixel software implementation available conduct core ghz execution c correspond regime ex example proposition theorem conjecture axiom million several video demonstrate applicability scalability become popular linear address combinatorial selection problem rely develop reference primarily encourage spatial hierarchical effort design induce capable allow successful bioinformatics topic vision sum involve nonsmooth prove penalize overlap induce within make show thereby scalable solve efficiently relevant various structure dictionary image patch differentiable
parallel distortion compression parallel worker operate rather minor degradation formal trade equation finite aspect experiment focus derive minimization cluster trade experiment alternate projection alternate derive minimization distortion ct writing provide box derivation x tc tc
choice come appeal distribution subgradient lipschitz subgradient objective simply write point reasoning distribution understand supremum range convex minimax minimax proceeding conclude past bind convexity supremum convexity technique basically theorem theorem repeatedly claim supremum briefly attain supremum attain loss lipschitz value rademacher combine inequality eqs observe choose depth rademacher conditional distribution irrelevant dimension prove proposition infinite say online learnable supervise finite bind online learnable online learnable immediately adversary jensen simply regret correspond randomized contain deterministic one minimax player dimensional rely fundamental order expansion convex also shall prove old inequality upper equal achieve carry mirror reverse two
change manually use learn name evaluate runtime yes fail syntactic ignore example b syntactic yes omit shorthand contain probabilistic pn otherwise determine probabilistic special case experiment name implicit kind choose choice later result em em pt dd kp p assign h r gp em abstract operational semantic resemble operational semantic execution state analogously additionally fail execution want fail symbol abstract operational semantic identify serve sequence tuple solve match note conjunction goal tuple number prevent trivial termination counter use state
calculate reconstruct network new subsequently give length generalise marker sir marker separately drop fashion fall section marker binary adjacency indicate direct enyi total average incoming per relatively sparse likely infected recovered begin I determine stochastically show table parameter marker reasonable frequency marker report lose parameter marker infect node infection recover infection infect naive covering explanation infect precede good guess therefore imply interpretation marker trace capable produce data marker explanation true since away marker ensure positive
dependent ica toolbox software mode thresholding individually I amount posteriori variance threshold pattern see equation actual implementation generative differ reduction step successive ica observe image common mixing pattern impose outer subject share set pattern map vary ica multi external course cognitive subject tensor ica low hand subject loading independent describe specific level group concatenation observe subject ica structure mix mix em group limitation ica compare provide statistical framework compare glm contribution comparison fit mixture ica approximate ratio compare discriminate model essentially variance systematically assume across voxel tractable computation loading different population important cognitive strong comparison complete perform difficulty stem ica global exist decomposition ic feature separate ic achieve subject variability bold independent component common span generative pattern via give pattern represent subject pattern
least inequality hold minimization go construct hold criterion definition excess r j rt r contrast depend strong correct conditionally dyadic tree dyadic recover true clearly rt two estimation note fine establish assumption assumption specifie dyadic either impossible boundary true element parent I small moreover cart hold minimization form result fine assumption control result synthetic dyadic integer
nevertheless segmentation class label try label match truth inefficient namely hence suggest give ground label index contribute maximum number local efficient produce outli receiver characteristic roc widely binary roc positive rate false value roc evaluate auc range r section em plus minus member member yu member engineering university china visual group microsoft china laboratory university china electrical laboratory usa electrical usa subspace union multiple subspace subspaces propose novel name lrr low candidate combination lrr solve clean structure contaminate certain corrupted lrr approximately row space theoretical membership provably determine lrr correction outli process often enable need parametric characterize know mainly effective type visual face texture subspace reproduce model much recent year principal pca establish completion recovery hypothesis draw lying namely consider show strictly draw subspace draw underlying subspace group subspace cluster numerous vision image clean draw subspace solve
commonly accept aggregate general scenario accept conclusion preference happen accept grow work economic political science survey field abstract field aggregate formalize need outside scope individual opinion answer issue j issue vector ease presentation like acceptable accept logic consistent conjunction example literature issue logical set consistent opinion assignment achievable logical search semantic proposition state prefer uniquely opinion aggregated issue issue majority present state member consistent nf return opinion conjunction issue majority consistency accept agent require decision aggregate member act unit independence guarantee wise aggregation justify
chart fast shift detect moderate jump call nonparametric covering far combine classic chart run chart shift inferior chart propose chart small shift normality comprehensive chart extensive carlo follow although power considerably increase extremely investigation surveillance detect result quality take analyze real detect enough engineering image grey mean dependent complex take account important behave important namely application statistic asset uncorrelated square correlate conditional see difference mind propose chart classic chart chart buffer store recent modify
u entry straightforward strength must differentiable jk easily derivative like quasi method minimize first method resolve several different add rule derivative mostly omit mark notation rest parameter predict make different extend source graph pair example slightly modify gradient independently alg optimize individual less likely final descent impact eigenvector speedup achieve position initialization walk bfgs derivative solver improve converge optima quasi synthetic graph edge triple try model free start create add exist create strength uv exp uv uv random way first already vector variable interested thing model classification accuracy whether edge deterministic create add perfect recover close weight area auc report mean auc mean perfect figure show model blue perfect news noise drop reach actually bad algorithm perfectly
process three raw decide text convert string character car normalize third raw text mark identical string annotate vb noun nn good linguistic linguistic surface syntactic syntactic attribute extraction english simple approximately work sufficiently well sophisticated accurate english know g versus recognize wish ignore stop word word include separate break text character match often correct accurate task language normalization string character want word reasonable normalize variation common word reduce case english problematic language difficult word distinguished mean problem sometimes information whereas common kind internal word compose form past compose ed kind stem english heuristic relatively concept combine word analysis language sentence english performance system truly say query truly normalization variation recognize similarity thing increase sometimes variation significance variation cause normalization also token corpus selective normalize text normalization information retrieval annotation string identical string depend annotation part sense accord intend parse sentence word sentence role annotation precision program noun able search act computer noun precision noun say even never may gain ir performance result syntactic syntactic annotation query segmentation speech semantic association annotation word discover row vector word contextual parse annotate first generate adjust element frequency matrix way similarity give mathematical transform raw smooth term word situation event practice complicate corpus step scan sequentially event hash engine idea surprising event event surprising measure semantic discriminative context like high expect document tf weight tf document tf weighting evaluated demonstrate tf weighting
symbol row correspond equality block radius establish thm observer machine state average symbol decay thm eq w entropy symbol although slightly formula publication hide machine suffice constant eq establish constant also eqs finally know constructive arbitrarily say distinct pair path every distinct mm convergent machine
non unseen generalise task generalised attribute biased ordering index give size else expression recursion n n n aa kind closed present unstable partial derivative thus lot remain figure tx pt distribution skew upper shape location horizontal spread pt expect detail size posteriori sample b bb symbol posteriori b bb n st almost sublinear roughly posteriori deviation approximately posteriori approximately b nb distribution follow size sample identically notation series series quite experimental evaluation close approach dirichlet behave like dirichlet exponent case behave factor parameter multinomial rather partition give assign n use symbol bring class sometimes subscript symbol use create far use create second technique
minus mu minus mu mu minus mu predictor generate interval standard variance note variance var correctly rate coefficient prediction score replication independent prediction manner var result various var job large model simulation summarize table note current consuming compare var work package design variable large var var mse var var var mse c var mse come mu plus mu mu mu plus minus mu minus mu mu simulate rest rank forward default estimate surprising probably discuss var tune select cross carry package maximization greedy widely pursuit suggest methodology efficient especially approximation mcmc problem framework direction simultaneous currently research extend group appendix mu mu plus minus mu mu minus nz z mu minus mu minus mu minus mu q mu mu evaluate variational posterior generating bind simplifie plus minus mu mu plus
viterbi reference result subsection section surely viterbi alignment analogue viterbi process alignment prove method run convergence different subsection imply alignment need general reasonable programming minimizer apply optimal coupling develop main use couple stationary ergodic limit viterbi main preliminary risk classical g notation shall let version z delay recall measurable hold chapter sub algebra
redundant variable everywhere gradient adequate performance presentation also approach similar demonstrate save impose htbp ex oracle redundant dimension increase ex greedy applicable dimension estimation calculate significance point however make prediction likely efficient grid calculation require useful establish power include variable selection oracle redundant linear interpret oracle two select converge property predictor may attention weak oracle regression converge error would advance property nonparametric converge order asymptotic variable achieve correct parametric scope
critical result table little limited study asymmetric alternative high asymmetric alternative test whenever high sided test often powerful
less week come involve consumption three implement version call package surveillance apply believe method list surveillance system indeed necessity modern surveillance system deal analyse handle count method treat past since count past particularly count period past method language allow automate intervention although et incorporate several
differ assume hypothesis alternative interact level differential need essentially relies identically distribute seem could marginal distribution still extend setting convergence establish proof rely formalism extend composite provide distribution function assumption spirit result recently suppose test infinity fdr drive shape whose centrality parameter result plug hypothesis infinity therefore rate near slowly enough may strategy estimate fast rate powerful incorporate plug estimator characteristic estimator location semi regularity specify recently lebesgue measure estimator extreme arise case occur one sided laplace x reach vanish nan little room non proposition reach distribution test laplace proof omit sided location statistic side one e side side g probability e concavity two
firstly give secondly method stop limit later run correspond control effect positive set common yield chart type regard testing viewpoint partly horizon horizon run substantially yield associate rejection length power ratio residual design simulating limit case stationary additional rejection chart robust parameter determine increment behavior consistent finding monitor early quite change detect quite may quite reliable regression rejection rate right change slope point change thank associate excellent review
huge set explore filter item identify user identify user compute recommendation introduce recommender way recommender system paper item recommendation moreover simple conducted system propose enhance quality recommender storage become increasingly cause people pour internet contain thousand article valuable resource item successful recommender technology predict particular item n user recommender system create collaborative successful create item preference recommend target similarity compute base common call recommend approach recommender many randomly item available randomly great selection great failure algorithm recommender example customer frequently pattern minimum require recommendation express rating recommendation preference merge item list user
unobserve view task must prediction draw parameterized write say row independent feature member row rating may natural ordinal prediction may pmf necessary explicitly include pmf rating nuisance dependence pmf parameterized representation capture idea side nonparametric latent ordinal bayesian often specification definite machine restrict gp prior deal reasonable scalar gp cholesky decomposition inter function relevant hyperparameter across member intend variation tend member learn
social network scalable evaluation synthetic superior especially overlap demonstrate social link student seed find algorithm complex facebook user red blue many diagram community visualize algorithm find community node assign graph accept community clique partition preserve recent overlap community repeat capable capable detect scale empirical graph subgraph compare edge method spirit exist objective community proceed use simple heuristic finding allow objective work overlap
column effect coefficient random coefficient normal matrix fix effect non state set table coefficient intercept previous example fit mean deviation run therein tp positive cc tp indicate fix cc ccccc tp indicate cccc model low appendix set whereas restrict biased covariance approach parameter gauss cycle fix without show whereas estimate variability around model table effect half also observe towards notably concern know drop variance use methodology group performance procedure validate validate lasso covariate measure generate table differ
standard consider rating add web content population may produce discrete application fourth characteristic site tag probable future high user visit visible conventional arrange sale ranking ranking customer ranking effect rare article sale rare interesting future idea investigate service structure possibility often long tail central limit theorem convolution statistical effectiveness consideration plan explore item besides sum expand release providing present analysis package internet small population limit theorem large effectiveness mine modern internet uniform information material manner report sale amazon com book conventional
etc notion repeat game opponent opponent period maximal period notion consider concept say opponent achieve infinitely game arbitrary rule play possible opponent dynamic maximizer payoff maximizer infinitely look importantly dynamic payoff know opponent strategy loop assumption certainly maximizer include absolute present subject repeatedly game paper exploit maximizer question paper submatrix number ordinal potential also provide sufficient essentially separability game gain q start play look opponent straight stay straight stay opponent stay suppose dynamic payoff maximizer payoff differential opponent sum payoff maximizer play case receive opponent heuristic theoretically experimentally mostly mistake allow al generalized games os payoff payoff
inspire estimator result converse
col integer polynomial expand integrate produce end density gibbs package sigma sigma clearly average rao grey n col define distribution conditional normal mean modification program l grey example conditional grey sigma bar sigma col example additional grey iteration col code type grey alpha col plain red original need markov see assumption miss shift cauchy schwarz ergodic almost surely small estimator necessary condition exercise regular enough modify program bootstrap j ts ts grey col figure variance iid iid exercise posterior closed form read show marginalization marginal albeit marginal gibbs simulate q use loop mu alpha mu alpha alpha sd alpha alpha alpha mu stationarity start var pass var pass nd var reproduce kolmogorov ks alpha alpha alpha alpha alpha seq ks visible pattern indicate uniformity compare sigma alpha exp
weight geometrically subsequence probability geometrically w I differ existence decrease w r pr hold condition bit bit fix equal pr invariance principle setting principle application contain derivative exist say bound ensemble variable match element invariance variable ensemble moment bound ensemble ensemble roughly second ingredient hardness distribution exactly bit distribution follow parameter specify bit independently bit know express linear feasible calculation bb ex accord eq q calculate conditioning coordinate match case replace moment conditioning reduce case integer random bit example positive test completeness pass h pass probability k x
normal update bayesian regression double prior quantile burn iteration prior burn probit first stage burn density sampling random metropolis algorithm burn correspond normal double normal data se university concern iterate variable mixture normal walk refinement perform multimodal compare scheme realistic prior metropolis walk copula good keyword hasting normal chain monte carlo extensively generate proposal example tune draw deal adaptation successive proposal converge theoretical adaptation make metropolis mixture normal proposal body theory adaptive construction adaptive sampler receive
computation time introduce new moderate square square approximate square computational feasibility similar address closeness capacity good superposition reliability least achieved mention dimensional setting discuss code discuss development reliability subsequent section framework code linear list book vector coordinate vector accordance keep terminology call arrange value choice term send small somewhat superposition message sign coefficient algebraic book section size likewise split indicate additional freedom partition code desirable arrange convenient size power bit length give index zero index say code per channel close superposition code number alternatively allow size match allow additional simplicity simplicity analysis partitioned code sign code string split section specify zero bit specify control computationally advantageous code decode number number codebook direct impractical extreme coding sign generator code combination subset concern converse channel capacity close independent result draw yield want obtain associate sum coefficient coefficient henceforth coordinate non magnitude decoding consist
adopt jeffreys include correction correction correction identical neither still however process band way channel permit jeffreys margin less obvious seem however imply fact set regard reconstruct spectrum result uncertainty dispersion correction way know sect construct accurate map peak asymptotic perfect never combine appropriate way improve correct permit moment end combine gaussians formal existence scope note accurately gaussians often approximate peak describe gaussian use use practice assume well pdf tail want distance kullback measure practically surrogate introduce un probability degree freedom enforce read
central metropolis available perform realization gibbs integral posterior mode conditional marginal mean produce extend solve balance factor seed system inference case survey come certain might discriminate specify bin range know range count fall depend cost aggregate ij conjugacy another approach inform censor dirichlet prior lack conjugacy regardless substitute dirichlet next update explicitly proportion clarity also joint posterior set informative dirichlet flat sampler two step alternate
order large similarly increasingly large eigenvalue w note result exponentially reason previous yield need show concentrate minor become get n psd measurement modification uniqueness program let recover negative eigenvalue n nn psd recover measurement program psd curve weak uniqueness give psd strong solve measurement although resolution consistent white success various give exist suboptimal lemma carefully paper generalize estimation tight tight significant compressed singular nan allow suggest
average number point need expensive special suppose stop word expect series rejection generate accept probability odd exactly correctly provide
l l expansion line sufficiently imply know machine several respective fix take relation mm x gx see eq combine eqs exactly know exist number sum expectation compute omit machine machine rv rv nb order convert need entropy decrease primary synchronization first use denote prop equivalently monotonically follow
take feature sign available regardless sparse easy use vector proximal overcomplete discriminative representation multiplication believe valid building robust expressive dictionary di universit scale di pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt pt algorithm universit di v http www devote learn
identical outli outlier outlier identify observation motivate let eq promise research understand case observe entry success rate vs observation report experimental outli pursuit anomaly dataset digit correctly note stage digit exactly decomposition large outli thresholding sample identify htb complete partial average consider decomposition outli pursuit pca setting outli pursuit exactly result innovation whenever concept believe description quite goal collaborative filtering partially obtain tight identification orthogonal pursuit condition orthogonal pursuit succeed pursuit succeed hold succeed choose first outli pursuit succeed pursuit
uniformly library routine number evaluate describe standard idea contiguous array call scatter scatter overhead fast straightforward implementation scatter give normal generator sx implementation method sx effective peak time two sx hardware root encourage table method
decision dynamic treatment rule clinical intervention map patient recommend treatment review approach area index dynamic nonsmooth functional unbiased arise treatment illustrate asymptotic sensitivity asymptotic perturbation treatment use child illustrative discuss regime policy create health relate compose sequence treatment decision regime decision via rule dynamically evolve patient recommend dynamic expectation cumulative population technical challenge problem clinical guide regime estimation nonsmooth functional generative consequently estimator asymptotically bias standard primary inferential distribute form denote subject collect receive outcome measure trajectory collect traditionally observational inference dominate method call illustration uniformly either drug intensity behavioral modification remainder month assess occur teacher concern behavior occur child current treatment behavioral modification long child meet figure trial rand cm txt edge south west north west node cm txt north west txt north ad ad txt rand ad node rand cm ad txt ad yes south txt cm txt aa txt
corollary upper third hold least defined let case true definition sx next due sequence martingale hold claim substitute prove result stage substitute eq corollary simplify exist universal low lemma lemma eq universal depend denote iteratively hold c cn cn cn cn cn tc cd therefore remark find contaminate observation arbitrarily contaminated result achieve unlike achieve limit proportion corrupt statistical dimension analysis robustness outli dimensionality comparable
small bottleneck flow start layer detail phase ai cell bi uniformly phase ai layer adjacent least cell consequently ai ai z z bi upper phase flow neighboring simplicity assume move argument ai contribution step bi ai bi last partial harmonic series parallel cell go carry contribute n completely analogous completely sum overall st proposition dimension give construction high step avoid everything hypercube always figure place really grid cell harmonic replace consist dimension contribution couple remark presentation simplify couple strictly flow reason direction leave flow magnitude final result stick rough consider turn loop contribution consider straight piecewise path constant add corner construction work bottleneck small diameter couple apart take care principle cf corollary decrease graph give
dy dy similarly abstract liu version generate forest length balance fitness simplicity constructing express acyclic require exponential attribute eventually choose even avoid address efficiently attribute liu liu
class whenever observe comprise proportion class stochastic analogously understand finally blockmodel force accordance maximize kullback fit stochastic trial average subset respective la z ij z former distance kind confidence finite blockmodel fit comprise trial blockmodel log least ab ab argument term assignment bernstein minimal restriction couple union hold growth restriction whereby blockmodel whenever
well working previously discuss dual inner product v u upper hold equality maximum v norm picture show cone still shape dual example term duality group lasso lead norm dual nuclear dual respect trace product correspond matrix reduce norm role specify differentiable decomposable follow decomposable belong geometry relation constraint cone star shape require requirement parameter nz fairly mild guarantee equivalently rather closeness depend curvature high curvature around excess desirable pt function illustrate loss error flat via notion strong since first taylor series expansion enforce require twice amount hold uniformly loss function mild population strongly hessian correspond statistic size argument show regularity neighborhood whenever way drastically
multinomial enumeration avoid iterative proportional fitting interaction fitting procedure require storage fit infeasible moderate dimension interaction family marginal sum chain sample degree marginal parameter degree taylor q parts come class family exponential quadratic goodness control repeatedly degree propagate use precisely define fit parameter adaptive carlo transitions fitting requirement parsimonious minimum family control
even train low attribute supervise irrelevant bring unnecessary effective projection bring well interpretation low list dimension face arrive good rate dimensional feature recognition dimensional representation reliable lda lda seven dimensionality dataset box plot method median box extreme robust property classification inter unstable lda label projection seven three face vector basis basis pca call basis et grouping grouping effect adopt basis interpretation face retain g contour part face contour face face fact explain face less lar loop list column zero set active increase norm track lar track coefficient path change another active keep make correlation equally along greedy proceed direction objective loop
constitute one control relate view delta rather cause intractable relaxation restriction show minimization result immediate policy ever decrease cost rather iteration let policy generate formulation schedule cover assumption obviously leave
sample ds ds reliable detection localization support possible problem amplitude exceed localization amplitude arbitrarily grow dimension nan decide methodology problem fundamental limitation set clear adaptive measurement differ flexible address flexibility gain arbitrary localization zero decide whether common entail give exceed exceed context thorough review process sequentially collect step measurement quantify adopt convention crucial sequentially adaptive observation precision measurement example collect observation sample collect exposure comparison non
computational little interestingly outperform alternative distribution recover without substantial noise explain observation extreme throughout offer improvement much signal valuable manual screen difficult offer clear simulation misspecification surprising similarity input weight covariance one think set weight conditional degree freedom go get assumed current weighted matter less gene normality fact gene assess adjust somewhat focus exclude graph exclude extreme recover edge
since play possibly grow happen match another probability c calculate still arm case optimal arm play several lot explore sure optimal arm present resource storage yield grow resource evolve resource discrete static consider hard explore scheme information remark edu combinatorial armed bandit bipartite user resource resource pair evolve irreducible markov occur user allocate resource receive depend
spin sure material show diagram ease presentation quantity calculate entropy joint entropy atom diagram everything want pass want ask know atom diagram involve enough atom delta information diagram atom diagram fast calculate entropy corollary theory observer states stochastic determine internal organization observer analyze convergence entropy compare block along introduce hierarchy integral hierarchy introduce entropy draw synchronization process keyword store rate complexity excess information dynamical generate device utility device chemical amount capability intrinsic remain intrinsic another practically service storage address processing characteristic system key aspect control concern internal stationary exactly know observer ref give design process series reliably desire internal design say synchronization synchronization observer incomplete typically start complete extract indirect set duality observer control key intrinsic computation dynamical leverage intrinsic useful computation circuit attempt circuit income essential digital must even amount digital change reliably reach elaborate device fail device digital operation concern memory line properly risk happen simultaneously component device quite
risk numerically popular taylor aggregated bootstrap cl deviation constant generate fluctuation text resample calculate residual estimator observe bootstrap conditional resample cl unconditional version generate cl calculate cl j repeat step obtain empirical bootstrap uncertainty frequentist mean choice parameter study fluctuation point uncertainty come formula appropriately scale correct fact introduce novel embed parameter capital present abc frequentist concentrate conditional term frequentist estimate estimator involve bootstrap estimation error variance observation tc setup choose unknown term since full distributional
equilibria ii nevertheless lb lb read number discrete subscript nf particle distribution velocity space obviously map velocity space transformation bold n lattice set discrete velocity cyclic permutation level besides freedom degree freedom correspond molecular original publication simulation evolution discrete
result clique clique ability induce clique sparsity prior clique increase increase mass put clique plot limit give small put mass cluster therefore select gain literature partition instance may select intuition regard verify carlo cross select potential computational form admit several connection prior put prior partition
interesting analyze curve basis link interesting recently variable group explanatory power reduce extremely provide analyst tool curve able select explanatory summary curve mask detail variable thank anonymous valuable improve paper functional function prototype g piecewise constant user relevance cluster programming problem finite know world spectrum online hardware good associate physical rate another consumption curve load describe load million france period year every summarize load typical daily period daily understand consumption weather new price monitor exploratory analysis curve analyst set classical cluster prototype map come symbol time transform contiguous interval interval actual guarantee error
generality without generality axis plane lie horizontal axis lie horizontal angle great circle radius lie projection fall z xx z contradiction let du intersect face contradict two geodesic geodesic within ball orthogonal point geodesic angle part must like g jx cut claim jx g jx g x eigenvalue qr exist angle manifold intersect bottom surfaces dx step create hausdorff ny xy lemma net
array k form gradient mention clearly expectation intractable evaluate make sampling binary stage field metropolis hasting meet remain stage randomly object currently select include criterion choose number distinct k wherein th step technique produce indicate estimate importantly full presentation probabilistic modelling order partition rank need return relate return unseen slightly standard object identity object find rank find unseen query q rank decrease establish return sort carry combination rank
simulation indirect monte application partial easy simulate calculate intensive simulate set least replace simulation consistent however negligible abc combine likelihood abc popular population possible increase importance see current amongst flexibility easily simulation evolutionary really require available recently implement within dimensional specific parameter value abc map iii posterior abc closeness closeness reason consider ability future exposition impose density bandwidth importance abc simulate assign possible introduce extra carlo thing valid value abc remove occur abc deterministic variable uniform ht mm h bandwidth define ng output implement abc input bandwidth integer ng abc error posterior
remain constrain inside place customer expression spherical component radius origin location surface show factor combine distribution sampler derivation step mdps sampler mdps allow relax many distributional assumption detail process reduce burden substantially specific model provide china mobile large phone operator china phone self report ideal base consist contact record phone call message member gold company customer contact date contact take purpose ignore customer customer divide period month month period statistic summarize nonempty dataset appear nonempty nonempty calibration customer empty coefficient mean geodesic distance call share friend non speak small large number coefficient assess extent small geodesic distance number friend person approximation geodesic expect size cluster coefficient mobile observe quite clustering see graph would possible reason distance cluster network type distribution notation contact end length period follow logic exponential core never first like close follow link way contact rate observe observation
keep rich expect positive accuracy positive typically entail semi always realize building easily impact pairwise approximation upper control number time approximately distortion projection leave much pairwise market price dynamic know profile tool quantify sample behavior demonstrate frequency profile throughout portfolio avoid ambiguity call risk optimal version stochastic latent log price brownian obeys integrate give ds ds analytic formula conditional decide brevity slightly keep make well model price scheme gaussian price gain asset price year us trading intensity mean trading time asset spread arithmetic benchmark portfolio portfolio grid second latent asset second simulation asset price trading day base price strategy assessment base frequency portfolio one price minute study observe minute use empirical position therefore price jump section latent price estimate covariance short regard method noise employ method
comment table activity row unlikely receive except column group member group highly group column message exception probability message send group belong last block evident primarily amongst although membership individual belong several tendency group tendency receive group inspection profile also ii membership rr estimate examine predictive link considerably due email communication communication receive message member message receive traffic group constitute member class group send member receive member calculation carry entire multiply column member display active expect size
learning maintain classified user click classify oppose server long load experience page loading web classifying type classification lexical name external acquire lexical external feature full feature lexical name introduce due life bandwidth nonetheless rely comprehensive lexical accuracy seek answer well one lexical lexical properly achieve achieve lexical lexical vs feature art specifically follow op weight adaptive lexical feature decrease full however lexical trade moreover lexical boost classification online impose less overhead outperform environment comprehensive working include google ii attack mis label insight gain classification operate use lexical implement desire
instead implication static prior follow wherein capacity observe induce strong linearity apply hand formulation much give unlikely population unlikely impossible term general useful way consideration constraint show restrictive presence error observation impose finally note value concern opinion mix efficiently complex surface least clear improve population dynamic reflect couple measuring device reflect particularly fitting beyond complex surface potentially slowly particularly metropolis hasting currently generalise smc incorporate realistic reason believe comment suggestion research visit contribution capability research proposal th weight evaluate j adaptive component proposal chain capable sampling state weak test examine observation error may effort novel involve combine sir hasting slowly consider er expression rao population growth notably multi modal
volatility transaction precisely change volatility transaction trading influence trading volatility nonlinear time transaction intensity financial frequency topic practical relevance participant execute high frequency market risk trade large market make volatility management often immediate strategy automate fact participant pricing option second frequency much complicated cause filter volatility paper describe efficient security treat relation price transaction computationally develop price transaction base filter volatility sequential algorithm work transaction contrary transaction price extensively zhang et al suggest localize version integrate wang schmidt noise essentially regression estimate side transform article change transaction conditional evolution unobserve log walk transaction time low frequency transaction treat lead decomposition volatility volatility trading intensity influence opinion advantage continuous aforementioned volatility volatility transaction constant volatility et relation unobserve observe transaction market noise round eq past addition maker complementary paper try round order book maker fairly possibly restrict deterministic cover complex situation close bid form observation transaction speak volatility view realize particle situation complicate particle article
reflect probabilistic vector modification et query take context fuzzy boolean determine boolean operator join text learn word type functional query free al es mainly view query set admissible admissible boolean set well follow syntactic atomic single composition composition admissible way semantic remain specify query boolean straightforward concept evident query find es difference base document query base inductive chen relevance feedback explicit relevance user guide sample document boolean obtain modify exist iteratively run learn execute boolean retrieval consider characteristic system relevance description architecture present system separate outside view space usually rely genetic gain robust choice framework examine mostly appear inherently application al popularity evolutionary largely parallelism argue less perform fail retrieve document probabilistic permit query evolutionary justify query rarely query try automated learning lot past year majority improve examine representation wikipedia enhance learn es evolutionary semantics automate query paradigm intelligence program induction evolutionary computer program individual get machine problem explicitly learn relevant document keep irrelevant document drive evolutionary pressure individual among query candidate es architecture
degree stable make accurate already unfold surface embed manifold randomly neighbor figure generate datum mapping fail variance entry lower similar new come first evenly manifold sample time low fail embed also select fig far validate performance procedure time versus test residual sample dimensional give b three method cost linearly increase sample experiment
suitable observation infer flexible try related gibbs conditioning alternate find perfect unobserved exponential exp covariance except amplitude hyperparameter hyperparameter covariance great train maximize laplace likelihood laplace discard pass variance variance sample gp laplace extensive review competitive sophisticated matlab toolbox implementation make forecast volatility historical volatility volatility cause fluctuation stock proxy truth predict
risk belief description sensitive show e define bellman condition element decrease p example verify elementary calculus geometric always provide stop ex algorithm optimal appendix poor convexity imply degenerate check theorem exponential recall conclusion delay straightforward social learning formulate threshold motivation subsequent subsection brief act order index also discrete act social previous section private agent current private observation let private define comprise state nature sec except instead agent record private observation iii private public time private belief action take minimize k ce incur pick agent choose social finally subsequent agent public action apart update eq pa ib result term cascade lead information cascade public agent pick make decision protocol system global system agent detection threshold social probability model q dominate sigma include history stop sigma agent belief social continue stop pick stop minimize delay decision choosing continue non incur stop public eq global decision learn policy bellman parametrize scalar partition interval belief readily verify observation tp main consider problem agent tp optimal b stop point interval iii interval region cascade cascade filter example illustrate stop construct double behavior perform learn due addition cost consider total social constitute total social public belief provide decrease set zero policy value double
observe opponent stage play generate mixed fp process player play pure mixed furthermore fp player nash player time repeat nonzero attack node payoff adjust fp player estimation see update opponent security couple exact weight
unfold divided initialization equation validation initialization experimental selecting bin pick step correlation bin must experimentally solve system choose generate expect row relate use square calculate accord matrix initial calculate variance calculate give adjacent combine reconstruct effect resolution leave procedure distribution procedure yield distribution error least transformation
attack attack attack accord token viterbi algorithm attack raw corpus attack contain attack test oracle target program previous attack string positive fp attack invoke false positive successful attack summary attack fp fail attack attack equation attack fp attack attack identify application capability identify achieve attack manual ignore database convert retrieve database convert page title adopt strict positive occur page would page two attack cm attack long plain text token limit
extension simple efficiency variety repeatedly rate numerous application mention publish paired comparison entry european build output classifier extension handle home advantage multiple popular name define prior permutation rank multiple possible rating simple refer algorithms converge excellent generalize guarantee towards bayesian approximated propagation develop suitable
replace dpp rl result policy algorithm mdp action benchmark time second lead solution approach file intel core gb specific library superior standard cpu performance run action preference interval cx correspond amount dpp achieve dpp rl outperform benchmark attain mdp grid respectively rl significant order magnitude mdp grid dpp well concern vi cause obtain table dpp rl vi deviation substantially small vi deviation lc lr lr mdp sec dpp rl show suggest dpp simulation cause rapidly dpp rl vi three benchmark illustrate limited sampling modification
combination number vector processor pool outline generalise generator uniformly transform way pool observe exponential standard uniform batch pair
quite mention architecture lead initially reason back neural deep deep deep deep network necessary realistic also architecture necessarily decision journal science author net use different architecture little decision ask article person answer tries pick entirely satisfactory think limitation deep advance learn problem scientific entirely satisfactory answer deep except
light idea context least contain context denote walk stop stage next observation tv precisely tractable recursion see follow give measure observation walk mainly straightforwardly recursion finally prove mention depend cover computational complexity case scenario context walk relate contain cover
regression provide coefficient estimate even presence multiple base graphic outli alternate estimate identify model clean initialize ol well something occur outlier know main challenge describe broad direct indirect procedure include backward selection indirect residual robust regression identify include median gm necessarily efficient fast publish example outli high exponentially py robustness although property worth closely quite identical problem coefficient still overlap coefficient two hand perfectly identify outlier task obtain support section predict
present illustrate key feature method hermitian e distribute value test hermitian part poor unnecessary error sure solution well first laplacian order eigenvalue interval negative component c denote associated converge cf otherwise relaxed condition total column list final reproduce classify linear solution twice summarize solver solution solver reproduce run eq truncation strategy justify orthogonal last element treat treat ar solver backward prevent bring back residual continue
computable identify hard unknown applicable bayesian partition single treat policy different multi bandit single selection policy well approach dynamic sensing secondary channel sense maximize primary model
conclude really outer convex project star belong idea topology close modification want give distance recognize tree bethe lattice mechanic length give reconstruct root topology character bethe lattice plane scale compare mutation trend phenomenon open plot could mutation data bootstrap mutation useful leverage detect jump context tree transfer subtree resolve tree event dna sequence since without gene tree exclude supplementary material distance tree basically group tree group rich effect overall give picture gene ht cx tree complete cat triangle thin compare satisfactory posterior tree give distance four four point large four three sum criterion among l ij kl jk er er
proportion compound empirical explanatory idea review paper zhang compound observation desired estimate major concentrate procedure bayes assume large note estimating resemble stein bayes member recent zhang advantage elementary grow high problem need stein problem estimation technique e rao seminal modern explanatory symmetric elaborate motivate certain addition one close area example economic size age exchangeable follow covariate statistically aspect sample rough permutation
kernel rbf derivation come embedding arrive rely upon model dependency discussion affinity fisher score parameterization computable without heat parameterization rarely hmms kernel another rational alphabet kernel treat distribution disjoint kernel produce leibl incorporate measure smooth property theoretical rkh induce kernel analysis kernel univariate probability distribution restriction ensure translation respect family parameterize parameter view space location bound hypothesis induce rkh corresponding family unit
great fully extremely small associate proceed finitely bin estimate number draw retain associated p proceed finitely say fraction experimental draw retain bin confident arise distribution compute ff third observation e table ccc table f k efficient computing confidence
score newly generate news especially little service individual distinct feature application partial nature observe article display exploration exploitation choose business article collect user feedback strategy contextual problem important application ad etc conduct serve recommendation expensive require substantial negative experience furthermore metric time contextual bandit valuable online recommendation benchmark supervise uci repository valuable collect benchmark reliable offline article recommendation yahoo front visit store click offline feedback article recommendation display candidate raise difficulty bandit algorithm algorithm create simulator run system unfortunately drawback simulator consume artificial reflect approximation contribution describe enjoy valuable guarantee result record yahoo
censor death occur occur unknown moment death occur censor occur unknown moment death occur censor poisson equal poisson process birth nonparametric likelihood parameter maximize proportional van van calculation ff nonparametric come observation window van datum behave
induce delay allocation compete collection number pay allocate type bid per engine click favorable maximize event occur contribution classical allocation ii click relaxation assumption I greedy refined click constant delay present study mab feedback instantaneous delay algorithm discount presence delay state feedback change play exponential finally delay balance along exploitation schedule delay across standard relaxation arm execution delay policy treat independently arm arm exponential delay decomposable efficiently question delay exist near optimum policy decomposable
doubly model von result draw sample end let denote sample permutation put index permutation draw identically marginal denote randomness therefore equality doubly marginal within underlie per shall assume satisfie exponential regularity approximate approximate well permutation manner inequality follow satisfie signature satisfy signature manner wish pe establish two condition belong family satisfied bound hypothesis pf pf item item map map map hypothesis fact signature existence signature theorem choice arbitrary shall think hand need implication start equal see permutation assign equal bounding involve choose permutation onto cardinality map turn pf pf j pf j item need item map mapping know conditioning assignment space permutation wish use therefore use argument complete theorem suppose signature signature signature solve program signature
process mechanic drift population exhibit population evolutionary structured population represent individual give reproduce study behavior graph find structure sometimes effect advantageous investigate direct self demonstrate range complementary evolve node neutral unlike sampling process combine approach examine affect population example desirable shift result nonlinear dynamic formation genetic drift coin stochastic allele develop drift memory examine substantial evolutionary population rather structural drift replacement genetic drift latter population propose tie language perspective population hide call infer generate population otherwise make latter much exploit architecture structural combine process generally structural occur form measure drift well structure outside subspace lead exhibit structural greatly simulation process dependence complicated theory nonetheless show decompose sum subspace spend entry predict component global diagram drift structured substantially short coin jump coin without matter spatial memory increase restrict sequential structural drift
empirically large therefore rate calculate parameter preserve take time figure value confirm optimal fix example h choice indicate negative present throughout trajectory total time constant apply rarely optimization determine nevertheless purpose outperform demonstrate really temperature annealing
generic singular index proposition py k py py pe pe q invoke indicate consistently estimate enough effective relative appropriate hope mean regard indicator strength large subgaussian interestingly matrix independent entry simple independent space equal pe consequence instance supremum noise list main rsc recover rank hold stay lemma remark theoretical paper remark need conclusion subgaussian give proposition ij corollary inequality remain independent subgaussian f pe eq inner operator
influence phenotype output gene pathway share genetic task activity brain word activity brain correlate necessary brain region region share stock prediction stock price correlate contribution fold fuse problem introduce fusion encourage sparsity optimization proximal smooth general fusion penalty employ fusion regression connectivity guide penalty parsimonious relevant subgraph graph structure output provide consistency snps phenotype adopt fuse fuse lasso learning fusion penalty develop challenge cone programming quadratic qp moderate chain structure input penalty addition point guarantee exact manuscript flow applicable addition make gradient
optimum rather derivative equation c extend angle plot measurement shape color dark region producing indicate region produce straightforward albeit optimize note underlie well change detailed derivative pair angle pair angle outline figure dependence angle plot angle indicate color dependence estimate give angle one energy model minima protein predict region serious problem state
dimension length typically validity intend slice select new state never current unless adapt cholesky subsequent compute drawing variate case elliptical slice hasting algorithm minor improvement previous need routine draw choose cm cm cm receive variate define slice draw whole edge reject make shrink slice return configuration discuss final move elliptical slice setting algorithm index visit angle
dataset likelihood leibl divergence negative motivation kl stem theory entropy cost would correspondingly kl divergence additional code cost code assume success setting unfortunately exactly image van learn train patch attract lot attention recent belief deep belief generative together greedy approach long network successfully task character recognition particularly patch image face image network natural belief figure relevant belief estimator estimator applicability evaluate image patch thorough belief consideration particularly respect even train offer explanation observation analyze commonly deep network chapter review remainder constitute construct deep relevant likelihood reader want skip notation statistical assume denote h bend bend left cm cm h v bend boltzmann restrict boltzmann form boltzmann contrast
certain concave interesting comprising parametric indeed lot reference estimator consistency behave aim deep approximation somewhat broad open class probability density density view fact integral rewrite minimizer leibler well unless estimator regularity bound convergence applicable identify unique want class goal mind denote small show converge weakly entail well show wasserstein sequence converge large entail strong maximum surely respect
reduction online scalable outperform challenge car play image human treating prediction degenerate competitive introducing consider horizon execute policy step furthermore average policy necessarily know observe base well expert surrogate learner action expert goal surrogate loss induce input policy convex previous train policy guarantee task problem grow sup hence traditional guarantee due instead prefer guarantee growth near surrogate upper forward train non stationary policy iteratively iteration policy train
similar principle spline propose modify additive fit conduct fuse covariate haar wavelet basis univariate allow additional constraint producing discuss explore leave let assume increase define term subtract intercept univariate monotonically covariate decrease function additive model involve half close space function k objective term allow neighbourhood origin objective solution value strictly unique likelihood covariate total variation take account covariate attain observe covariate flat beyond value observe covariate monotonically mean optimisation space fit simplicity represent knot shall equal identifiability term component without previously focu univariate optimisation ordinary linear pn amongst
column iterate relate familiar linear algebra information gradient prescribe show least computation perform linear algebra package equation bad vector compute zero complement norm subspace computation time overall subspace orthogonal entry predict compute case converge satisfy analysis ode guarantee convergence
often lasso lasso additional regressor define thresholded corresponding ol selection gain lasso section thresholding coefficient separate oracle separate nonparametric small nonzero may goodness fit motivate goodness maintain depend select ols gauss apply select thresholded third fitness thresholded property post model circumstance post oracle selection occur interest application key quantity value noise choose small level possible dominate namely propose drive datum note literature impose obeys possibly dependent mild upper possibly datum drive example satisfy bc sa last ols finite sample traditional theoretically motivated selection
pt pt wasserstein lipschitz ks distance wasserstein ks estimate see follow give complete side equation converge pt e bound weakly deduce n pt thus lemma complete rigorous estimate coefficient proof quantify make cauchy schwarz qx qx therefore hence make index n used mention go therefore prove claim j acknowledgment thank anonymous reading clarity presentation nsf grant dms diffusion stationarity date diffusion naturally occur measure hilbert absolutely gaussian metropolis infinite hilbert value sde complex inherent structure quantify naturally study behavior dimensional space focus metropolis hasting metropolis rwm target rwm move propose walk although pt somewhat naive within hasting rwm application cost acceptance langevin mala needs evaluate gradient analysis complexity mcmc metropolis one author consider proposal space
node biological e significantly connection degree pair value purely graph simulate strongly affect traversal technique indeed densely purely discover inherently interestingly explanation node wave surprisingly opposite degree tends degree discovery former contrast walks expect distribution graph simulate cluster extension graph however turn preliminary result decide work planning incorporate ccccc rw absolute length collect life example facebook idea life system facebook social million facebook social topology facebook make implement collect facebook rw technique
capture output demonstrate technique model comparison show nonlinear strongly nonlinear dynamical direction definition sketch remark recent balanced truncation carry reduction belong reproduce kernel hilbert characteristic benefit simulate may simple lead circuit light process reduction dynamical system success date detail linear reduce essential input behavior nonlinear involve scheme nonlinear control exist idea space balanced reduction work convenient analytical theoretical understanding
algorithm matrix slide state could strong requirement adaptation study recent fit proposal covariance observe
subspace generalize principal reduction mixture space focus pure mathematic sample address homology specifically group relate sample topological belong local persistent homology condition sample topological characterization topological intuition reach finite bound
pose technical challenge heuristic investigation future sake conjugate conjugate marginal sufficient image finite interpret suppose statistic censor observation affect repeat describe individual interesting regularity admit finitely similar evy bayesian finitely remain precise projective limit limit measure use neither though appendix brief survey relevant projective limit limit projective measure projective theory object subspace construct marginal specify connect object generalize space large object proper infinity index direct set relation let di mapping projective set limit canonical mapping projective product canonical restriction projection projective structure canonical preserve induce projective space relevant measurable carry topology algebra system projective projective make mapping analogously projective measurable measurable borel topology define projective generate canonical manner projective limit structure family mapping projective respective projective limit g projective index mapping unique I word diagram diagram preserve projective limit projective system continuity projective system preserve projective algebraic structure notable even manner projective domain range kolmogorov projective countable j exist uniquely projective generalize kolmogorov originally space automatically satisfied projective limit countable index construct subset point negativity continuity address set aa occur
pose prediction compression would pattern recognition statistical prediction interpret principle follow let symbol represent datum symbol require short additional description short basis description obvious compression let concatenation symbol symbol denote sequence box symbol candidate represent predict next symbol xy interpretation paradigm paper check interpretation reality capable binary source b transition respectively distribution
slightly restriction show sd gap completeness protocol point fast start hard start strong circuit markov ergodic yes measured point let let q hard second part unlikely polynomial hardness hardness unlikely contain hard restriction rule relevant research physics sampler spin convergence follow circuit state yes difference thus informally case additionally restriction diagnostic decide decide hard least hierarchy circuit assign distribution associate pair circuit constant sd circuit satisfy satisfy constant theorem
linearization alm k fx way iteration function current approximation alm add prox term step fx subdifferential replace qx qx kx identical fail lipschitz constant thus converge iteration function algorithm iteration improve nesterov obtain acceleration combination iterate extend et
function asymptotic normality asymptotic parameter relative triangle expansion triangle triangle segregation alternative density q edge segregation relative score segregation alternative expansion multiple proportional solid central dash vertical axis leave conditional give unlike one triangle pe sm j pe sm pe sm triangle figure triangle case cs sm cs sm sm segregation alternative relative skewness recommend central large asymptotic skewness moderate around recommend edge similarity segregation alternative relative central cs sm pe since cs sm pe sm compare segregation present realization figure pe r triangle density tend asymptotic relative testing skewness compare similarity association central pe central proportional finite carlo carlo finite require design nan hypothesis possible nan case simulation mild severe deviation extremely result comparison agree segregation association furthermore compare agree carlo similarity segregation segregation proportional parameter agree conclusion power parameter central mild segregation respectively alternative central segregation agreement optimal extension edge proximity region higher restrictive might information denote suggest effect restriction proportional similar correction outside repeat outside carlo adjust proportion p adjustment affect estimate rectangular hand segregation correction favor right sided alternative segregation alternative expect hull favor side e illustrate
replica broken spin theory confirm large predict achievable strategy increase expect perceptron huge different two successive single configuration classify act local hard reach technique landscape simulate annealing also study landscape ising perceptron annealing indicate delta limit make advantage weight perceptron value uncertain weight enumeration adopt pass ise perceptron able propagation study hide discrete add system
remarkable pre entirely fundamentally change easy sis ensemble enhance point design characterize diversity give phenomenon false false reduce mechanism list care false false necessarily negative versa balance objective valuable practical algorithm mechanism objective mechanism find leave future main give description ensemble point ensemble tradeoff strength ensemble well st demonstrate many theorem article mechanism ensemble pick care construct compare numerous boost bagging solving shall selection ensemble ensemble variable candidate independent entry typically variable vote type considerably
easily integer pseudo sign vector follow take transpose fix choice recurrence give recurrence define left shift similarly matrix idea product take see recurrence bit length shift operation
numerous every indexing structure sensible parametrize compute computation take type condition essentially compute particular bounding bins string length asymptotically negligible assume generality indexing every bin family vc bins skewed concentration neighbourhood neighbourhood bin centre verify empty lead contradiction course applicable indexing scheme wants validate curse hamming cube exact space structure understand cell probe abstract indexing consist mm cell index alphabet view rooted cell selector computable define think string value hold bit leaf near weak problem whereby range distance cell yes preprocesse consist store bit string indexing initialize leaf level begin
serve communication module filter environmental control responsible computational module serve serve consideration capacity individual module module location relevant environmental input sort brain planning resource separately architecture agent build environment permit agent observe decide act conceptual analyze observation capacity unit computational measure bit
endow determine transformation leave gain endowed entity way element configuration certain describe ultrametric topology provide comprehensive number agglomerative ultrametric chemical section rise deal embed target dendrogram embed haar dendrogram wavelet filter transform deal relate especially set example financial exchange trading introduction hierarchical cluster family automatic discovery fundamental unsupervised classification generalizing decision making statistical generalizing unsupervise classify event phenomenon huge cluster algorithm visualization ii partitioning include article last mention agglomerative hierarchy consideration pairwise common approach comprehensive useful number deal euclidean geometry start fact number natural take consideration follow view carry quantitative survey approximate topology convenience fine take alternative norm besides infinity label p theorem via endow locally compact multiplicative haar p zero set p integer
iteration consist iteration alternate macro iteration index macro cyclic rule randomly macro computed component variation introduce coordinate x pz iv scale linearly happen speed technique implement svm one loss observe theory coordinate objective functional point differentiable optimality coordinate descent prove reference therein notice lemma state next theorem entry matrix coordinate descent recursively rule solution functional also denote provide machine
lemma note bind trace acknowledgement thank proposition matrix recover individual observe application numerical latent analysis via trace minimization programming strong assume spatial pattern outlier stand contrast analysis decomposition outlier decomposition modeling setting matrix low corrupted matrix observe latent visible precision visible condition general dense dependency visible hidden conditioning hide small exactly matrix impossible sparse
problem instability post prevent show difficult spin ise test system satisfactory acknowledgement thank valuable discussion com describe property choose poor especially ise spin phase transition start careful iterative system appear enhance pt know ise spin worker molecular interested optimize serious eliminate replica suppose experiment difficulty make ease presentation
intercept average flat intercept implement center treat generally proper bayes nan hyper use prior standard close constant computation derive however simultaneous hyperparameter gamma precision matrix normal scalar variance factor produce fix hyperparameter moderately inference link observational fisher precision eq except hyperparameter nuisance nan center covariate zero ml prior approach author maximize marginal eliminate
estimate error true vanish dominate yield even e g nonparametric elegant solution optimal tree nonparametric nonparametric estimate finite consistency qualitative property clearly quantitative characterization structure size graph optimize free interval estimate choose direct optimize early compare class graph bipartite use represent consider gx factor seek set kullback leibl distribution induce geometry configuration kl cross specific instance discover markov cross pairwise mutual empirically mutual take information estimate effect finite factor denote graph cross follow eq shannon entropy give disjoint cross entropy test estimate entropy term nn plug illustrate entropy test replace cross entropy elaborate v entropy partitioning variable theorem statistic variance entropy estimate independence individual weak surrogate term dependent factor factor furthermore mse grow graph entropy test mse surrogate test factor graph want bias dimension dominate case rise notion dimension graph index integer count factor equivalence graph model zero factor equivalence model high graph discovery knowledge translate maintain use expression fix choice mse weak law theoretically entropy implication denote beta let x x dd ensure dependence f
row norm maximum notion relate focus context column original variety exactly low presence present algorithm coherence arbitrary ultimately interested contain singular mention impact algorithm focus closely subsequently leave apply low matrix perturb submatrix ht store u rank u onto denote projection orthogonal complement statement monotonically furthermore
allocation loose tight upper allocation regret centralize however distribute account among optimality scaling explore impact secondary policy channel increase centralized predict centralized number bad decrease result increase channel increase bad channel evaluate performance different channel varied fixing channel channel along channel channel situation quality increase agreement number bad channel keep fix logarithmic nature number channel availability statistic know channel agreement theoretical number gap know increase cumulative statistic scheme differ simulation optimal performance enable provide central allocation allocation scheme centralize low eps favor user chance channel evaluate characteristic assume channel depict allocation approximately allocation fair learning channel availability statistic channel secondary cognitive secondary provable guarantee combine low optimal analysis
finally predict constant sized final purpose scheme produce predict sequence later basis stage predict predict real peak height preliminary contain preliminary step compute adjust statistic use represent peak position local context database generation table peak process perform require match process specificity depend incorporation therefore dna peak precede local context result statistic sequence run obtain sequence present diverse long peak position height choose top four trace represent basis peak peak model context precede nucleotide different represent nucleotide nucleotide occurrence sequence approximately instance possible accurate prediction formally peak statistic measure first peak peak measure peak peak eq peak height table position height context height range position independent linearity peak consecutive peak move nucleotide represent peak peak position next peak peak distance distribution around show local
connectivity careful strong build road nearby indirect disadvantage apart say nothing length discussion length long straight sensible provide short technical level advantage disadvantage measure efficiency meaningful different calculate realistic criterion depend issue population incorporate optimal create short weight tailor sense exactly distance city form average interval width recall take area calculate proxy shorter assign regular study problem explain
approach support machine couple feature adaboost preliminary appear basic notion learn characterize hypothesis upper alternate encode bayes explicit sparsity propose learn evaluate select case publish finding dimensional gene set classification example input draw otherwise measure risk attribute attribute input conjunction direction decision finally label example towards conjunction code bit ultimately guide zero binomial classifier observing least observe point tight also bind decision suitable moreover valid discrete value continuous bit let conjunction prior string let denote possible dimension pd motivate come final specify choose decrease increase reason decision small eq q
build cholesky check near singular number right panel figure contour average h behave proportion contour leave panel rapidly somewhat intuitive region dimension design need conditioning correlation high design design follow number behave choose criterion lead behave correlation behave accurate compare section force model undesirable smoothing singular increment methodology simulator enough interpolation may ill simulator noisy bias mis require attention smoothing undesirable computer simulator confident desire interpolation depend develop behave choose iteration attain desire interpolation recommend inaccurate ill conditioning new rewrite profile replace ill behave get parameter behaved substitute spirit evaluate accurately
frequent value individual basic procedure treat final set snp imputation snps snps causal range snps correspond correlation snps indicate snps equally distribute individual range aware set phenomenon large snps play procedure multiple testing wise adjusted approximately procedure perform fdr model see calculation control approximately correction correlate regressor adopt detect snp causal snp absolute threshold false decide threshold absolute snps procedure frequently snps correlate causal snp threshold detection individual causal snp fdr tp figure fdr replicate threshold procedure fdr direct consequence fact selection procedure much large power discovery rate though small effect fdr procedure reduce positive section tb detect level respectively snp evident also power
deal reformulate distribute know underlie describe far small distribution realization get p n section frobenius completion imply logarithmic completion norm set corollary immediate let q constant completion matrix introduce distinct contradict small numerical already mention remark obtain inequality achieve lasso trace regression q ia diag without gram rescaling become usual little abuse large
distance transformation express patch assume adaptive single closed approximated linearization around put component include speed employ expectation maximization step video patch frame hold parameter latent use note degeneracy due eigenvector rescale remain unchanged scaling row row length ill condition detail rescale degeneracy scaling prevent multiply correctness know transformation construct apply specify construct
respect impose gradient natural interpretation variable irrelevant norm derivative respect norm importance selection motivate encourage add variable automatically derivative sparse learn data major innovation construct organized section certain proof delay describe implementation infinite backward splitting effectiveness gradient extraction world value draw response variable direction dimensional span zero depend function respect quantity et al gradient span development method direction regularization datum fit x f x jj taylor observations ensure locality example control bandwidth implement tune fitting might meaningful dimensional datum explain lie relative specifically embed well give taylor expansion expect j iv v yield sparsity motivation
converge discuss example appear stop order retain sufficient therefore simple powerful rule determine shift establish simple evaluation denote rule fourth shift quite estimator shift distribution parameter
remark evaluate section attain roughly digit good unlike alternative number point depend bin easy quadrature range rather quadrature double roughly digit subtract contour draw proceed form define cumulative positive sum center positive remark describe evaluate numerically illustrate via numerical example statistic limit statistic draw use cumulative
code experiment remain challenge model scalar depend high sensitivity propagation avoid huge calculation computer code mathematical polynomial neural gaussian gp extend krige principle two response numerous powerful code response particularly difficult design fit range hypercube discrepancy moment theoretical lead fit numerical answer fundamental important address set simulation accurate degree two approach paper gp analytical example evaluate performance fourth look validation problem minimize test sequential summarize
spectra classification mean similar model spectra incorrectly fail actual failed incorrectly spectra subtract spectra spectra fail highlight wrong miss update object real spectra obtain spectra model detail select school pattern show valid application evaluation discuss wish comment suggestion thank due centre provide computing support organization grant algorithm base primary school methodology continuously repeatedly uci produce ability rare neural network algorithm reflect generalization unseen depend explicitly quality training good hereafter implementation play crucial role popular large close logic machine
analysis gradient several interesting question explore kind combine different acknowledgment aa microsoft fellowship fellowship careful reading provide helpful basic distribute averaging recall without indicator function let conjugate uniquely strongly convex uniquely two important first bind integration upper q product note thus use eq q rearrange early thereby obtain upper bound eq exploit fact q second bind continuity triangle lipschitz tt lipschitz continuity projection state completeness continuity give short proof completeness denote obtain imply review singular gap connection tp tp remove calculation tx simplex chain node volume graph laplacian e path normalize k invoke conclude k node minimum easy assume loss generality calculate assumption low note direction pick direction node path adjacent adjacent since let leave right end share k kk result section generalize dual incorporate objective derivation brevity work conceptually equally write negative define composite operator conjugate lipschitz dual
sum square pool return routine free much q cubic buffer provide sufficiently exact suppose return pool differ pool previous pool undesirable pool square transformation could subtle simplify uniformly write unit occur although undesirable pool
vice versa drop instead refer various form recurrence fourier consider version bias ranking vice versa bias control one reflect preference subset object vice versa recover straightforward generalization bias rich drop prefer discuss certain operating single rather entire consequence conditioning operation certain map posteriori decompose straightforward inference decomposition independent pointwise respect bayes compose subset come form pairwise prefer argue pairwise depend whether prefer object object say factor rule less observation prefer affect intuitive conditioning rank like subset respect respect except post prior sketch factor uniform relative factor distribution conditioning item require associated rank condition condition modify show comparison involve update cover intuitive ranking estimating item would independent raw statistic second probability discrete consist form count thus mle simply give formula eq like compute interested estimating definition may interested know marginal compute item rank ranking quickly intractable main remainder compute explicitly compute perspective raw probability reader jump directly fouri theoretic exposition unlike analog real fourier transform take fouri order respect discuss analog coefficient order rough frequency reconstruct marginal example first marginal reconstruct familiar continue paper fourier ii fourier scalar paper consider approximate truncate set fourier inference operate fourier marginalization conditioning ever fourier generalization tackle generality discard later section follow fourier domain join conversely compute return h refer theoretic b I domain fourier join tell fouri convolution
network different degree homogeneity semantic concept inter versus intra behave differently relevant algorithmic statistical I discuss chapter role determine network extremely enyi roughly concentrate sufficiently fully close node simple mechanism reproduce qualitative deep cut well balanced deep cut law due variability analogous qualitative new randomly mechanism reproduce qualitative implication ensemble return spectral versus manner know operate piece set interpret scale nod piece interpretation flow permit piece bag combine flow require piece flow become spectral find piece long path cut obtain coherent substantially small figure local spectral consist piece conclude section verify understanding course cluster smooth recall solution pose theory view bias substantially variability computed quantity typically natural regularization optimize avoid fit lead hard think regularize least regression discuss interestingly large intuitive notion tend optimize community score instance obtain intuitive rather notion implement introduce systematic compare much intractable bias case spectral regularize community incorporate cut internal implicitly incorporate statistical benefit formalize note algorithmic ask prominent include statistical probabilistic central recent case otherwise
improve submodular result even bound distributional et game theory economic build al efficient new section structural et submodular show possible obtain computationally inefficient within within coverage e prove expressive submodular problem preserve analysis learn answer submodular technical piece build allow lipschitz monotone access appropriately normalize stability obtain algorithm statistical query target error appropriately consider submodular represent pseudo membership ok simplify graph also chi van grant grant fa discovery microsoft fellowship ex plus minus ex ex minus plus ex minus plus minus ex plus ex ex ex plus align ne ne also interesting property illustrate via example collection lie lie
subsequence imply imply infinitely sign infinitely conclusion singleton neighborhood versa indeed know continuous notice sign monotone l remain jump c parallelism subspace translation versa left exist c cp j c cc hc hc ce ce p uniqueness origin move segment continuous paragraph continuity theorem financial engineering york ny well perform lead due spectra accumulation researcher sparse accumulation biological responsible extent misclassification discriminant independence finite regularize optimal discriminant road road select narrow pool road coordinate study theoretical advantage variety structure result piecewise road interpolation impact scientific throughput thousand sample challenge overview challenge dimension discriminant rule perform
suitable configuration simulate towards accept eq accept construction dark either light configuration attractive whereas configuration add large plot follow moderate scale scale among plot wish multiscale give result use use value moderate scale seem slightly large scale possible interaction envelope solid transform remain point datum remarkably small scale provide appear define analogue process transform fourth subtract yield value process plot thing firstly choose perhaps scale ignore gain behaviour nature main advantage perfect exponential feasible
automatically publicly assess apply differently shape normalizing produce fully result simulate several variate allow variation case asymmetric multimodal obtain apply normalize different shape real implementation marginal involve substitution branch define building integrate hence method need two tp evaluated ingredient monte simulation code prune straightforward prescribe density must kind obstacle lie half eq become obviously call jacobian like substitution follow simplex set constraint entry rate constrain space call call additive transformation nucleotide
base act transition conditionally tx probability consider set choose probability respectively estimate transition rely kullback divergence seminal prior mdp need kullback leibler divergence convention sequel dramatically behavior significantly performance section advantage use kl divergence instead norm illustrate mdps optimistic detail tn jx tn
resort possible approximate requirement simply reduce noise gaussian mean co eigen contain q let submatrix correspond eigenvalue al ml mix ml arbitrary estimate source source et et suppose entropy worth constrain correlation seminal et approximation entropy work expectation solve together al differential constraint give normalize version differential case odd value average et entropy density nh entropy empirical minimize constraint date sometimes correlation modify maximal introduce transform maximal impose therefore development square source exactly exactly uncorrelated word vector ignore correlate nd term drop correspond correlate case estimate I subject element orthogonality important orthogonality enforce orthogonality
sequence sequence sequence indicate progress frequent team thank collaborative overcome towards goal area configuration action recognize sequence take predicate meaningful pattern team use per configuration regard configuration make meaningful team fisher use power datum average occurrence pattern class discriminative measure gain fisher discriminative two discriminative power limited power suitable mining represent frequency calculate among different frequent team reason enough adequate discover frequent team fisher team b c b b c action collaborative behaviour sequence team purpose ball relational action behaviour team predicate description agent act consist predicate action represent considerable state situation abstract difference behaviour robot front front pass typical significant forward front forward characteristic team front
convergence ts ts imply q obtain eq yield assertion use c lag west estimator satisfie weight limit along proof since continuous chart e df df control chart particularly weakly ks dr dr correction alternative interest situation unity depend horizon limit asymptotic unity alternative affect nuisance account array observation ar satisfy brevity drive brownian motion assume unity process process crucial argument weak numerator sketch tr ts converge uniformly exponential
unknown except plug measurement probably become dimensional bayes pure symmetry one base explanation neighbourhood prove beyond scope discussion support ep theorem quantum school mathematical sciences ng rd united science mail ac pattern theory numerous recognition image diagnosis follow feature unknown feature quantum copy outcome classification future find asymptotically typically estimation excess excess theory broad research statistic devise pattern practical recognition computer diagnosis paradigm information quantum carry potentially fast engineering develop accurately quantum engineering statistical procedure pass purely status practically orient interface put inspired
diameter metric embed hence proposition yield satisfy metric exist input proposition agglomerative problem note every cluster equal diameter corollary norm input time compute norm follow time embed without agglomerative instance state upper bound factor simultaneously precisely sublinear raise doubly factor center diameter limitation extend general distance improve lower bind proposition observation appear publication available department bl foundation grant bl subset subset cluster compute approximate agglomerative complete linkage decade practitioner agglomerative dimension norm analyze closely center consider give maintain input show agglomerative agglomerative agglomerative notation doubly exponential approximation level compute optimal clustering necessarily hierarchy approximation independent technique prove center center logarithmic
policy respectively core however unclear whether ever hypothesis undesirable could right htbp hypothesis operation mode policy select region dynamic know region analysis policy agree operation iff theorem law mt tw mt pm almost surely choose probability q proceed member similar inequality apply multiplicative
e negativity become throughout loading decrease also borel paper stating imply load lead new loading loading monotonicity specify outline research consideration answer subsequently number interesting mathematical house hence depend form might equality exist whether answer naturally rely turn variable sort goal distortion pose justify section definition whenever facilitate submodular
penalty beyond avoid instability function nuisance demonstrate penalization illustration constrain programming constraint symmetric quadratic minimizing tend calculation define convenient appear quadratic handle diagonal sign check reduce h ij mi avoid ignore numerical x minimum quadratic program x list iteration acceleration constant fortunately acceleration conditioning problem convergence inner loop
need specification role convenient conditionally change u gibbs consist alternate draw update update directly possibility often attractive alternative categorical probability full conditional full conditional mean full
connect remove remove edge nan connect span vector component connect projection group coordinate fuse boundary graph change dual part coordinate value boundary group correspondence note primal dual fuse coordinate leave dual primal fuse apply boundary remove corresponding edge boundary remove connectivity happen remove coordinate path leave dash general last focus covariate problem much solution suppose py u u u definition fourth rewrite transform rewrite new problem dual study column constraint treat px px nd xu path dual mean also piecewise logic fit conclude working would need expand newly hold fuse still boundary coordinate leave graph x deal fix p apply put aside concern may preferable instead may example fuse prediction analogous perhaps look rough sketch computation rewrite constraint svd basis kkt modify incorporate become subject instead appropriate block satisfy leave consideration discuss solution primal rely path expression
rotation g gs p q p gr rt q translation note p order pair within order matter argue one pair grid ir jj ir composition rotation affect translation error subsequent show recursion affect rotation point since runtime order give term recursion grid power size rotation multiply approximately preserve problem surface range space continuous kp weighting capture surface compute metric computer image
product assumption write hence discriminant decide function relevance determine boolean present classify optimisation formulate construct pattern find subset examine impractical obtain explore subset find adopt discriminant initial optimisation characteristic pattern make give seek method quality solution combinatorial greedy empty candidate add good local improve search well local optimum greedy adaptive iteration local construction neighbourhood algorithm procedure perform use search
surely almost surely cf correct precise apply fact law number triangular array identically distribute variable form state immediately triangular nx algebraic conjunction sequence number next distribute ni dx stein jointly zero converge kn lemma function independent absolutely conditional random denote immediate follow prove strictly variable convenient span sequence limit element wise characterize algebra p recall formula vector I I onto random dd mn rotation see dd b lemma lemma appear note argument easy non ta tx equivalent simplify drop conditional denote multiplier lagrangian usual impose yield multiply multiply obtain leave f ff aa correct vanish induction let property inductive hold hold
refer correctly foreground represent distinguish video omit almost performance foreground video regard comparable low score rank lrr application stock clustering make try solve follow derivation z rank structure cluster effectiveness application apply platform video summarize cluster video correspond task lrr consensus affinity subspace ssc presents assume linear ssc lrr liu model comparison norm lrr provide thorough default suggest lrr segmentation post low seek good lrr get processing enhance low sc contribution subspace exclude post effectiveness result method video three report error two know slight improvement lrr lrr already promise highlight problem consider htbp
markovian markovian reader point book single perspective computational challenge generalise standard selection goal impose count response generalise explanatory partly conditional transpose sample explanatory generalized specify belong family expectation parameter possibly dispersion link relate bayesian proceed outside model prior distribution extension discuss chapter motivation factor scale avoid exploit generalise binary corresponding thus easily handle chapter illustrative diabetes probit presence diabetes predictor bp pressure diabetes residual min coefficient std pr bp degree freedom degree fisher column star characteristic exist covariate strength bayesian impact special associate discuss competition say bayes associate hypothesis approximation link competition exclude covariate matrix bp individuals bp notation
nonlinear gaussian particular wind wind contain chain zero jump nevertheless outperform forecast forecast review consider forecast hour forecast mention wind time highly nonlinear wind power appropriately aggregated generation relevant power wind forecast aggregated generation study consider aggregated wind series argue utilize correlation wind forecast aggregated wind power multiple time unless generate aggregate wind univariate series ahead density forecast wind wind wind demonstrate normalize aggregated wind describe density truncate forecast simultaneously step forecast normal although perform power forecast computationally forecast obvious normalize organized wind explain approach concern logistic benchmark performance various proper paper benefit research direction aggregate wind generate every total wind wind facilitate comparison wind
remark ergodic consistency exist parametric notion homogeneity achieve assumption neither assumption case example quadratic goal formalize concerned generating often cluster similarity assume similarity good even one reason cumulative seem goal np financial observation case ergodicity consistency achieve within distribution datum ergodic assumption stationarity intuitively
discuss mutually might code significance utilize child represent probabilistic take x utilize node nx joint probabilistic density neural x minimize make property rule nonzero contribute update come child child activity
occurrence em last sum allocate update update nu jj nu nu nu nu jj ji jj jj nu ji ji ii computation replicate last overall show cccc sufficient correspond correspond large statistic computational requirement normal
contain state distribute accord mild restriction converge portion ask independently transpose relate complete thus get integral xt rt residual term shrink structure refer stand maximum small non row diffusion lyapunov equation regularize state assumption submatrix incoherence let defined observe enable reconstruct network state support trajectory square recover support logarithmic polynomial roughly
exploitation epoch lead th exploitation epoch reward exploitation epoch upper cause denote arrive arm regret arm exploration cause mistake epoch play exploration epoch cause play time spend arm dt cause mistake exploitation part total f incur incorrectly identify player explore occur occurrence event slot mistake player logarithmic cardinality exploration thus expect logarithmic occurrence logarithmic prove also logarithmic consider contiguous consist player correctly identify good two arm randomize player good arm order singular period logarithmic theorem transition edu liu player play play evolve accord unknown passive reward player know arm good construct exploitation structure achieve nontrivial certain available regret player exchange show decentralized preserve centralize adaptive
assign nx irrelevant asymptotic consideration k arbitrarily minimum distance md mapping minimize whenever arbitrarily otherwise course closed serve auxiliary device asymptotic furthermore whenever satisfied view well turn md estimator fact tend follow strong suppose minimize n furthermore converge minimize assumption event value compact kx early p b remark tend proposition b remarks iii observe tend coincide therefore b subsequent immediately n nx event tend event md md coincide md underlying satisfied attain particular mod inclusion satisfy minimizer outer md remark outer detail see assumption always minimizer uniqueness consistency result md estimator lie show md asymptotically normally matrix md show carry estimator standard non partially differentiable condition hold always possess proposition theorem furthermore hessian minimizer mi additionally sense proposition ii open space probability tend mean value evaluate mean invertible outer proposition proposition continuity satisfie central lemma n p appendix interior view consequently converge proposition eq
iteration diameter begin agent distribute deviation agent observe neighbor action normally distribute graph know linear coefficient value add memory already span space span belief memory keep unbiased calculation involve invert linearly independent take alternative improper measure private different q finite normally since turn give otherwise
simplex rest symbol well value space functional become analogy functional space note elementary elementary elementary calculus view often differential discretized define order situation abstract absence geometric vertex need geometric differential capture star operator first take either purpose edge edge elementary triangle correspond triangle entry match appear triangle approximation manifold e partial differential play important boundary arrange zero operator operator decomposition describe chain arrange reverse deal value operator require chain write relation write elementary clearly analogous topology determine recall definition tool distinguish space homology notion topology value boundary come diagram call equivalence cycle cycle say chain coefficient homology respectively homology homology real homology one piece information integer homology turn space consequence coefficient simply side dimension algebra fact four subspace chain combine laplacian combine diagram duality one box leave geometry operator denote act numerical analysis increasingly diagram geometry consist form vertical identifying via duality abuse three clique definition operator study clique homology number algebra combine vector space desire recall clique decomposition analog analog negative divergence two two analogous
learn intuitively every agnostic learner sample technical consider penalty weight method close update step predict expectation qx qx qx distribution universe px qx step reach return recall multiplicative weight qx b qx remainder far else indistinguishable qx tx qx qx finish long proof equation every union stage occur probability assume equation satisfied put reach point subroutine concept q assumption whenever conclude concept clarity probability release grow call union agnostic reverse let least label use simulate condition return approximation qx np qx x simulate run obtain answer except prove monotone polynomially monotone learn
future pt symmetric rectangular occur problem economic statistically meaningful especially factor involve possible systematically find
cardinality arbitrarily set denote intersect complement depend dependence family finitely x family ii extend restrict case simplicity work conjunction vc sequence measure establish conclusion corollary mutually exclusive family vc depend borel measurable clearly infinite vc finitely
level explanation actor path know relax place prior allow membership mm actor identify level interact actor stick prior actor number stick construction stick length let remainder stick length remainder stick fraction general draw influence mean influence hierarchy learn stick interaction fine level expressive expressive actor indicator level mm vector pair specify hierarchy denote contribute actor propose full conditionally goal finite depth community entity connect node link level actor involve identity level interaction community identity specify
lin school business chen computer science school business optimization compose smooth smooth expectation type interesting propose incorporate method first proof problem decade program lasso lasso optimization especially bad exact formulate objective problem reason mention solve unbiased fx gx subscript present constant convergence rate show convexity function utilize inspire
journal another lead symmetric work display structure left side figure homogeneous intra weighted find homogeneous intra inter distinction interpretation find exhibit small intra link play reference indeed top left find american review journal journal economic journal political economic review economic review economic statistic health health economics journal economics natural journal economics economics environmental economics management economic history exploration economic journal economic economic history dedicate topic economic less reading decompose let moreover number sure given establish identically event q establishe come prove limit central apply appear side establish limit decompose sum product denote denote map obtain singleton give whereas indeed value negligible term converge infinity suffice first term rate indeed go
define exist test universal hoeffding test family support sequence family next bound exponentially answer ask approximately design suppose function dimension divergence approximate alternate keep universal tradeoff propose class
seem te suitable maintain te give worse order give low strength te couple system embed time small time te unstable within radius te small discrimination coupling embed show three coupling ahead noisy seem flow increase couple te three reach discrimination still reach te whereas small pattern small strong free te te decrease lower obtain give good b r driving strength vary
examine arrival job service time invert detailed request receive request day reason build library source performance system system possibility whenever probability choose call record call arrival queue job whenever task easy time job task observe sophisticated observation certainly example appear order detail information random task percentile approach work key difficulty interest job divide service represent delay service service queue capture response job cause missing goal mcmc time invert equation approach alternate likelihood approach design sampler complex subsection difficulty rejection metropolis design even simple conditional varie arrival see two processor horizontal vertical arrive enter service finish service time service exponential rate arrival condition wish sampler queue dominant service excellent inexact change poor heavily conditional proposal decay exponentially precisely consider unobserved arrival modify job force modification service later arrival keep terminology graphical model markov processor queue panel depict value
alternative variable shorter continuous understand essence may wish shorter real continuous x g use take g iff f x x vanish denote g h g x g g g g g g g g negative iff identical prove another p x x x pt g g I x easy x b theorem pt x
done evaluate statistic compare variability within chain latent ht cccc two sparse robust variant measurement sparse instead parent box latent embed dependent initialization issue inference practical sparse model statistic structural parent gaussian term interaction parent perhaps use rbf determination pseudo study predictive respective set height indicator variable variable indicator original neighborhood correspond within fourth air due data file non give use neighborhood refine neighborhood neighborhood skew indicator among direct accord
zero contribute kind new step new arrive moment make neighbor examine examine material perhaps effective one illustrative atom atom atom atom atom atom atom empty slot non atom atom
fold cross kernel page threshold affect support classification purpose experiment set testing full full first fourth band band band band moment see text variety way vary object object result decompose object decompose experiment r text r r result mean deviation text reflect object identify object depend visible observer point testing raw repeat object decompose object result result well raw visible datum distinguish
single add outlier average call function node see right presence submodular depend fw w behavior decrease explore term cluster together show allow allow strongly plot scalar replace cm lead variable piecewise great middle figure leave outli present optimization regularize lead problem tackle proximal subgradient inefficient piecewise affine consequence present subgradient one proximal cm cm proximal cardinality set piece level total case still affine agglomerative
prior plot expand history expand gibbs sampler prior expand figure three px sampler sampler pseudo default start seed code reader replicate unfortunately expand offer explore small improvement show shrinkage prior appear model bayesian quantify sampler run combination
course facilitate use estimation condition vector strong point augmentation problem piece point nuisance regard result nuisance parameter useful discussion shall detail motivation present present methodology extension main suppose sample point weighted write dispersion almost form estimator unknown need estimation simplicity throughout quasi likelihood linear normality taylor presentation quasi estimation residual regularity condition simplify next subsection remove section identifiability stage describe one assume parameter dimension matrix great nothing quasi title subsection
n line w en p n n n dt dt put everything pz k tc p tc outperform define small problem scale go estimator normalization substitute become behavior behave assume outperform separation estimate corollary proposition conjecture multiscale multiscale system review result
snp association case status significant genetic play disease genomic together association jointly locate snp feature selection identify feature interest weighting near construct one whether neighbor neighbor value randomly neighbor weight strongly distance attribute drawback significance far find interact instead exhibit snp average one propose idea subtle systematic difference individual population identify pool breast close pool pool control suggest control building idea technique pathway set control pathway distinction analysis snp relationships pathway expert biological closeness pathway snps show genetic disease pathway pathway seek identify differential heterogeneity control return quantify case relative pathway snps control pathway quantify manuscript detail first permit pathway issue snp abundance significant marker well pathway whose leave return odd conjecture pathway disease exhibit distinction design snps close control close systematically snp snps compute remain snp relative distance statistic question amongst adjust hypothesis find snp
mean truncate specify redundant parameterization length burn phase length value random effect poisson largely major uncertainty due evident pooling uncertainty shrinkage evident estimate uncertainty shrinkage evident year either credible theoretically practical describe describe section material evaluate look diagnostic quadrature
finite approximation mle depend tradeoff nonparametric detail process admit eq kn principal infinite sequence finite model ny kk rest vector purpose notational b b rewrite mle cutoff accord variance maximize likelihood cutoff note much exist universal characterize slope assume course essence stage lemma serve build establish main proof section unknown write lemma notation straightforward parameter lemma state
nonconvex overall objective mkl early set relevance correspondence process structure start strategy kernel weight furthermore framework numerically mkl categorization task contribution classifier extend framework jointly literature concern formulation theorem formulation mkl belong space classification kernel specifically rkh combine rkhs rkhs represent rkhs correspond function zero sec lemma intuition supervise combination write hinge might introduce
sec drop distribute accord translate function localize mark three without sense treatment incorporate induce incorporate virtue pure wiener case add uncertainty add operation spectra filter finite amount pure filter correction describe wiener filter pure spatially statistical inference discover independently theory quantum mechanic dependence theoretical extract partly smoothness controlling derive smoothness gaussian signal covariance know wiener filter spectrum characteristic wiener filtering gap spectrum extension exist problem assumption implicitly beneficial answer several optimally incorporate filter really assume deal relevant answer question accurately example wiener amplitude signal pixel free distribution infer spectrum prominent rigorously spectra scheme trivial analytically interesting work uncertainty problem spectrum filter principle go beyond pure fourth generic provide specific application pure spectral fidelity sec smoothness present finally conclude sec briefly extend uncertainty well terminology find spatially signal measure treat spatial abstract vector observational dealing precise field freedom therefore call usually bayes marginalization field function
compatibility eigenvalue depending restrict follow adjacency unbalanced degree quantity c eigenvalue parallel give selector construct graph precisely say design requirement hence design satisfy distortion universal show unbalanced satisfie adjacency unbalanced degree satisfy however subsection
e follow weak epoch condition put weak large number hold restrictive class indeed take cross conduct illustrate approach real module quantity assessment simulate power module regard acceptable denote standard represent hypothesis validate sequential smooth depict approximation apply detector conduct monitoring rule obtain monte ensure run length simulation mean real chart jump l ccccc delay dr uv financial foundation exchange service remark anonymous presentation ts ts k
flow dimension obtain decrease submodular include cover include source notation b net get max flow cut theorem decrease type decrease belong flow flow get method submodular induce submodular function one fa g g compose subset connect contain capacity figure submodular fa hx extend consider differential lead matrix relate positive function concavity lead submodular p submodular thus decrease submodular find rectangular consider covariate unit consider non increase k certain subset cycle component subgraph column support grant de european author like thank discussion pt pt pt plus minus pt plus minus pt plus pc appear area computer science apply vision operation submodular space principle good knowledge book article material mostly order present material relate research paper
kp ir ks lrr variable dl atom integration notion every variable least atom force variable atom reason generality adapt dl kb student mix dl say student say notice weakly safe safe semantic semantic nm semantic distinguish head atom body h r semantic stable dl predicate still predicate particular ground query analogously answer perform problem rule atom problem linear nm semantic treat verified ground rule always applicable act sure rule inconsistent conclusion dl predicate moreover semantic nm aforementioned boolean dl dl adaptation model semantics gr obtain dl dl appear rule dl replace similarly partial except rule occur atom denote
fig active information average agreement offer new deal computationally expensive however expect truly question mi aa use scalable selecting centrality performance type deal network heterogeneou low mi aa correct block acknowledgment helpful web j support cs edu com com edu find node integrate independent
know impossible monte integration condition weighting q allowed condition partition combine unconditional conditional weight nearly therefore perform integration q show gibbs markov monte obtain partition constructing equal implement gibbs sampler collapse include iteration sequentially remove chain follow base measure partition observation cluster partition gibbs markov chain limit change hold let fix normalize base conjugate criterion rule number burn discard burn thorough sampler n base initialize discuss weight convergence concern dirichlet produce distribution neighborhood true like examine numerous conjugate normal continuous condition provide simplex base weakly true weakly convergent almost surely weakly convergent
affect scenario challenge perturb scale brain signal intensity need paper know insensitive perturbation cp small problem
dual conjugate translate employ substitution translate hand translate combine lead unweighted sum regularizer call sparsity induce regularizer considerably entry hence mkl favor solution note recall side optimization translate maximum subsequently expand slack result mkl mkl kernel employ norm mix identity norm dual special hinge optimization study primal sketch equivalently express group dual small formulation handle solver however quadratic constraint non demanding propose unconstrained norm verify notice differentiable memory quasi descent anomaly detection svms description cast rise align c often expert instance interaction form compute frobenius dot product scenario pilot kernel inferior one scenario handle consider isotropic regularizer verify desire isotropic straight fashion instance standard approach mkl optimize remain recent mkl solver set svm commonly albeit appear kernel cache pre compute carry repeatedly induce heavy memory always optimize end many require certainly suboptimal suffice master mkl kernels set commonly application bioinformatics database security present mkl subproblem decomposition embed extend sparse problem analytical update typical version defer algorithm first strategy op group hand derive operate coordinate descent gauss thereby carry basic
bank identify mesh ht ht accumulate extract information bank var break plotted policy quantity provide bank capital equality ht observe attribute percentile subsequently risk extreme significant value begin coincide dependence divergence sufficiently significant risk provide discrepancy figure attribute call understand provide set derivation subsequently great occur risk equality policy reduce tail tail justification frequency distinct difference fact extreme decrease comparative capability extreme loss extreme decrease decrease probability dominate become event consequently two bi variate figure low aggregate impact coverage exposure limit rare low figure figure display exposure dot line bank exposure dash risk bank demonstrate bank dot line coincide light
straight line necessary source point calculation ref rewrite intersection six fig rewrite may express rewrite nonconvex expression nonconvex body explain expectation us b due coefficient nonconvex distribution measure straight constant absence analogue nonconvex body bl l bl bl body average multi early sec prove possible preferable voxel small surface smooth surface sphere carlo straight boundary tangent distance along straight intersection consideration orientation distinguish often
lead denote q denote thus solve advantageous generalized reason restrict absolutely ordinary convolution via fourier transform restrict transform choice regression subject
interval increase verify pooling unit deviation estimate value around second second intel cpu expansion determined parameter simulation average ci skewness skewness skewness ci skewness ci skewness mean ci skewness estimation population incorporating propose simulation rarely study trivial estimate difficulty derive density numerically transition approximation inference estimator several decade e review suggest multidimensional result study move quadrature integral effect latter grow need count statistical quadrature option quickly grow reference review mixed framework devote integral g decide symbolic calculus long symbolic calculus software ad already
converge perform decomposition find couple original formulation inherently nmf formulation write one way update value fig
snr range limited complexity use show cause contrast purpose cm phase perform perform classifier sensitive function kolmogorov achieve
powerful suppose computed wish discard strong sequential giving motivation demonstrate example safe strong value panel population negative four panel zero value plot stage safe exclude begin remarkably rule scenario dense bottom panel zero plot along plot screen slope plot penalty panel predictor panel outperform still winner except standardize give motivation strong rule sequential kkt subgradient let derivative ignore conclude eq
distinguished computing stay distribution p bayes time approximate x approximation compute analytically symbol predict predictor measurement next measurement prop predictive tt transition predictive approximated exact exploit compute filter closed eq integrate p measurement exact mean marginal measurement exact marginal measurement distribution give respectively
mutually therefore induction start calculate conditional q iii edu family square properly provably square deviation expression choose advantage convergence assumption regularization least identification interference rate ease implementation robustness scenario example impulse performance several early active
behave map define lack degree exhibit interesting seek could category cf manner simple example sep collection space eq empty construction follow generalize seem make rich addition particularly quite constant finite equivalence define check yx x xx xx yx x xx yx name come construction metric later construction context group restrict ambiguity desirable call existence pick choice context refer partition subsequent application ax xx ix family invariant recall x xx xx yx xx ix ix yx explicit apply space depict metric non permit turn collection isometry metric space clearly specify manner statement state obvious
keyword nonparametric prior increasingly popular last wide interest bayesian approach inferential nonparametric group see specie ss allocation non atomic collecting distinct non weight exchangeable ss characterize predictive characterize dp exchangeable appear literature g covariate derive predictor dependent generalize mechanism relax exchangeability predictor implicitly special maintain exchangeable sequence notably hence marginally stationary representation introduce rule reduce known case choice instance coincide characterize beta random call choice specification specie simplicity later section allocation also scheme individual characterize mark individual determine cluster assign tag tag subject tag cluster observe tag discuss induced specification weight define block
apart replace exchange move computationally calculate exchange acceptance pair one l maintain balance condition must chain whose select pair denote temperature chain I accord relatively instance finally adopt monitor delay choose batch add delay rejection liu end period precise first total batch burn laplace approximation specification case line easy drop subscript response distinguish posterior guarantee recall posterior transformation avoid boundary square calculated laplace find throughout mode root de jacobian calculus log ty r real solution moreover c c real solution one summation pair product coefficient especially become proved guarantee large middle tend zero fulfil solution equation c define c great two root order admit positive equation unless mle bound positive axis propose logarithm current every acceptance distribution batch arbitrary batch period burn able set impose batch burn restrict inside indicate batch figure report ess analyse ess independent enable implement independent fair perform fix ess ess secondly ess define implement former latter proper exponential ess challenging
objective ratio quite restrictive modeling perspective homogeneous quite interesting standard eigenvalue relation unsupervise application spectral cluster prior work nonlinear graph cut recent improve considerably cut oppose eigenvector second principal motivation pca pca difficult interpret nonzero kind study recent reference pca natural characterize min principle several modeling restriction functional positively pg gx sf homogeneous eigenvector easy functional standard functional gain differentiable operator
give explanation rate partly al ensure numerical shall establish strict monotonicity yu show rate eigenvalue combine convergence yu monotonicity iteration strict monotonicity monotonicity emphasis strict
fast theory introduce discuss possible transpose symbol one write identity write give vector say compose old trace financial observed construct optimally manner completeness theory matrix return objective problem portfolio weight straight multiplier covariance data portfolio literature write et minimum portfolio construct subsequently arrive early often computational delay trade stream allocation information allocation assign constant portfolio portfolio allocation technique study al outperform allocation study multipli fix window slide ensure length selection conversely slide q ridge formulation detail fall give recursive estimation able devise asset allocation
feature censor collect mixture writing seek current conditional step ei iteration maintain monotone log likelihood mle ar potentially exist heavy auxiliary eq eq appear twice formulation indicator component component proportion become step routine algebra reveal mixture proportion equivalently derivation indicator collapse version density observation proceed calculate formula expectation iteration correspond fraction datum augmentation van improving call obtain subtract component slight show much
third third argument upper finite payoff hold subsection average throughout additive norm side lemma sequential martingale difference sign close invoke prove general lemma generalize two integral bound wish scope upper bound sharp start obtain bound particular time q smooth norm identical lipschitz smooth suppose proving upper lemma martingale banach space tree concentration smooth smooth tt calibration existence arbitrarily outcome norm numerical exception control quantity consider game norm maximal packing belong minimax upper upper bound mass element packing cover second theorem supremum let theory note indicator radius parametrize length arithmetic vc case parametrize value membership vc effective let place action finite supremum value pack everything give rewrite deviation q almost result sure guarantee confidence show uniformly fairly calibration play player adversary lift instead closely value trick almost sure guarantee discussion grant dms
impose tensor unit mixture unit unit optimal choice beyond focus tractable select subspace model code amplitude model model radial edge produce dependency spatial least implicitly angle dependency phase angle edge c v v phase phase couple denote matrix hide contribute dependency cosine wise cosine identity difference phase dependency may
formalize random want especially make everywhere everywhere unique support argue hope exist proceed classic bayes possible computable density exist literature hoc technique circumstance constructive definition rao sensitive issue recall measure theoretic approach notion computable sense conditioning computable probability computable clearly computable question average string david theory extend general ai set detail set fail call return sequence everywhere elementary concentrate argue restrict conditional admit continuous everywhere study computable kolmogorov complexity conditional rather character point l respect conditioning setting computable present paper work study continuity derivative integrable computable show equivalence case moreover ideal provide detailed situation section negative conditional construct computable probability everywhere even elementary dirichlet computable continuous fundamental barrier construction possibility natural question whether conditioning
infer applicable notable case arise expert ideal net resource precisely expert make fail independence situation attractive observe belief make assessment belief reduce set long probability belief suited independence notion imply inference interpretation justify net give fully answer consideration direct finitely towards tree chain address uncertainty tree play build chain base define tree general model joint way conservative define number conservative model marginal independence alone show natural extension crucial tree section turn recursive leave show one global one consistency criterion bayesian net separately see important requirement separation criterion go develop justify make inference compute belief conditional call treat system remarkable work node compute expectation net formalism condition lead inference inconsistent start inference comment separation paper amount rather
several interpretation provide encourage sparsity base approach instead use suggest problem formulation begin regression satisfy ii depend characterization add q element present recall take algorithmic randomly formulate seek subset x b view randomize approximately subset column describe optimum multivariate regression approximate
near neighbor near approximate binomial distribution distribution much negligible estimate nan join event repeat say nan idea also great choice question
follow binomial link correspond ff j ji ki f integrate latent approximation fast mcmc exceed comparison assess fundamental health service management map category clinical difference variable modification happen road access structure age range patient identify place may allow subsequent determine match patient reliability group address patient determine detailed
obeys constraints balanced balanced upper exactly magnitude complement simulation design I fix high max compare available predict asymptotically sufficiently risk trial plot scales simulation sparsity correspond average case trial empirical sum figure n n indicate trial red square plot scale identity improve quickly dominate improve small accordance opposite interestingly competitive across sparsity correlate error section multiply brief issue generalization design grow infinity variable weakly correlate predictor design matrix researcher research toward acknowledgment discussion early manuscript van help anonymous support part testing significance assumption modern actually contribute response instance dna level
read probability pdf sources
probability q p cp follow l conditioning value level relative chernoff e recursion cd p cp upper bind canonical sparsity q transform orthonormal see false alarm p e c alternate empirical transform bound x x p x fact also arbitrary low employ coefficient coefficient correspond taking hold covariance recall covariance auto covariance small pairwise leaf cluster covariance covariance less covariance covariance first integer moment odd moment
thank van point proof corollary work nsf grant theorem set family vc uniformly approximate immediate corollary fact vc number law average result law
maker instantaneous function convert learn set expert algorithm instantaneous security share share security learn valid market must property loss expert bound loss market maker learn bind trade maker price show standard bind converse result bind learn restriction market market price quickly suffer naive capture stability define price differentiable j allow slowly price price point quantity price maker collect security market instead share price ready derive assume behind partition period exponentially increase period lead extra factor stable price let maker equation sequence expert loss simulate outcome share step simulated market completely control simulate market set share
pdf parametrization reconstruct monte event pdf event formula monte event choose bin bin histogram investigate quantity calculate parameter statistical positive bind contain realization bind test fraction program calculation asymmetric reduce event
consider evolution function f ps ps maps restriction markov countable state almost sense convergence positive policy recurrent construct nonempty probability state must state reach call composition correspond nonzero partition q recurrence transition nonempty nonzero chain eq positivity constant lemma direct finitely many finitely depend thus entry independent copy homogeneous property dropping condition distribution convenience banach sign define finite sum ps ps ps fc map inequality pick positive condition convergence invariant ps chain irreducible recurrent hence measure pick let restriction mass divide since variation norm pick triangle norm q information variation ergodic transition still take increment lie take step increment small example information converge make sense consider sum work finite quite restrict remainder state observe exclude contain correspond
tail central probably wide relate generalization instead deterministic include call present strictly financial mathematic strictly literature depend stable hold index normal property definition strictly infinitely exist powerful tool investigate distribution generate
cancer pathway investigate optimization problem estimate pattern structure induce penalty structure penalty fuse lasso difficulty show optimization penalty proximal problem show enjoy desirable scalability direction reduce convergence rate hard gradient easily careful investigation method far boost accelerate jacobian framework jacobian strategy idea nonsmooth function conjugate proper function strictly lasso separability nonsmooth composite gradient method leverage directly separable coordinate leverage order algorithmic structure impose example order tree group fuse idea group thereby introduce enable incorporation prior structure standard fuse extend fused graph structure tailor fuse readily apply optimization solver interior could always solve either second cone qp prohibitive great devise numerous propose induce survey short reach unified induce structure fuse lasso penalty motivate although common form optimize directly use
solve cpu roughly quadratic appear htp total relative equal next learn gene gene gene profile microarray diverse chemical across lead final summarize spend consistently derive graphical objective algorithm converge solver htp c sample much free select explain idea matrix maximum problem nontrivial likelihood propose bregman
include interpretable since small factor observe allow ibp generative many assume isotropic option noise complete graphical b derive define infinite simple dimension tell whether contribute source independently contribute binomial conjugate beta take ibp model conjugacy binomial integrate find limit nonzero take limit feature nonzero harmonic binary correspond infinite source arrange line customer observe sample customer right customer sample reach previously
reliable question acknowledgment thank manuscript management innovation innovation thm thm david institute biology department road normalize justify discrimination hypothesis namely result di bayes evidence vanish asymptotically weak regularity condition hypothesis unlike di require minimax averaging pseudo extend unweighted leverage side hand nan involve involve di robust weight suggest sample indirect selection maximum reduce weighted quantify area science
acceleration growth growth acceleration observer feature trait development depth framework comprehensive reader refer website use quantitative movie assign step towards quantitative trait classify mention appropriate signature mutation learn come call machine flexibility svm tailor rbf refer radial rbf movie describe calculate show class movie precision employ growth acceleration htb understand
sampler treat arbitrary make proposal multiply jacobian give propose metropolis acceptance confirm acceptance acceptance probability similar apply hasting propose new pseudo posterior rather implement surrogate current require poorly fix true gaussian point various gp hyperparameter four one cox full code supplementary material summarize follow section run different seed iteration
max location finite either accuracy measure regression well baseline edge base distribution b segmentation assume paragraph ignore consider minimum gap information measure close boundary direction aggregate distinct segment base proximity location cluster base geometry weight balance contribution display boundary segmentation apply largely final future smoothed present predict concentrate
intuition numerous deterministic outperform valuable early measure motivation work later exponential family leibler divergence logarithmic precisely reason composite imply conditional probability exclusive family geometrically interior positive mention variational information employ geometric introduce optimization recall fact abstract represent functional distance establish prove relate mutual continuity strict convexity resource convexity separate measure requirement information generalizing axiom continuity apply set quantum several fact channel separate transition deterministic broadly constrain utility unbounde deterministic utility end work represent transition space output utility conclude theory measurable measure expect argument kullback linear follow program figure measure package conjunction option either package graphic terminal graphic macro ltb lt ltb lt lt lt lt lt lt lt bp r show dash belong maximized physic call represent lower irreducible
modify stop satisfie present rule inner surely consider discrepancy stop threshold follow bernstein discrepancy stop outer except replace assumption bound sc discrepancy satisfy material surely criteria intrinsic parameter bind bad rate regularization knowledge dimensionality inner factor observe obtain outer lie reproduce hilbert
stochastic program markov directly define proof query investigate promise top typical fact program use module fact unlabele ground interested answer interested query ask likely would approach exact approximation explicitly reason probabilitie individual carlo program execute probabilistic execute call require fact stop current substitution proceed next tool implementation execute goal logical fashion similar logic programming logical perspective pose fact efficiently call program approximation transform discuss mechanism label fact example logarithm create specific ground retrieve update path queue benefit source scalability fact engine allow access indexing contain order call non probabilistic fact ground maintain within query would queue update
problem graph application tend quite matrix definition powerful extract sort norm constraint optimize computationally original clearly problematic one large heuristic sometimes exactly nontrivial three viewed implicitly compute interestingly regularization form semidefinite powerful extract useful smooth interest meaningful solution pose exist datum generalize
put admit proposition solve summarize follow q inverse eq enough operator series write eq follow appropriate hilbert q maps height follow fact last simple compactly support right scale prop lemma prop open open st hence study hamiltonian c hamiltonian range energy example surface dimensional show first explain operator eq classical flow generate vector associate classical shift focus easy quantum energy analyze reference lead evolution recent refer green operator absolutely interval continue disk sense integral view flow ensemble abstract topological mean intersection totally tangent surface split neutral stable decomposition property find flow call fig formulation stability also neighbourhood eq neighbourhood characterize quantum map family open fourier integral rank q precise version involve space respectively give full control cutoff operator family reason often hamiltonian simple existence
class course dimensionality arise microarray evaluate knowledge extensive address microarray reduction widely use base handle gene multivariate large therefore unsupervised supervised use score class prediction performance average test consider reach although benefit preliminary gene aim value performance confirm discretized response level indicate discretization minimal expression work demonstrate effectiveness microarray may improve prediction follow discretization gene three low low medium medium etc ii mechanism qualitative statement discretization introduce high microarray potential
behind semidefinite global optima function manner nice define instance polynomial nonnegative interval optimality coefficient polynomial nonnegative semidefinite reasonably well statistic completeness appendix assertion even characterization necessary criterion handle level exist optimizer semidefinite begin relationship main semidefinite respect side inequality positive definite admissible non empty representation matrix optimality function admissible every information criterion see semidefinite quasi representation admissible empty optimality hermitian matrix rp semidefinite let plug k yield representation form optimality fit framework limitation literature respect geometric eigenvalue mean rational semidefinite invoke symmetric also admissible set fisher optimality criterion criterion approximately design recent introduction
fu write c pf x x pn n x pn pa n surely right go fy cn pn pf f pz fashion consider pz pz pf f pf n n pf pn x f pn pn argument go life writing consider ft lx lx immediately since converse since expression fr cx tc recall prove fourth identity tail medium tail immediately tail result tail vary vary represent rapidly vary rapidly vary prove case follow slowly technical challenge smoothly smoothly consequence neighborhood represent expression one expansion u h h
true noise constant version estimator follow uniform probability aggregate upper risk page give good magnitude temperature improve oracle sharp sense front require procedure selector impose impose design p eigenvalue coordinate complement
well initialization figure pick realization initialization lead misclassification simple select neighborhood determine prove local neighborhood state accuracy speed hybrid manifold handle manifold expect together group fit theoretical guarantee hope quantitative alternative dimensional affine outlier affine fraction neighborhood good look locally though globally curvature pure optimal represent affine rh fact immediately obtain whenever point observation conclude verify generality rhs satisfie orthonormal pass span one eigenvector prove notice combine direct applying inequality use integration w r r rhs simplify proof q follow last indeed rather sufficiently answer question regard ssc code provide version code public edu mathematical sciences affine begin form affine subspace size automatically geometric
choose indicate obvious sample pca similar learning sum square close dictionary location dictionary multiple represent sample address relevant coefficient representation np hard understand despite decade aware generalization learn address discuss section relate identifiability dictionary give dictionary recent somewhat kind identifiability result except otherwise contribution bound complementary used length uniform order q case polynomially minimization compatible main significance achieve rate regime question leave due learn class bound result
function family path parameterize viterbi parameterization fractional discuss wide principled compare decoder efficiently newly define usual memory viterbi recent advance theory main risk review practice discuss conclude hmm decode sound statistical notably broadly prominent also property viterbi discover several claim suggestion explain give leave within framework within forward backward framework viterbi argue analytic family possibility analysis cm align leave clear show accuracy path still path maximize correctly recognize block size propose design rich family establish key general functional markov newly family explain idea viterbi fail viterbi establishe regard algorithm viterbi optimal incorrect forward scale operational decode hybrid propose replace transform operational algorithm viterbi optimal accuracy viterbi decoder give risk specify incur decision predict actual popular shall risk minimize viterbi sequence stand correspond suitable break viterbi advantageous logarithmic lead generalization section pointwise additive form element every commonly stand set related distance stand bayes relative misclassification maximizing maximize risk subsection explicit definition validity give refer path prior aware publication state path positivity prior understand positivity positivity guarantee positivity probability often decoder constrain priori happen entire emission alphabet enforce properly admissible minimization prior assume classical forward recursion viterbi constrain path constraint long equivalent term viterbi decode
value record surface view function value trait extend trait along role consideration indexing consideration indicate branch covariance unique natural evolution see two statistically trait topology evolutionary root call process generalise
f cause give condition define expectation formula index realization uniformly average random object index receiver jointly receiver logarithm entropy clustering side logarithm receiver clustering logarithm joint entropy size intersection error asymptotically free communication error suggest control derive expressive clustering rate resolve grain fashion select
mapping could linearity working become classical amount system possibly particularly machine euclidean hilbert offer discussion hilbert unknown way disagreement quantify order popularity optimality robustness wide variety employ design nature stress well apart training employ system approach close attack minimization sequence call initial stand metric projection denote previous recursion time classical deal smooth besides online indeed let substitute identity previous recursion well knowledge robust priori close convex usually empty set analytical intersection avoid popular solution strongly demonstrate potential wide learn task classical adaptive classification priori motivation couple observation nonempty
highlight resolve critical proxy minimize proxy keyword fusion assimilation numerical spatio spline substantial proxy information sense particularly air term fusion combine goal air management exposure sense spatially surface spatially correlate contamination cause spatially correlate term proxy relate proxy effort reason model relate gold proxy latent process I support define multiplicative identify scalar take coordinate additive bias moderate scale structure sense play proxy spline discrepancy fitting function fit function computational recent moderately specification quantity number basis dimension flexibility spatial properly short spatially discrepancy proxy spurious implicitly fusion proxy reflect discrepancy gold help assess potential discrepancy scale relative discrepancy small scale proxy effort lie flexible discrepancy discrepancy efficient mrf specification sufficiently flexible scale hold sec thin precision mrfs autoregressive proxy critical mrf specification variation scale stand contrast basis omit efficiency car variability possible analysis realistic spatially correlated discrepancy unfortunately latent discrepancy scale flexible may poor attempt proxy signal open question proxy sufficiently inherent constraint improve improve prediction improvement prediction krige find
full write distribution likelihood may less response term vector length section say correspond original scalar conditional unchanged diabetes machine include outcome diabete value reasonably missing remain treatment treat follow estimator posterior fixing apply obtain mle glm estimate apparent mle panel converge map increase accurate particularly intercept term considerable near highest surprising map ht illustrate panel posterior rapid confident way consider infer ig typical default convergence spread half binomial cdf binomial logistic sample rr
basis indicator treat fractional factorial distinction design designs scope fractional design application gr basis theory design topic relatively algebraic algebraic recent computation become feasible statistical fundamental field fractional factorial part factorial design properly factorial design orthogonality design algebra factorial design regular fractional factorial design algebraic statistic simply distinguish fact algebraic treatment resolution
topic model also hierarchy combine live hazard practical base demonstrate datum increasingly evolutionary diffusion author thank provide cifar valuable inference helpful institute advanced name propose flexible prior nested stick break allow live infinitely provide pseudo require understand infinite interpret part evolutionary structure example area
design begin quickly ga turn get maximum maximum function contour ga static design add average hypercube display contour ei ga algorithm accurately decrease contour rapidly branch competitive ga estimation contour well simultaneous minimum ga par function ei explain feature interest branch contour compare two approach demonstrate careful ei save significant amount simulator version bound rectangle investigate use branch bind computer still time consume explore surface simulator computer sampling strategy use study implement contour main contribution generalize expect estimation develop branch contour simultaneous
respect actual payoff actual payoff convex rely payoff linear indeed optimistic get evaluate consistent move player conclude monitoring improvement calibration aim link notion extend calibration appropriate derive full monitoring thank great acknowledge appear proposition perform auxiliary construct auxiliary game converse calibrate tool framework game monitoring player action define internal monitoring calibration introduction notion link calibration repeat prediction outcome probability calibrate empirical outcome predictor specific forecast close payoff great payoff call set characterization convex set
extend first framework work special beta case suitable purpose due necessary procedure follow beta likelihood eq approximated forget initial know initial autoregressive process constant belong set specification consider multivariate indicator chart identity easy uniform positivity truncation simplex return parameter constraint carlo procedure prevent take boundary parameter boundary simplex distribution fig bivariate create boundary simulation real contribute considerably density hessian need carlo fact behave term allow algorithm constrain modify beta naturally independent multivariate stochastic way easy representation beta satisfied chart
lastly pt lastly standardize covariate motivate mention option adopt instance binary interestingly yield similar via slightly matrix great variability performance r acc score cn cm pre acc score cn cm cn l save space via bic diag show lastly rely approximate ise likelihood save case pt consistent r c pre acc cm c n l cn measure select intersection two denote edge compare agreement disagreement agree lot graph extreme ratio expect respective agreement resp disagreement model high model quite different question model
mu minus mu plus minus minus mu mu mu mu mu mu plus mu minus mu minus mu minus mu plus mu formulation bayesian formulation normal gamma mu minus mu minus mu mu mu minus minus hierarchical mu hierarchy mu plus mu mu mu mu minus mu j mu plus mu mu mu mu mu j r mu mu plus minus mu derive conditional mu minus mu mu mu minus mu minus plus mu mu mu mu assess usefulness framework three brevity bernoulli compare performance coefficient weight shrinkage validation present result sample coefficient linear signal adopt
deconvolution factor case error slightly non coherent concern close fall evaluate accurately hyperparameter compute estimation exhaustive choose wiener solution sec report smooth variation optimum value good wiener improvement negligible report tool work true knowledge true image characteristic histogram histogram fig marginal report image parameter uncertainty histogram concentrate around non explain system system law reliably estimate second small consequence non value knowledge histogram quite hyperparameter fig interval histogram parameter manner informative parameter finally visible interpret ambiguity primitive deconvolution manner shape
six gp sim well canonical scope experiment gp sim sim modular section generalize exception extension already r package gp sim sim fitting suggestion package add sim add poor motivated idea bayesian tree infer partition region divide package far come convenient gp model enjoy extension give rise sim possibility upon run sep sim benefit align counterpart partition lead order unfortunately interpretation aspect sim translate interpretation nonparametric try sim lead improvement partition reduced sim partitioning partitioning categorical pair flexible nonparametric categorical predictor sim detail application gps response consume thus minimize subsequently extract experiment common design heuristic fit update simulation repeat maximize relationship
apply set value namely define define analogously easy dc follow concept observe contain contain function setting distribution basic agnostic recent agnostic recover advantage approximate h specific agnostic equivalence weak agnostic concept weakly learnable agnostic boost easily translate characterize strong dim polynomial pn pn dc weak briefly completeness agnostic contain hypothesis polynomial tolerance follow tolerance approximate gx finally convert agnostic agnostic dim characterization pn b dc dc concept let monotone learnable index conjunction variable n agnostic uniform learnable analogous agnostic noise theoretical hardness reduction hardness agnostic simple survey agnostic characterization present brief detailed behind mechanism resource performance mechanism measure evaluate mechanism ideal ideal
cardinality partition binomial base notational simplicity corresponding denote ii scale invariant study exist facilitate specifically lipschitz expect function cf n lemma focus ignore yield q substitute nontrivial value explicitly expect imply uniqueness scenario acquire corrupt unknown sensor stand zero ambient precision analog digital communication measure link group originally propose recover setup present solver noise useful uniqueness identifiability issue practically high sensing application suitable additional sensor noisy counterpart rs state aforementioned minimum incur combinatorial since problem solve outer relate l small cf solution satisfy readily show solution problem subsection establish robust estimation building remark subsection sensor rise outlier corrupt decade huber huber cast cutoff
tree exponential tree cascade present cascade edge si graph dag order construction triangular determinant upper product instead super exponential time require build determinant cascade super graph goal find search graph propose ad hoc heuristics hill combinatorial nature maxima leave alternative optimize cascade formation cascade tree moreover devise provably find optimal section informally approximation concept edge get infected influence network mass media cascade similarly influence tv phenomenon think create external small connect external influence every probability get medium additional influence infer hard capture external influence introduce concept edge add make clique union create clique edge role thus connect disjoint edge via analog transmission diffusion first solid line role edge edge fail label bold dash bold iii edge fail solid fail cascade product eq edge real cascade magnitude benefit introduction edge consider edge still diffusion later become monotonic treat edge possible cascade cascade aim simplify consider likely cascade instead consider possible cascade compete concentrate might case concentrate extensive structure exhaustive indistinguishable approximation equivalently maximize log occur
explain low harmonic population conversely profile count detector periodic input distribute neuron drive produce action fluctuation hyper potential delay pool active event existence ensemble either active depicted fig component fraction nature event arrival potential neuron exceed release assume happen stochastically generation indicate fig detector actual model describe fix duration poisson duration randomly
em factor correspond eigenvector bivariate em display increase corner represent bottom nine dimensional information effect display increase display smoothed curve thm thm effective acceleration propose allow acceleration choose dynamically lead widely applicable theoretical neighbourhood numerical two factor cpu popularity publication expand scope area time run widely stability recent various method accelerate tu liu wu extension r pl compute observe estimate replace step
appropriate formulation stochastic analyze approximation follow dedicated motivation introduction formulation proceed analyze vector finally address dedicated discussion extension follow deal real matrix letter write singular orthonormal value semidefinite vector singular eigenvalue instead nonzero frobenius also convenient introduce norm ball radius lebesgue first approach may form together restrictive one polytope return introduce invariance row long apart avoid polytope ball consider far twice domain sphere get require satisfy row actually unnecessary rewrite particular direct g q c row orthonormal
valuable share package markov integer phase thus construct valuable criterion matrix optimal share package break moment choice bank choose calculate function share package transition analyze subsection let rewrite fact respective trace distribute conditional analogously expression sequence employ threshold lk l accordance kolmogorov obtain markov desirable share first convenience function obtain quantity inequality algebraic result procedure impose fine agree volume markov sequence sum find strategy respect share package transaction portfolio
task task update observer response retrieval observer model whether consistent discussion realized maintain consider observer practical observing thought maintain database purpose reveal question fire observer provide reality represent observer neuron maintain label label labeling vertex nothing edge label join element non detector feature mechanism observe algorithmic implication observer model observation situation reaction observer date change coherent incoherent close containing begin switch coherent coherent false prove coherent iff contradiction raise turn situation actual meaning decide necessarily right current case easy satisfy intuitively think candidate take cell realization graph equal physical signal propagate imply chain hope course propagation play processor unbounded computation reader notice propagation signal think upon therefore exponentially possible element satisfy remain wave contradiction detector coherent incoherent throughout reason rarely coherent ultimately remain observer also update increase wave though instead approximation update retrieval database management believe possible answer way observer I partial neuron every immediate fire neuron connection every wave propagate tool probabilistic algebra appear range definition observer algebra
ps ps perform bayes factor bayes ps sc ps whether example try impact unable change respectively model n ta ta still improve see bf ps bf bf explore ratio example link concern could conditional laplace satisfactory considerably method worth possibly modification propagation ps mcmc path fail smoothly marginal due poor estimation also fail ps gm intensive mcmc standard implement moderately choose correct bayes ps sc sc factor comparison c ps sc efficient real life real life long processing explore whether ps reduce ps sc behavior high real factor notice set early discussion reliable bf set study also model ps ps sc differ lot rigorously laplace analytical difficulty point approximation method search narrow field model use ps sc ps see ps ps wrong partly grid path
genomic dependency independent region unlikely region combine testing testing allow significance cluster control family wise rate prove permutation permutation cancer array comparative genomic array design resolution copy number image voxel snr ratio gain fmri spatial feasible voxel array possible splitting independence offer fdr value smooth realistic meaningful rate apply suitable hierarchical procedure control testing array cancer datum set dimensionality common throughput genomic large traditional note generating allow suggest
section second perform factorization minute speed ghz ram gb special code sequel application guide namely concern classifier notation xx specify th element zero henceforth xx brevity expression leave estimate otherwise bx data x reduce bx bx bx bx point optimistic bx side bx replace bx estimate
differ obtain whereas result dense advantage approach function image classification combination factor sift namely scale sift scale sift px sift van de al
rate consideration splitting decode successive decoding send procedure case let statistic associate positive threshold step likely quantify fit step inner form quantity easy somewhat indeed direction maximize wrong complete decode previously pick need step briefly describe increase reliability vector receive associate
relationship show family suitable finite process conversely connection notion find reference show cover bound weak paper cite satisfy variety characterize note uniformity class respect provide condition uniformity early element function subset vc result family show f theorem direct construction technique uniform core contained state theorem outline key diagram provide let ergodic process f replace generality elementary lemma omit ergodic exist ergodic process marginal suffice case equip real analysis measure x precisely
recently vary formulation tailor inherent characteristic approach ii fact solve mkl primal insight difference purpose various compare empirically formulate regularizer regularization incorporate formulation modular dual criterion separate practitioner plug adjust flexible mkl mkl elastic match cast kernel unify supervise label sample n iy unseen return minimizer f
rotation parameter simulate reliably likely variance method effective match false essence posterior similar pairwise configuration extend al green way mcmc exclude possibility match et match run convergence match match adopt tool way find optimal correspondence surface surface shape region correspond bind et among comparison consist site protein protein protein beta site protein site protein protein suggest protein protein probability often match represent run
apply gs lebesgue along net generalize include order large integer large dy exist proof immediately estimator hence need b iy iy iy sf subsection digital construct net coefficient digital net b q proof n subsection depend
specify reaction connect task furthermore sampling along advanced quite force capable dynamically utilize trajectory current facilitate set discover reaction dependent priori reach although work langevin free landscape kullback appropriately framework provide energy obtain collective analyst free energy landscape efficient scheme rely carlo enable potentially multi modal clarity generalize later reaction molecular boltzmann role temperature free additive estimate parametrize adopt statistical approach functional successful reproduce kernel hilbert definite adopt order fix z literature type thin spline function
goal apply walk size community member node run walk length bottom decrease walk alternatively way example close clique argue er good capturing real member quite member participant technique clique combine random clique metric qualitatively benefit approach graph consider cluster confirm fm orient user community interest fm build internet service user fm fm popular social site relation fm mainly music use social make reach likewise music music empirical despite challenge show sampling representative via relation well mutually something group connect last fm connect fm match activity direct consider adjacent collect user random friend event
identically covariance identify observation address conjugacy day innovation transform covariance product covariance eigen decomposition eigen orthonormal eigen vector trivially vector give recover independence property identify transformation identity lemma give identity mean factorization solution mean block factorization decomposition theorem tensor obtain decompose admit handle impose situation factor th dimension th element element choice ensure conjugate transformation variate matrix matrix comprise give whose remain element fact specifically select diagonal identity show explicit identify uniquely transform eigen invertible unique utilize conjugacy fit price shift asset conjugacy wishart define q define conjugacy direct consequence develop conjugate choice equation later demonstrate conjugacy utilize stable novel algorithm intractable variate admit model series markov mcmc set mcmc sampler literature
think predictive though distribution posterior density interpret universal code base estimator code plug code estimator code establish redundancy closely code plug conjecture code essentially conjecture family redundancy become large conjecture fact location look like hence code look plug code bayes
blue link figure present roc procedure mention cp high cp auc score last return accurate period investigate cp experiment vary amount value experiment instance test score show auc link training randomly increase advantage cp period depict predict correctly link decrease figure varied cp correct top detect level perform extremely large percentage signal crucial computer lose context service auc tp tp average result size forecast auc predict seven tensor tensor generate median box middle outli red conclusion prediction cp accurate link link numerous task include temporal suggest summary graph time incorporate seminal co network except link comparable variant note truncate approximate recommend
attribute detector detector difference total amount profile patient preprocesse htp error htp fuse fuse lasso site predictor vector solve fuse spend ten fold approximately ten case able solution tumor cell large dna segment phenomena technique dna differently intensity dna calculate gain copy probe fuse signal detect array datum array htp parameter htp brain tumor show cpu spend regularization fast improvement art copy detect
minimize prediction competitive free prior valuable nonetheless acceptable selection tool validity model grateful associate valuable comment suggestion greatly improve quality comparative es es es author de grateful participant bioinformatic comment pass influence field continue universit paris france mod de universit universit paris france benchmark bayesian frequentist simulate real comparison build free proposal numerical highlight
maximal subgraph keep construction firstly grow clique cdf variable unbounded graph probability restrict family advantage extra extend acyclic direct describe factorization cumulative member definition kind former call parent parent subgraph set respective remove relationship graph connect connect entirely bi direct trivial associate consist node I pa gx gx x x notice external singleton c subgraphs density mass parent already pa gx
star angular momentum evolution clear physical interpretation appropriate detailed extensive currently apart initial plan angular momentum analysis lead understand fractional generator similar repository compare autoregressive inferior describe long determine arise angular momentum toy essential momentum statistical simulation dynamic centre dynamical mass section characteristic existence position
sg sg c twice differentiable g show bounded lipschitz lipschitz bound set theorem induction iteratively prescribe induction bound ray proceed h h h f side non iteratively entire wolfe together cauchy schwarz continuity angle always combine angle wolfe h h induction finish proof gs low bound banach modeling occurrence genome neuron claim financial activity occurrence mention mark seq model genomic organization seq process linear model possible computation terminology furthermore spike train consider train
whereas equivalence iv transformation linear univariate admit satisfactory consistent relationship residual natural decomposable preferred individually significance take emphasize comprehensive extension multivariate consistency prefer individual equivalence transfer robust wider univariate variable contaminate multivariate differently scale constraint additional robustness practical example eeg detect neural differentially satisfactory establish numerically frequency result obtained form preferable high existence univariate counter unstable simulation confirm derive novel potential substantial new light complex density system dynamic integration extension dynamical causal measure dynamic capture aspect non visual capture world effectively prominent compete measure multivariate causal significant deal application third helpful consider one approach respect micro parsimonious macro function relate micro g idea macro micro independence characterize macro
place person cluster perturb perturbation eigenvector present section justification affinity section completion compress measurement small coordinate span perturbation section dedicated synthetic reveal object linearly pattern make spectral eigenvector form reveal structure choice kernel eq walk graph normalize symmetry lose find eigenvector spectral prove eigenvector extend clustering use multiple early use second perturbation follow perturbation provide eigenvalue equal separate form connect cluster separate second affinity ideal underlie point eq perturbation
previously context graph show final graph sensible trading european join link single clique connect notably large connect integrate european increase variability impose evidence effectiveness graph method author low variability exchange day datum run burn five completion assess verify figure belong comprise roughly regime comprise deal difference display model chance inclusion regime structure broadly tight grouping quite clique connection clique amongst economic area similar european track long european exchange mechanism erm
intuitively would regret time average reward armed bandit linear maximize slot reward arm across store independently information alone show policy require storage grow grow exponentially number intuitively dependencie motivate efficiently exploit refer reward np nn nm maximization q accordingly htbp l map index slot slot observe description store random rather whole operate exploitation gain arm two vector slot value slot arm one time slot
eq distribute much appear calculate reason unknown sufficient appropriate approximation define take give along line take remain set conclude least key nz b k nc nc bc nc use p bc bc bc range uniformly finally formula form model provide asymptotic joint
dimensional family indicate distribution blue measure appendix necessarily family greater equal believe many model ham minimum distance binary cube hamming individual simplex every cardinality distance vector odd one x binary every contain distribution cardinality furthermore n convex sufficient statistic ham remain statement assume
lrr psd scheme instead sdp solver poorly derivation psd scheme lrr lrr future proposal validate theoretic optimization real affinity thresholding deal setting formally segmentation dense k vector respective regard vast range basic mean method elegant sophisticated spectral sc towards strong believe exploiting enhance basic framework sc extensively review employ image segmentation vision remarkable fail simple method sc partially explain connection method easy freedom method sc form affinity sc affinity kernel affinity affinity eigen analysis laplacian normal sc intuitive perhaps fitting aspect classic expectation
popular empirical asymptotic well establish bandwidth et basically distinguished choice follow fractional period multiple assumption sequence satisfy eq ks otherwise assumption q j simple expression reach mx k ol theorem ray one ar parameter display replication carry short dynamic case ar bandwidth bandwidth use short component investigation approach ft possess memory harmonic frequency frequency exclude frequency spectral ft see fractional ft estimate parameter summarize first
well provide constructive correlation advance marginal mixture hierarchical rely straightforward initially example costly inversion rely order limit additive poisson hybrid produce source otherwise function ask whether attain answer back fr introduce set minimum coefficient call upper hoeffding characterize follow f h h inverse
multiplicative optimal design implement criterion yu reference therein important optimality accommodate uncertainty naturally bayesian simple context et monotonicity
learn base learn gap bind ng actually easily establish instance coordinate bernoulli gaussian spherical x learn calculate sample estimate center near gap generative summarize margin rich characterize adapt algorithm semi feature construction characterize sample believe obtain answer thank therefore k theorem complete sequence every column mt necessity theorem set column exist matrix rank
derive obtain derivative respectively asymptotic expansion fact lagrange hereafter drop drop quasi tend kl regularity n expectation well argument expansion share generalization incorporate effect misspecification maximum indeed term n predictor glm reflect aic extension work cross weak cross log suppose set compete popular put nonzero bound principle choose I correctly specify exponential n seminal include make argument assumption response lebesgue divergence lead regardless additive choose divergence
I dynamic eight web randomly make rare word consist l l business name th frequency moment propose geometric mean moment find stable geometric implement function gamma matlab fortunately sufficiently estimator together estimator projection dash overlap numerically new numerically present result normalized decrease panel roughly flat capture trivial geometric new middle decrease latter moment shannon entropy entropy
force parametric however learn something extent external neuron subscript fire firing describe mis indicate apparent actually external driving quantify bit drive force stimulus stimulus drive relative leibler theoretically neuron run actually distinct predict future stimulus quantify predict neuron technique neuron spike train input entropy appropriate stimulus drive entropy predict fire neuron spike record neuron iii train without external begin neuron whose rate hz spike follow hard period spike twice baseline decay see figure history dependence intuitively period reconstruct resolution bit internal entropy residual randomness bit bic exception spike transition spike happen transition however transition state fire fire internal entropy spike conditional subsequent emission compare fire solid line square model generate spike dash calculate spike except spike fire discrepancy arise firing plot since spike emission panel entropy complexity high emission baseline state probable entropy third panel spike impose
mle communication interesting mle symbol symbol good suboptimal mle use training symbol estimator symbol inferior low snr mle know sensor see enough compare analysis unknown estimator approach mse lead evaluate record carlo show second step mle suboptimal computation suboptimal snr mle implementation exponential simulation r db mle communication fc final base receive communication error traditional assume perfect communication reference therein exploit redundant sensor dramatically ignore wireless aim achieve capacity improve reliability example bit communication signal appear motivate communication orient diversity combination fusion orient fc strict decentralized sensor bit quantization discuss channel noiseless noiseless channel propose universal isotropic quantization adaptive method mle gaussian introduce suboptimal decentralized snr
tuple value triple formal useful concern ergodic although summation approximation consistent suitable empirical estimate cluster point maximize minimal point assign contain close assign simply put mix rate finite probability incorrect notion assumption try optimize parameter may interesting preserve hypothesis concern without
iii submodular contraction soon equivalent stability equivalence straightforward n jj jj jj jj jj jj jj j jj x otherwise complement thus non affine representation support show path surprising finite point submodular associated maximizer minimizer j jj moreover non algebra exercise desire j j c jj c j jj jj jj jj thus w jj jj x property eigenvalue j z
framework extend support separable slack parameter extension change characteristic algorithm semidefinite program equally simplicity discussion gaussian hard margin modify margin satisfy perturbation objective lead classifier preserve differential privacy size perturbation frobenius norm method norm direction random propose minimize function
display variability associate value variability hypothesis relate matrix assume form value correction account variance asymptotic hypothesis assume g correction associate instead significance hypothesis become example significance value associate multivariate specify covariance completely marginal estimate monte significance test statistic distortion
recovery frame frame guarantee entry contrast recovery literature around section detail proceed develop facilitate regard recall concatenation generality permutation equivalent state measurement express collect general notion relate orthogonality permutation orthogonality respect derive name trivially norm relate case average heavily specifically x I follow theory next apply tuple thing linearity condition define easily order obtain every obtain similarly j regardless establish martingale assumption see inequality difference routine k identically rely begin note j ii construct martingale thing suitably km complex combine fact follow identically uniform ready provide writing proxy let note need verify notice trivially event follow use threshold establish claim regard therefore together obtain combine previously finally eq q conditioning fact loss
decay clustered overlap auto correlation walk inside volume surprising counter intuitive since close neuron similarity understand property ensure zero unit potential pattern equal probability potential induce evaluate obeys distribution whose factor threshold tc ct supplementary
objective optimize argument maximize iterate reach point
relevant use isolate use use web language game game object among scene another guess object need accurate user guess far provide description allow semantic perform sentence correct sentence average provide sentence rest paper structure soft description main inspire present description scene compose overlap square nevertheless turn realistic aspect description user familiar include shape triangle circle allow
variable path add outcome connect belief propagation update newly acquire terminate marginal intuitively similarly well close step probe likelihood value co jointly constant single good fit minimize sigmoid rapidly decay available prefer less outcome procedure sensitive specifie decay sigmoid moreover topology day observe available interval observation execute latter procedure slow rate swap swap fx fx fx fx fy xshift mm fy mm yshift fy xshift fy xshift yshift fy fy fy fy fy fy fy south fy north south fy north south fy north south fy north south fy north fy north south
nf author interpret either imply research laboratory reproduce purpose cm il mail edu rao usa mail university california berkeley mail berkeley edu berkeley computing microsoft china partitioning natural texture commonly image widely accept crucial understanding content segmentation application largely visual e object understand dominant qualitative quantitative comparison segmentation explore principle good segmentation texture image multiscale cut cut ms seek color distribution contour edge shape method combine color contour segmentation local homogeneity salient image feature scale area image segmentation practitioner mainly two
verify claim nonlinear act counter claim always method estimate degree freedom well behave taylor expand number freedom nonlinear taylor freedom view issue really help sect sect degree reason thereby upon consequently uncertainty propagate draw degree freedom argument far see absolutely nontrivial degree usually number degree however
mode comparable et supervise inference hold truth parameterize ps supervision learn nonetheless performance hdp manual supervision hdp var performance seq seq seq ar seq seq ar hamming sequence g represent red hdp var hmm viterbi variation sequence truth inaccurate specifically typical turning note tracking pattern dramatically affect indeed create specific mode attribute mode within achieve reasonably discrepancy performance approach al hdp var mode five six jointly infer sequence hold see partially improve head head pre unsupervise specific modify switch vary body noise place detail hyperparameter prior c var hmm unsupervise var partially dd dd ard ccc ard seq ard seq ard seq affect ard assume switch var likelihood compare hdp conjugate hdp var ard prior g ard avoid ard switch dynamical consider
scheme transition sphere algorithm birth death algorithm inspire canonical label regime pack canonical birth death critical spin hard configuration feasible application perfect method much patch entire nature coupling chain start configuration probability couple construct may lower naive correspondence paris france couple
far promising optimisation designing sometimes interest use widely standard diameter length weight constraint stress limit detailed please study compactly limit use solution design optimisation four length area cost constraint stress stress load problem limit bound exactly et al cs find
moderate test accuracy gap promise abc boost classification criterion order find se expression weight response mp end split shrinkage normally terminal alg
relative benchmark solving would interesting study method procedure suitable even behind procedure paper goal splitting probability analysis contribution properly splitting evaluation relative complexity small theoretical justification believe beyond recently generality class addition dependent connection user event algorithms split total propose base depend state weakly turn construct recent base lyapunov evaluation achieve guarantee run time choose importance bottleneck sharp believe understand connection organize efficiency deviation asymptotic require splitting theory concept efficiency event discussion context event p without exponentially design rare construction number
else x b diag diag return double double double return else max diag return diag help call code choice purpose else besides exp two everything else elementary
invariant adaptive problem setting standard htp c avg positive avg negative htp avg error positive negative capable give setting however average evident mainly penalization run table result superior incorporate penalization improve performance drastically likely small htp correct avg positive false negative avg avg negative ie n hierarchical iteratively solve let run repetition hierarchical adaptive clearly
simultaneously complexity increase advantage topology degree degree simply prescribe distribution define stage approach define generate next generating couple link draw link generate measure axis splitting must normalize next multiply take associated product generate convention stand generate link measure point interval generation fig show small p leave interval respectively height box multiply middle
frequently encounter computer vision present analyze fast typical svm model I classical svm ls prove complexity refer nesterov acknowledge fact nesterov compressive covariance differentiable hinge primal solve iteration round auxiliary weighted combination current historical two multiplication easily census categorization scene classification scene event four solver light experimental indicate short svm solver write svms minimize hinge nonlinear framework saddle subtract
repetition correlate covariate sis well worth sis reduce serious correlation covariate implement cox adapt code recall cox convex subroutine system laboratory university convex quadratic take much long especially dimensionality confirm strictly sure screen median large absolute select coefficient size nonzero yet case performance c survival ensure marginally sis confirm select rarely marginal screening challenge nevertheless demonstrate van take hour van finish minute huge lasso case demonstrate performance
element element unnecessary compressed look stable method measurement assume vector know measurement additionally call take generate difference ordinary compress large modify perhaps prominent identically distribute force gaussian entry realization role communication compressive set I compressive generalize er rao bind asymptotically achievable matrix condition depict sense literature correspondence rao noisy compress accurately lemma compare tail building equivalence
formalism framework error function input true spectrum choice calculation discuss simplified equation formalism discrete optical factor choose eq quantity use give probable weighting calculate term sufficiently logarithm low due solve equation define noise deviation default exact write equality term contribute average data systematic eqs apply iterate iterative figure illustrate behave ill behave statistical huge
discussion theorem axiom conjecture exercise notation ny essential argument belief rational agent logarithmic unique relative two entropy bayesian scheme tackle question nature information discuss concept directly rational agent argue uniqueness entropy I relation entropy bayes hand design form whether compatible situation beyond individually explore I
co integrate van include intercept time trend intercept include methodology co integrate series denote integrate relationship vector gaussian covariance error correction give lag express produce detail parameterization trend multiplier multiplier important quantity model root integration integration relationship univariate adjustment adjustment equilibria likelihood long multiplier indistinguishable standard unique restriction van still enter chain variate occur product direct carlo var informative var illustrate van present conditional use conjugate h variate prior degree freedom definite variate likelihood trivial variate trivial marginal variate
nature generation count across letter make classifier document summarize accuracy result suggest naive high accuracy plug character character leibler distance chi seem leibler new dirichlet kullback leibler high degree smoothing see pool naive dirichlet large chi classifiers accuracy random plug classifier classifier rate character suggest write accuracy accuracy match exceed rate currently publish identification summarize near accuracy leave evaluate determination document actual grow interest determination system tend known verification determination document refer know list search database
improve however next make life well task analogue previous approximate active denoise overlap patch image simple consistently improve depend method patch compare encode via repeat variant regularizer two variant variant summarize figure although quantitative improvement coincide one report noisy provide case extra denoise obtained produce reconstruction appear white l cc code c code average cc cc show coding combination clean recover result patch value average cccc cubic tool right summary cubic result reflect improvement summary noise previous image tool simulate miss technique wiener filter real averaging fill miss pixel summarize improve apply propose class use regularizer repeat baseline universal improve accuracy begin classic texture patch belong database actually
proposition theorem corollary technical report ann mi locally university analyze heterogeneity multiple infer arise covariate domain formalism dirichlet relate notion global cluster provide efficient inference utility include analysis local process model common group cluster observation group often interest change covariate primary extract sort cluster aggregated tracking covariate snapshot group cluster really movement individual evolve primary path global infer locally observe functional information mean functional reality conceptually individual behavior number day cycle local group cluster typical behavior evolve throughout aggregate day cycle might typical medical cluster privacy subject neither collection level example index different consider individual gold substantial body sequel unless specify covariate heterogeneity group assume cluster covariate handle variant process measure call concentration base center dp stick
hold candidate minimum negative increase derivative approach zero never equal transform heavy squared condition mle large therefore showing must line proving decompose monotonically decrease input monotonically attain maximum unique mode obtain theoretical example time tail back remove heavy tail flat estimator remove tail equal match property assume result back transform formally skew important lose transformation w monotonically also identity slope decrease transform unbounded rarely also back transformation affect triple iteratively package divide deviation k step normalize output pass obtain update new back transform well k start pass tail w double replace remain rv slight excess estimate median rr gaussian w mle na c c base
find lattice cardinality lattice vertex lattice da db c db cluster linkage pairwise linkage text formal lattice discussion base hierarchical abstract lead cluster dissimilarity measure note see computational logic ultrametric embedding topology ultrametric hierarchy tree linkage logic ultrametric generalize ultrametric logic review ultrametric ultrametric review allow geometry topology information start develop metric ultrametric generalized ultrametric logic particular chain ultrametric ultrametric
view htbp htbp show infer functional general additionally dominant class show take random treat whether principle kind formulation strength unit parameter control capture idea may nevertheless something option follow specific rather hyperparameter turn variation determine flexible several approach might nonparametric together related nature modelling approach nonparametric begin early intend nonparametric yet capable tractable posterior arbitrary eq simplex density proportional give discrete draw exhibit
strict taking numerator attain equal thus I nonnegative nonnegative consequently strictly change part take account establishe theorem case substitute stem value except accord account give leave zero equivalently inequality hold function monotone bound regard attain stem establishe follow equivalent small meet condition satisfy sufficiently
obtain reasonably good split small block put splitting block equation reduce integer equal block reduce computational time necessary carry reduce evaluate deal kernel density copula normal cdf cdf equation represent use simulation example compare sampler define autocorrelation iterate tend factor autocorrelation analysis decay lag simulation mathematically factor ratio accuracy preferred may account sampling fast need less second iteration ratio parameter note although factor sampler univariate delay replicate sample step iteration true nest prior specification range volatility positively time shape scale
coupling delay select difference always driving also drive htb system use time show legend b driving driving also behave similarly subtract vector though couple component couple drawback neighboring frequency scheme observational variable vector couple system system embed mostly component another component criterion choose couple wrong criterion bit measure give take small predict scheme transfer measure generalize synchronization equation drive drive complexity individual mean great try regard investigation first limited realization three f n lag give strength conclusion proper choice especially great lot strong drive driving give reasonable system third choice respective value increase multidimensional lag computationally demand alternative ultimately restrict mutual length component form see coupling drive reduce drive
base actual realistic background galaxy run search completeness firstly sort match match match cluster exclusive match detect match completeness fall bin cluster match number cluster match total completeness plot plot cluster sample input give look gm recover though systematically low primarily due low cluster essentially rich retain low consist cluster dr survey cover small perform cluster uncertainty difference scatter similarity galaxy careful cut make center ambiguity time minimize match due projection yield accommodate scatter appear counterpart match ultimately determine quantify agreement uncertainty difference appropriate selection window radial separation separation appropriate generally speak matching match low cluster therefore match respect cluster cluster word completeness execute match yield match place match leave panel separation create result figure compare match color large among new lambda perform original estimate lambda hereafter dr
algorithm discretize grid previous multi behavior social social threshold ph model yield threshold kolmogorov criterion comprise result condition characterize hyperplane ii characterize ph simplex use preserve involve social definition order consequence appendix describe sufficient condition policy iv sec optimization ph ph constrain belief similarly decision social learning section make following polytope say treat outside stop agent treat identically leave hyperplane obviously ph change would degenerate result maker hyperplane straightforwardly vertex introduce follow assumption p relevance apparent q ph sufficient ph begin sec social ph geometric entry note linear function non unlike stop numerical illustrate region denote bellman say region ph polytope states lie line correspond region lie segment region depict polytope assumption theorem numerical show fig concave policy social detection example though single detection single classical problem observation ph subsection curve stochastic decision maker vertice social base detection ph ii convex polytope almost everywhere iii geometric distribute identical kolmogorov polytope boundary coincide classical optimal incur always threshold polytope consequence obvious together ph polytope state mapping trivially cost deal concave ii stop argument straightforward decision state thus detection kolmogorov uniformly distribute proof iv
give recurrence pack max formal ball max disjoint radius great intersect structure give bind behave r dr rd show probability radius intersect ball invoke theorem aspect ratio radius show split one argue see cover radius clearly ball suffice without center ball lie get get split cell radius split separate show useful whereas tell split cell get get bad get split time
network differentially gene essentially differentially associate locate extreme none express differentially express reduce suggest classify disjoint way gene construct connect clique decomposable since adopt convention vertex intersection two gene opposite arbitrary classify reciprocal convention would maximize clique classify differentially gene clique express differentially express number gene network resemble entropy measure uncertainty uncertainty standardize divide find way uncertainty differentially represent illustrate might differentially gene decomposable model however allow identify gene visualize pattern hold capacity differentially express proportion differentially present situation
signal write make model analyze standardized iteratively scale row mean covariance column capture keep separate process two test decomposition model instead sample population gene iid whether shift throughout variance common n go know statistic n l ij n vector center state array correlate statistic column correlation within assume microarray alternative centrality correlation distribution square pool long random denominator correlation small four structure scenario block within block j j within ij column distribution see correlation dispersion statistic distribution explanation dispersion see microarray statistic statistic appear affect confirm table variance
assume file procedure might record record consequently extra necessary linear preliminary estimation alternative incorporate assume become unknown record record linkage knowledge linkage literature matter dependent suppose necessarily dependency among redundant comparison function independence key fail disagreement key introduce correlation absence independence mean sophisticated dependence specify analysis diagnostic gold propose introduce subject categorical categorical account measurement structure linkage conclude stage explicitly introduce measurement misclassifie record parameter reasonably simple call level variable conditionally record model second one see assume replacement unobserved natural count due determine frequency correspond hold write role matrix variable unit linkage problem match note exactly denote match true need facilitate section illustration
original use importance correct case rare importance gray line show varies use uniformly follow though sample statistical illustrate testing uniformity specie order sum sum relevant select occurrence sequence sum statistic denote transpose statistic competition specie generate approximate approximate importance sample new reporting even error guarantee guarantee national foundation dms author statistic university nsf dms lee comment example importance sampling common include correction value create create nan correction use original observation give valuable problem evaluate accuracy approximation invert create nominal significance large exact conditional logistic nuisance technique include carlo approximation besides sampling
op k l hence eq eigenvalue argument similar q proof theorem ti p fm f establish model arrive use series dimension many vector practically proper make reduction observation retain dynamical frequently way dimension multiple early include result large time effort mainly focus economic financial phenomenon identify common factor common series factor series white identification asymptotic series infinity identifiable adopt consider depend substantially different aforementioned series part dynamic conceptually bring identifiable furthermore
return mind failure highly reliable markovian rare small make need reduce simulation importance system largely measure rare importance implement change markov chain transition bad
two possible discuss stop know scalar necessarily max max flow reach optimal constraint feasible arc j new solve satisfies arc replace compute cut describe fista duality stop without look datum duality obtain satisfied pair variable consequently nonsmooth one often choose encode knowledge prediction interpretability learn penalty index index scalar use literature norm piecewise individually overlap still set set consider general criterion context group proximal sum continuously differentiable lipschitz smooth therein linearize current w w keep solution hold constant call unique convexity iterate proximal obtained develop extend overlap wide spectrum norm propose compute
bind side five split training constitute slightly consider iteration test perfectly tight mainly meaningful cluster result could improved chosen loss minor cluster grow considerably due lower much tight bind scale experiment tell help problem enable graph
n number online represent cover scale pointwise easy verify claim depth simple ensure sufficient covering bound fix learnable learnable consider value vc lemma ensure small argument call sim model describe sim du lipschitz iteratively make mistake possible round elegant computationally perceptron variant even ask basic sim learnable deal question interest particular necessarily decrease evident composition square lipschitz decrease proposition covering increase class learnable improper regret computationally method exist theory rademacher classical integral prediction round information static put binary side write loss independent loss rademacher appear hand side tree verify scenario appear need pay tool minimization work
observation answer query q iff refer q derivation form allow right shorthand execute choice query q result learn base occur shorthand repeat observation number first toy program coin head head chance query head head tail head tail result query head outcome head match program paper fix player distribution however thing player third cf interaction dr ex dl dl play player play make situation partly observable game derive move information less straightforward estimation presence plausible learn build indeed
marker report element relate marker trace bind initially less opposed logarithmic marker different continue marker marker trace rapidly tend marker propagation network amount heuristic alternative false positive determine marker hence reconstruction match set use distance identical value match bad upper number true positive limit marker trace constitute case reconstruct perfect free reconstruct reality adaptation datum assume consistent perfect perfect report marker assume infection infected require presence marker marker incorrectly suggest presence edge really increase quantity reconstruction
maximally direction nan select exceed specificity distribution develop instance mixture modeling voxel salient pattern method currently ica model determination generate select device separate group variability thresholde absolute voxel mostly discuss expression describe variability ica motivated fmri display validation criterion ica unclear feature subject across subject split subject ica map overlap thresholded map map overlap quantify study identical quantify reliability dataset extract full subset compute validation coefficient average measure compare concatenation implementation toolbox tensor ica software effect separate cca group equation component level independent component run perform non thresholded ica implementation thresholde separate thresholding thresholded thresholded thresholded little interest voxel specificity voxel statistic map define stability subspace span map overlap subspace group frobenius different normalize dimension quantify subspace span map span dimension dimension instability
dimensional design generate previous space two os enyi maximum construct precisely either edge select care maximum underlie graph generate learn dyadic glasso entire partitioning achieve apply note middle line connect great variability method b proposition graphical encode dependency vector estimating value paper build space cart optimize cart cart cart dyadic partitioning establish oracle risk consistency cart tool analyze complex let way
u tv form simplify z tu u tx tx ta lemma four compatible cc f bind proof optimizer problem ij vector subspace uniqueness minimizer uniqueness minimizer ij lrr solution theorem z e v z way classification problem challenge possibly exist follow cluster sample contain error draw correct segment respective noise perturb perturb whose sample fraction phenomena fraction subspace generally subspace exhibit outlier three subspace shall concern b belong way recover term lrr basis lrr representation computational procedure regularize problem convex solved dictionary lrr solve clean lrr exactly recover lrr original well data error approximately subspace membership lrr term lrr image segmentation face recognition subspace subspace handle subspace clean theoretical robust share considerably exist four main category mix model mixture distribution degenerate
connection scenario characterize aggregation consistency mild opinion profile aggregated opinion aggregation issue iff consistent characterization find aggregation mechanism point think even result quantify number profile conjunction prove five condition close necessarily consistent aggregation return aggregate opinion return profile far return return opinion profile aggregate opinion profile aggregate opinion conjunction truth functional divide conclusion conclusion opinion attain conjunction mark restrict mechanism almost almost define consistency aggregation aggregation similarly quantify define mechanism version ham certainly trivially consistent whether aggregation aggregation aggregation mechanism question equivalent
uniformly shift still normality assumption computer system control chart reduce slow shift classic chart shift argue question arise hold classic chart drop interpretation dependent namely triangular array array respect chart also effective monitoring dependent organize introduce control chart chart appropriate change viewpoint main appendix chart extensive present providing recently chart detect shift allow chart tune reduce control lead quick jump limit alarm refer chart powerful change order move average chart satisfactory shift cite
curve ph unweighted curve high predict area care facebook user top error bottom compare walk plain common decision tree pt friend feature seed table method facebook network random auc surprisingly recommend nearly supervise significant unweighted walk gain logistic unsupervise logistic term auc walk near facebook near walk improvement supervise extraction many relate edge random determine network make supervised walk compare logistic regression art outperform hoc feature examine assign sense recently friend co amongst walk graph feature rough run ghz processor facebook put category learn take iteration quasi converge minimize compute graph partial parameter iteration converge derivative iteration derivative minute increase facebook new link recommendation utilize edge walk co random walk improvement random walk machine require
integrate gate semantic semantic implement measure similarity create term document reduction use example projection view operation vector package emphasize efficiency argue indexing updating encourage research development create source convenient modify software incorporate stanford project projection vector modular design kind instead support building addition build operation calculate search operation search search document operation include quantum provide package public http google com module source implement library semantic lexical pattern together corpus word pair follow sentence effect channel vs sentence pattern frequency smoothed occurrence expand seed alternative expand south south store detailed survey aim reader reference scope group involve pattern matrix semantic document cosine document retrieval develop devote core idea document cosine angle vector vector variation retrieval retrieve document technical discovery although system smoothing expense document collection indexing help collaborative document similarity across cluster flat may hierarchical group group soft differ similarity minimum average similarity document task label sentiment positive versus spam infer thus classification notion document involve document matrix similarity automatically student assign proportional student highly get focus series cosine topic shift drop view document document block question answer answer find question corpus typical many big retrieval extraction retrieval present automatically base question direct automatically question measure semantic similarity similarity two word cosine row document choice question achieve human word average english us college document seem long researcher shorter prefer word center word remove word context important distant
reach machine control hmm alternate alternate think alternatively bias depict hmm break isolated leave fig depict hmm inspection redundant machine analyze synchronization edge hide machine symbol symbol pair x markov machine give even abc np example machine note machine ref machine whose infinite definition definition present state apparent
university increasingly model image rich distribution chinese article review dirichlet build partition infinite partially use chinese crp article theory notion partition distribution crp conjugacy well relate property consistency interpretation improper infinite dirichlet posterior conceptually dirichlet present refer non distribution remarkable conjugacy property basic chinese restaurant process distribution bayesian dirichlet know name chinese restaurant crp elegant analogy incremental model prove way tool modelling want flexible useful phenomenon compression well perspective generally task associate model partition tree hierarchical structure concept along statistical develop general form partition crp crp tree e nest bayesian improper infinite improper dirichlet readily obtain sampling science partition express posterior second play role occur important reasoning mathematical community kind consider require
mu mu mse validation var properly compare seem transformation account potential compositional positive impact overall result agree literature diabetes ten body pressure six response disease necessary regression model ten accuracy predictor variable predictor input five age evidence solution forward implement inclusion quite interestingly var show evidence literature diabetes select form remainder predictor var include assumption predictor var respectively average respectively number deviation poorly selection backward move generalize aic package parameter information present approximation approximate var selection default impact encourage variational describing consider aspect consider fit device conduct gram four variable initial pressure pressure mu mu minus mu minus mu mu plus minus mu minus mu minus mu x mu mu minus range fitting term respect indicator correspond center multivariate zero estimate posterior mean estimate posterior construct variational variance use hasting draw burn
statement iii hmms viterbi cluster primitive state meet satisfied since irreducible primitive condition general mixing imply versa infinite alignment possibility hmms assumption relax finite viterbi alignment side hmm viterbi alignment viterbi satisfied alignment positively recurrent process respect time infinite alignment piecewise recurrence construction r theorem obviously consistent viterbi predefine break property chapter vi argument identically necessarily
fitting regression relationship response make accurate technique attention particular response also investigation notably smooth spline exist nature exclude naturally estimate local influence may elsewhere level significance definition local close one would predictor influential treat treat incorporate influence accommodate significance certain discount treat locally know ignore predictor lie final consideration particular particularly informed often cause nonparametric interaction vanish neighbourhood consideration regression adjustment locally algorithm motivate example property
positive skewness large negative alternative similarly normality reject alternative tail clear perform compare tailed test comparison power nevertheless logistic mixture alternative
week six year order alarm could case adopt alarm week observe apply disease surveillance center al count author compare generate week national reference equal past seem quite nevertheless comparison replace systematically compare week gold represent count triangle alarm triangle line represent method c line see
achieve procedure increase asymptotic slow parametric oracle prove plug whether fdr critical procedure depend apply model one side study laplace double laplace may appendix correspond situation fulfil specifically decrease side additionally appendix arise test gaussian unknown variance central degree freedom location write translation ratio convergent formula dominate well integral central distribution dx student test imply test side test proposition model parametrize centrality ii tail student freedom behavior crucially characterize primary interested note location student likelihood test centrality tail yield xx laplace location parameter ht value satisfied panel bottom panel slope cumulative panel fdr fraction procedure laplace panel ratio f laplace location g pg laplace statistic appear laplace h
condition invertible ax vector sequence theorem explicit process appear case process residual observation sequentially extend tr ts r q notice limit sequentially appear influence summarize us r dr r base feasible use simulate appear statistic brevity exposition sequentially residual modification agree put residual calculate sum definition reasonable plausible eq formulate fix e brownian residual chart appear consequence chart corollary mind worth discuss issue nuisance
movie movie would rated recommend recommend movie experiment recommender try reproduce provide age gender student yes child category science h movie name movie yes movie follow use build recommender user scalable recommender system mainly way scalability combine rough slope smoothing build user system recommender privacy issue deal recommender system system add business keep dr paper produce template recommender technique available item vast growth internet system recommendation handle efficiently vast growth recommender system high even list express opinion algorithm attempt recommend item past treat recommendation problem used select
covariance determination ard scale ard prior popular situation periodic unit allow variation perform restrict via restrict sign improve model less expressive clarity generative come denote prior uninformative function transform appropriate cross transformation apply inner product compute model ignore model view pca pmf view inner observe movie collaborative filter link latent dependent another factorization stochastic provide participant dependencie model
four summary facebook dataset runtime bp community network belong community exist low overlap facebook seven community demonstrate node community demonstrate community community thank table subsection b section clique social need algorithm capable overlap recent assigning performance contain highly overlap highly structure introduce facebook five benchmark benchmark great consider community undirecte loop letter realization denote letter g realization letter
effect vary induce important cope surprisingly establish understand likelihood present estimation size penalization smooth convex log hand could discovery method design coordinate numerically focus covariate model fairly intercept pre specification covariate penalize effect develop perspective setting address truly scenario scope present datum model effect illustrate empirically effect comparison penalize mixed describe detail simulation procedure proof defer observation group let grouping observation group response effect fix unconstrained positive possible may multiple remark fit nonetheless sake
difficult detect continue forward payment add tag inferential subject able tag tag text thing social service help tag development allow visualization tag importance frequently tag see customer pay see previous correct correction view user speech recognition correction case correction contribution service extraction multiple user refinement independence well kind want currently service overlap characteristic tag create convolution unique carry basic book store addition case service aggregate material service framework medium business diverse avoid idea trend web tail public historical method social
round hard prove hold hard class symmetric game highly economic game public games pool game nash organize give section provides two player endow payoff player relative payoff player payoff relative every relative game since introduce player opponent action strictly formally period action player period action dynamic maximize payoff give maximizer strategy accordingly prefer maximizer payoff payoff yet maximizer even number use maximizer prefer payoff begin payoff feature rational opponent would advantage assumption regard maximizer maximizer infinitely patient look mistake importantly assume opponent maximizer decision absolute maximization action run period essentially maximal one concept evolutionary stable prominent role
implementation consideration cf moreover
le iterations b col slow mean exp graphical metropolis hasting involve produce alpha alpha last alpha exp alpha exp alpha rate exp alpha b c est col gold give simulation le last high acceptance b sd sd type normal high seq le j eps eps b last last type main laplace walk increase however random acceptance performance decompose depend moreover truly chain normal conditional implement sampler I I col main col duration time impose truncate component eq chain constraint program alternative solution keep plain normal conditional constraint j prop p prop p prop p prop j prop marginal long normal coherence exercise constitute e proportional density z n acknowledge dim I sd sd col grey false sd col grey false add posterior information table individual multinomial whose clearly break complete extend sampler
prove let I suffice eq rewrite inner fact degree fact therefore establish lemma absolute constant approximate bound xx q state bipartite smoothness I e labeling say satisfied maximum absolute distinguish pick neighbor projection pick labeling generate know ji opt define label assign every vertex fraction pick vertex least vertex neighbor recommend label nice recommended recommended fraction nice side fraction contradiction pick pick pick projection define analytical corollary remark university wu work university hardness hard constant agnostic hard allow big hardness result previous include immediate corollary result weak agnostic decision list previous hardness positive first invariance regular
mixture normal mn cl mn sa multivariate normal third term accept draw stage give detail burn update update successive acceptance logistic datum normal burn update mixture less fill gap proposal also acceptance refine hasting heavy tailed mode easily well example refine sampler initialize hasting proposal normal proposal challenge realistic distribution long alternative version difficult way generate generate take condition iterate converge regularity iterate draw describe walk dimensional initial constant take identity
choice sign power enough near david helpful edu channel constraint decode least square tailor assume communication show reliable probability small shannon capacity channel shannon superposition code superposition code achieve small communication capacity provide development merge linear information theory familiar problem require bit string string real number constrain across add receive string event bit event analogous reliability requirement sufficiently error small average communication communication channel traditional white interest mathematic version sphere pack code codebook moderate prior probability sum square problem value partition superposition range give solution probability identify produce bit heart differ unlikely provide polynomially sufficient determined partition superposition code rate capacity probability mistake fraction require section polynomially reciprocal reciprocal undesirable less complete task sufficient outer code arrange tailor partitioned code outer code fraction mistake end rs rate correct associate superposition total rate mistake distribution fraction mistake superposition regard composite superposition least separation decode
describe location rate separate signal whereas recover former diagonal hamiltonian diagonal internal calculate close due expand logarithm separable first term reduce vanish general taylor expand logarithm expansion stay small percent uncertainty locate position instrumental low expect mostly informative solution substantial location reconstruction expect simplify hamiltonian dropping reconstruction suffer region affect poorly already ignore term find minimize correction keep ignore change point spread sign term always suffer knowledge ray ray point neutral fortunately signature knowledge formalism take background count physical space isotropic
eq mean average respective oppose simply pick linear combination feasible normalize z assignment address offer motivation margin intra practice however simple posterior adopt walk actual position leave metropolis summarize obtain execute otherwise accept reject follow desire q obtain carlo use mode unimodal interestingly probable fraction configuration come grow single role proportion guide configuration however principled relation proportion origin margin distribution along margin feasible
desire hermitian replace intersect nan rewrite theorem draw equivalent product use previously hermitian firstly entry denote increasingly order h w w previous program entry program maximize whenever word result consequently program except repeat step previous show follow lemma decay use psd figure analyze h ct concentrate around get sufficient psd threshold gaussian operator nn q formulation function semidefinite threshold gaussian
idea give generate number development mention background people mathematic project gene uk take away finish stanford day communication
contexts ref completeness somewhat ref separately calculate directly step rule step entropy say observer predict intuitively suggest observer vanish mm kp h lm lm concavity state statement establish proof exist let unit word respect distinguish length know step follow eqs contradiction distinguish rv rv eq contradiction
impact dictionary build assign specific class practice compute c result obtain dictionary encourage computational code fix image second overcomplete learn require code order drawback propose aim leverage jointly code sparsity representation penalty real image capable recover dictionary art intensive superposition every orthonormal would seem signal
u c u step lemma first represent whose coordinate fact continue u strict q need lemma incoherence incoherence incoherence recall hold cv space orthonormal I cv orthonormal orthonormal I thus last prove condition indeed certainly since hand depend establish main outlier strictly succeed note imply namely prove observe copy show modification specifically observe approximately follow replace q essentially noiseless outlier succeed succeed column counterpart succeed slightly strong requirement noiseless constructing outli succeed noiseless solution pair define x g n cn hold
implement processor require implementation generator restriction split rejection would implementation even slow box normal slightly box consider normally ratio show processor dr comment implementation generator computer agreement pt plus plus minus plus
estimate nominal level effect intercept estimate stage example nr regular estimate denote intercept estimate construct draw stage nr r algorithmic use bootstrap draw turn bootstrap compute interval estimate cover rgb draw thin fill blue draw thin green em minimum em draw thin draw corner sep fill blue rectangle corner sep thick corner sep inner draw thick height em sep inner draw none none proposition thm thm grant science university support sciences engineering university york city ny ann support mh da treatment regime grow clinical science clinical aa south west txt aa north cm ab ab ab north west ad txt south west north west node cm b txt ba txt ba south west txt ba north cm r bb txt bb south west txt bb north west problem asymptotic alternative bad method estimation propose index clinical trial school simple estimation stage salient challenge furthermore subject
analysis tractable optimal pc pca subsequently variance theoretical favorable performance effort robust remain algorithm design dimensional observation apply consistency lack sufficient motivate propose new robust pca take inherent difficulty high proof support inequality let q upper due equivalent last prof set eq side inequality hold decrease hold substitute decrease property I increase attention decade spectra observation range thousand practical high microarray financial trick linearly infinite transform effort extend tool design regime fact cope
c h last px factor term quantity h plug assumption graph minimal order volume constant proposition ij constant eigenvalue converge plug spectral np corollary probability j graph approximate soon disjoint path minimal degeneracy time random operator underlie even though limit evaluation limit distance degree speed fast raw moderate might remainder explore grid lead tight distance grid explicit grid variance good distance collect implicitly tailor definition tool geometric graph know appear let p pn n n compute maximum graph ball serve template later count collection px ib resp b graph often expect distance quantity proposition rule degree valid region maximal n converge px px part part come argument
various approximate leibler utilize maximize terminate result liu mutual large connect connect make connect large terminate process candidate loop mm task rather use maximum
meet follow go present investigate blockmodel moderate instead explore visit theorem empirical sufficiently os enyi blockmodel achieving likelihood identifiable achieve identical empirically remainder investigate os comprise independent ab ab ab record respective respective loose error whether necessary sufficient os network growth edge number fig prescribe density closely require size shown dot solid test generate correspond
guarantee property low matrix satisfy analysis extend substantial column simplify entail generality since noise vector mean instance corollary consider sparse normalization although error bound form past g norm complement entry compatibility final w specify choice x n valid claim corollary regression well one formalize ball radius weakly form case cone rather center compare panel ensure tolerance term state allow rsc ball conclusion normalization sub belong parameter strict generalization naturally toward al consequence lasso generally rate function state rsc thresholded follow supplement sufficient convexity subspace rsc hold corollary approximation q r ensure claim outline family model glm suppose parameter
might repeatedly boundary preferable gaussian switch di elliptical attain idea evenly correlation feasible implement end product minimize matrix propose alternate algorithm problem far projection expensive experience parsimonious might effort importance sampling carlo evaluation q combinatorial binary vector rely usually combination copulas family sum family auxiliary eq cross
loop induce sparse step reconstruction patch definition manifold calculate whole coordinate calculate scale new indicator column calculation fast calculation update projection matrix patch calculate patch part optimization accord unified alignment compute accord pca class center define eigenvalue conduct lar loop matrix define consider lar conduct several column sparse ready sample lar subsection nonzero lar lar computation sparse via unified popular learning patch alignment local retain penalty add sparse combine term elastic net elastic superior reduction algorithm powerful consideration well square apply lar find regularize lar lar optimize general regularize special lar solve problem asymmetric constant lasso least penalty special define lar optimize penalty loop change consecutive lar
weak policy relate present great development control history part discover reward history interpretation probability proportional utility discuss recall formulate describe aim pass expectation propagation original close message claim optimality control specifically make
also converge z tail p r last infinity z z chebyshev inequality expression rp probability necessary convergence finitely many capable control discovery discovery proportion conclude proof let sequence fp gp last conclude fp established fp fp fp fp rely account establish fix kp kp gp kp gp j kp gp kp formula q theorem theorem lemma initialize edu noise measurement precision budget hypothesis satisfy adaptive
illustrated variate coordinate alternative symbol coordinate covariance imply correlation constant surprising appear marginal correlation imply interpretation allow latent simulation confirm consider correlation inference alternative model present difficulty likelihood however em conditional trace entry cycle index draw iteration full q normalizing use detail simple formula integer freedom form divide write cumulative standard moderate lead instrumental drawing accept repeat suitable multiplier important ingredient rejection tail focus multipli minimize rejection draw
grow decrease grow achieve logarithmic present decrease expect specify stand sequence lt show depend system consist resource evolve irreducible resource table c stationary resource state reward pair arm optimal arm great reward figure logarithm bind proof empirically however allocate resource learn reward maximize minimize define reward expect reward per step achieve logarithmic user resource broadly
l h difference entropy claim sum multivariate measure vanish fact h limit actually generate conditioning current alone determine future atom vanish vanish past vanish generate acknowledgment intelligence helpful manuscript include currently bit tool dynamical system analyze classify dynamical kolmogorov shannon lyapunov characteristic invariant logic result short fast bridge theory characterize short synchronization connection synchronization nature give duality synchronization control term one notion synchronization though interpretation different converge differently either process representation give insight elaborate apparent employ related review shannon asymptotic aspect hierarchy call systematic convergence short possible description note description model analyze entropy entropy entropy introduce block latter entropy synchronization summarize synchronization relate back introduce derive notion order spread control synchronization explore increasingly emphasis conversely class widely statistic elsewhere start optimally relax redundant synchronization class relax stage convergence property entropy turn induce contribute largely apparent tool advantage various sign display diagram dynamical history see control dynamical broadly go symbolic dynamical synchronization
increase step chain window recommendation diagnostic mean divide converge chain early st window lie within material chain city diagnostic run parallel simulation post rate statistic demonstrate distance superior secondly test aspect markov abc euclidean markov discard sample cl model associate map mmse j posteriori posterior knowledge mmse quantity unconditional joint f acceptance markov mmse factor estimate marginally obtain standard estimate bootstrap classical cl corresponding chain example datum loss turn claim divide analysis previous justify simulation coefficient variation
xy xu yu xu yu xu yu heat equilibrium tune belong reason stability ns lb sound instability compare pressure pressure ratio propagate sound pressure jump use visually subscript indicate numerical heat ratio relaxation boundary reach end equilibrium
vote occur vote party visualize voting question role determine operate explore posterior binomial graphical prior binomial string graphical place grouping sc al ga tx arise close ar tn ny ks consistently node variable consider indicator vote bayesian approach place sparse close examine therefore exhibit neither
finally embed prototype refine summary alternate flexible quality aggregation strategy show meaningful summary help greatly analyst get quickly good curve list variant illustrate numerous distant note look unique support summary piecewise contiguous interval series average turn label step segmentation basis solution merge two build mean prototype interval prototype prototype assume obtain prototype optimally respect specify total number piecewise represent section segmentation overview segmentation introduce segment mostly independent standard technique build connect building view goal section material relation segmentation point approximate simplicity concept article capacity learn rather visual analyst smoothness e base
entropy relate hausdorff distance hellinger hausdorff contain restrict pilot control hellinger hausdorff vary greatly hence pilot within nice normal cut stay parallel second define distribution lemma entropy quickly hellinger hellinger distance hausdorff imply get thin tight hellinger hausdorff true thin hellinger within rate hausdorff simplicity data pilot let estimator step relate allow entropy finite cover manifold net cover small covering call hausdorff hausdorff cover dc hypercube cube within manifold translate natural coordinate domain
learn hold yahoo competitive know human express preference consequently statistical comprehensive survey intersection retrieval area learn review learn retrieval broadly speak rank order natural phenomena movie book web page wherein focus model tie previous pair see ignore simultaneous interaction strong return query alternative object partition amongst amongst permutation partition wherein partition object generative partition object partition specify partition substitute potential choice potential potential normalise potential mcmc demonstrate application hold yahoo besides second
either ignore stationarity come average algorithm acceptance rejection posterior transition p come make change condition imply taylor give k k modulus right suffice expectation choice case denominator numerator k integral argument combine give ex rgb rgb abstract many modern easy calculate abc method replace calculation involve simulate simulated summary semi manner aim parameter mean parameter mean within empirical analysis ad choice summary literature demonstrate advantage affect implement abc implementation turn understand alternative posterior variable summary density give conditional carlo function numerator calculation importance mcmc rejection summary discrete summary statistic marginal summary finite ii
aggregate latent course scalability bayesian hierarchical parameter line observe connection among customer panel several adopt new product around customer customer likelihood model choose person position individual share realization locate coordinate aid visualization add label include interpretation considerable distinct super space cluster customer quite customer figure represent configuration small space provide become examine segmentation central strategy behind e price copy study two interaction practice interaction customer service help potential customer result observe basis graphical highlight among customer quite identify line individual place circle common link line potential customer useful reach link depend customer homogeneous segment represent similarity proximity circle customer observe occur certainly interaction among well example college people college share trait age education gender close relationship nothing else common college share common friend place student close close friend aware represent base observe alone unlikely behave way incur
low instability realize exposure portfolio increase drastically high reason fact produce low attain local portfolio risk moderately exposure financial comparative need exposure constraint high frequency profile improve e position short approach maximum portfolio exposure constraint reach comparative portfolio concentration daily stock carry return period avoid hence affect price jump financial portfolio numerous jump jump news move hold exposure sensitive accumulation expect hold approximated accumulation pairwise precisely element pairwise semi definite projection need pairwise strategy financial volatility estimation window theoretical observation simulation outperform one exposure constraint portfolio allocation theoretically far need integrate integrate simply time process process v xy condition impose little estimate integrated volatility need random variable finite generation realize volatility proof generate consequently hence variable specified continuously bound note part term far c paragraph eq calculation
raw relation node message correspond simplification simplification message model link strength facebook comparative feature binary relation transaction work block frequency relation take account group assess measure situation truth assess soft developed clustering measure apply later mixed membership membership normalize propose evaluation measure overlap output soft propose overlap cluster metric precision data data cluster belong true extension assign count number vector estimate membership membership precision measure present network corpus another website simulate recover interested
online overcome base propose name compare approach online online enable continuously evolve become increasingly tool sensitive internet report anti record target payment service cause internet community put mechanism currently service site safe service microsoft provide already appear else spam email provide full low impose overhead classify new I clicks rely maintain discuss describe dataset extraction describe algorithms dataset feature discuss explain base attempt include account somewhat adopt party trust party find distinct technique manually select lexical feature google page work build identify feature common string design automatically google classifier lexical contain google content describe maintain server side
understand posterior denote metropolis gibbs vector latent highly inefficient correlate moderate sized mix poorly metropolis irrespective proposal simple would gibbs markov mix slowly problematic require estimator complicate sequential positive evident markov path sampler mode adapt mode demonstrate methodology effect rare accounting moderate dependent suggest negative easier parsimonious include chance decrease observation absence include noise effect find evidence datum strong agree observation numerical inspire multiple mode suggest relationship growth sampling methodology mode static instance analyse hand scale algorithm surface dependence highlight ability reproduce search evidence metropolis recursive low model space time avoid jointly estimate latent jointly exclude special realistic sub kalman filter linear
rd ed american statistical association realize market journal economic statistic quadratic create stock variance serial covariance quantitative security bid ask journal financial infer quantitative trading finance li approach process normal journal american quadratic variation journal liu imputation miss journal american gibbs generation truncate conference realize volatility j quadratic price increment manuscript li bernoulli simulation limited journal schmidt frequency corrupt g kalman ed fluctuation market trading quantitative finance volatility simultaneous bernoulli normal high volatility volatility manuscript g efficient constrain regression technical volatility round convergence line jump business economic incomplete journal integrate estimation noise journal financial wang almost convergence recursive asset price filter finance zhang time volatility noisy journal american association frequency data exchange journal business economic volatility estimation volatility frequency technical thm proposition mm pc pc cm volatility transaction volatility occurrence transaction computationally price volatility neither transaction also volatility noise particle sequential volatility transaction volatility implementation result application could replace notation realization decide stick notation exist log additive zhang
es es fitness see define learn problem es stem reason correspond document thereby explore direction es voting solution make decision es gp therefore individual population rely solve evolutionary es detail aim propose fit inspire behind population pressure cause rise fitness fitness improve mutation ensure diversity evolve es rule form collect population vote remainder structure begin query query population fitness production new query evolutionary present formation es express act concept figure act four concept boolean operator document contain concept tree depth representative depth branch reach terminal terminal depth meaning boolean operator drive principle population randomly set depth individual start root place single create repeat grow depth terminal terminate growth tree place terminal complete individual equal size assign fitness individual mention create relevant concept give compose member boolean query document predict irrelevant fitness search maximum possibility therefore fitness produce query mutation parent choose create subtree root parent subtree root parent parent figure produce operation produce mutation operation mutation consider primitive primitive mutation operation mutation primitive
nystr om come investigate principal reduce extend approximated use theorem extension theorem reformulate address explicit nonlinear mapping low mapping follow manner integer stand indexing vector coefficient assume polynomial mapping unknown unknown lie manifold kronecker hadamard define prove framework find embed high reduce use polynomial explicit substitute substitute eq equivalent simplify th entry solution
define bivariate cdf cdf copula default substitute gaussian marginal long capture dependency reason contain copula imagine choose drawing encode specify univariate cdf gaussian process value beta obtain draw beta gaussian inverse variable dependency encode generate many distribution dependency call describe copula index distribution let joint base inverse previously gaussian copula copula mapping
structure underlying parameter still chain ph introduce substantial parametrize scalar completely distribution order ideally suit preserve expectation trivial characterize partially order within totally order subset threshold show delay alarm existence compute hyperplane several iv reader description various starting jump detection existence switch exponential sec generalize poor time consider novel stop threshold mild delay stochastic geometric time ph change belief control use finance seek optimize accumulate seek term risk control delay generalize lattice vi stop problem learn amongst economic market agent optimize local private observation social agent make stop automated system sensor decision reveal decision subsequent agent time belief belief strong threshold policy result decision local decision involve deal constrain social social rational eventually pick irrespective private stop enhance relate reveal observation stop stop intuitively stop state accurate private constrained propose switch public belief stop implement system scheduling change bound transition probability question markov stochastic problem formulate consider agent scheduling optimal policy rigorous intractable markovian large formulation allow stop problem subsequent comprise chain distribution phase ph ph hence construct markov chain denote assume evolve markov state correspond view composite one determine transition equivalent power appropriately occur special geometrically q integration counting measure integer decision choose
application network relevant play comprehensive exposition organized follow provide overview fp process fp invariant frequency update time fp conclude player game concept fp fp subsection security game pure simplex take first static select mixed instant player boost mix
fluctuation error component investigate unfold procedure measure calculate ix bin number ii cm present fig comparison demonstrate superiority previous comparison literature unfold h cm algorithm eight table perform robustness bias eight problem widely measure frequency pass resolution event acceptance unfold replace original use slide unfold problem low smoothed filter unfold
attack automatic collect public web information security attack token handling first attack seek attack attack output token attack attack attack attack follow subsection responsible attack familiar attack detection code double attack directly also attack decode character decode attack attack accord splitting recognize six token responsible transform token token attribute plain text comment token receive attack attack original raw corpus information token attack attack increasingly attack attack profile attack vector automatic sequential attack regard
iterative estimation bayesian monte tailor hasting show associate set property understand poorly scenario demonstrate fold introduce suitable home potentially like differ interpretation efficient idea generalization model include comparison base em sampler perform inference knowledge ever context great allowing perform well organize follow basic
explicit require possible simulate optimal loss norm dpp standard approximate dpp dpp preferences round preference approximately dpp monte cx nan begin dpp presence establish dpp bind integer accumulated dpp performance bind dpp define asymptotic error unlike critical dpp dpp near large consider law number dpp converge result dpp simulation carlo well presence sampling variant dpp dpp simulation surely need problem instead simulate dpp sampling dpp dpp dpp rl replace bellman nan nan ax kp cx pseudo code dpp rl algorithm h cx nx cx nan rx nan equation result dpp
processor processor outline idea variation time box uniformly distribute parallel computer require normally since translation give
belong another management status list management create relatively balanced unbalanced use average ap ap community unbalanced balance receiver characteristic roc auc reason auc ap affected table summarize c task adjacency ap auc machine hyperplane fit svm specify slack often cost parameter package fit wide benchmark understand benchmark conclusion reality svm cross
fall write conditional estimation mention pseudo normal wishart conjugate density set initially inference form construct structure online partitioned large easy depth branching depth maximal leaf reach consequently examine depth consequently ht figure demonstrate firstly gaussians bivariate angle consequently highly normality
able extensive technique inherently detection promise adopt nonconvex paper operator iterative substitute appropriate thresholding multiple procedure use somewhat surprisingly soft thresholding hard thresholding hard challenging pick start eq coefficient hard directly ol clean question arise try arbitrary converge condition converge answer question discussion rule threshold include soft thresholding define odd unbounded rule version vector assume huber soft thresholding correspond generally multiple threshold hard thresholding find small throughout pilot j cm penalty q
previous singular preserve symmetry length system avoid must positive symmetric compatible system possible equivalent denote vector root multiply equation get consecutive term recurrence addition recurrence multiply original new write also output associate sign equivalent solve z produce c second last element k k k x w x compatible length compatible return solution necessarily ax compatible dy ax bx minimum
duration novel meta policy spectrum access version problem provably near logarithmic achieve reward optimal system user try access availability evolve chain see channel user select channel sense channel
fair character evolve tree pass edge probability long branch mutation molecular thus throughout length mutation character character leave little mutation h factor tree two generate balanced tree sequence larger easier precise estimation method group non parametric method presentation remarkable year problem implementation posteriori estimation tree designed take extra computational viewpoint strictly rate markovian criterion individual favor distance markovian simple hamming distance compute building procedure follow one heuristic agglomerative tree procedure linkage update pair find build distance simple dissimilarity leave drive evolutionary dissimilarity either gene word hierarchical clustering inform case tree clustering create score distributional clustering tree thank bootstrap aligned tree resample character
decision bayes spatial stein bayes paper ann place stein small area proportion design value high dimension journal theory decision compound rd rao berkeley berkeley statistical ann zhang h zhang compound method ann remark assumption compound spatio incorporate covariate normally involve empirical view generalization method prove compare spatio certain relation identity exchangeable procedure permutation permutation q abstract space exchangeable realization belong function loss
kernel work two briefly describe kernel distribution rkhs iid fix induce clique empirical map maximal clique clique universal kernel induce clique q specify exponential limitation map operate handle dependency note zhang optimize explicitly document necessary well modify mean modification induce q pd completeness positive definite pd feature map product element pd benefit distribution currently open algebra identity mean iid extension dependency assumption whereas generative mean
convergence straight slope provide numerical create take specify confidence sort level sort vertical horizontal number draw straight slope dot slope correctly mean square compute equal distribution continuous cumulative list describe conduct parameter generate class
generalize example china yahoo com contextual bandit popular systems yahoo news online user practice simulator environment hand algorithm simulator create simulator paper contextual different simulator easy adapt provably unbiased empirical article recommendation yahoo front page offline contextual algorithm show accuracy effectiveness computing web recommendation service yahoo yahoo user activity clicks identify front page suggest solution create real unbiased thorough investigation improve theoretical study exploration medical web different classic arm particularly interesting contextual information make contextual call reinforcement bandit side bandit arm contextual trial trial know context arbitrary context unknown ta emphasize feedback payoff expectation define
delay left offer scheme observe leave complicate survey possibility restrict use consider process equilibrium absolutely minimal hazard hazard apparent later fix say recurrence bias proportional triple result let correspond distribution marginal marginal length consider truncate follow similar truncation model
subsequent decomposable technical policy make initially load play reward develop outcome index without delay final show decomposable complete characterization somewhat index highlight accounting martingale prior bayes traditionally result play yield reward current encode play accounting policy clarity approximation extensive mab know compare scenario typically comparable motivate instead moreover play true delay delay exploit outcome however constant regret arm underlie draw distribution prior specify input reward play make arm play available step observation posterior serve
within increase dominate encounter effectively computational lemma property claim capture preference key outcome effectively view permutation datum turn permutation choice distribution permutation near consistent approach seek choice permutation marginal seek establish choice admit exist choice whose relative agree signature relative force empirical american set useful choice signature road internet arise human choice particular behavior assume individual rational maximizer collective behavioral entire population guide g build country product put interest refer simply discussion choice crucial input make effective reveal preference via assume behavioral mis choice ideally behavioral structural absence need criterion select preference natural criterion simplicity model sense seek sort consistent simple choice model marginal literature devote partial parametric diverse name follow provide history reference simple set permutation parameter distribution probit realize logit
symbol ref identify role isolate symbol write english notably structure responsible dynamic symbol amount way shift architecture drift correspond change diffusion within subspace jump subspace correspond loss drift behavior explore periodic formal short human chain communication though distant begin periodic capture semantic sequential language drift mutation future prevent preference extend iterate evolution evolve production acquisition agent stimulus effectively force bottleneck agent generalize pressure bias traditional view human language brain propagation valuable evolution language approach neural bayesian drift serve agent linguistic latter framework quantify trade learner transmission bottleneck evolution ce decomposition linguistic present serial communication channel structural give drift population light kind behavior extend scope phenomenon exhibit derive population environmental political lie population area proper drift question model evolution progress rapid fisher drift process view pose fisher fitness connect fitness
superiority linear top pointing although anneal significantly priori constant need investigation conjunction determine concerned langevin sampling canonical ensemble choose local hamiltonian also temperature function propose accelerate langevin boltzmann gibbs follow potential positive wiener boltzmann gibbs canonical geometrically natural thing enable
denote procedure statement theorem remain unchanged projection section construct adaptive term employ notice challenge value p complement p nm appendix remark value decrease rsc employ rsc minimize work provide corollary constant error subgaussian replace suitably generalization lemma next obtain oracle resemble rsc stress fact entry furthermore inequality hold constant x random plan model thorough assumption nuclear rsc comparable point restriction estimator rank recovery show rsc achieve trade
real world closely likely relevant account effective genetic goal discover genetic proximal exploit fusion accelerate desire scalable solve involve fusion penalty univariate fusion penalty structure briefly review proximal optimization preliminary result task regression space flexible module qp descent grid work real world datum mining mapping example genetic problem quantitative trait attempt discover association snps million candidate clinical phenotype g problem formulate problem regression regression identify parsimonious multivariate treat independently adopt structure among response
analytical work location sparse x kx let kx kx interpret space densely kx real posterior distribution derive choice ensure agree satisfy b inspire dot solid increase become gradient carry induce lead p derivation intuitively marginal x figure g linearly yx yx analyze relevant interested cluster location
propose angle satisfy reversible edge place opposite randomly could lie acceptable recommend elliptical slice accept effectively place check location slice cover representative posterior proposing point one another width tuning method proposal away current towards slice move previous weak limit prior algorithm initialize draw angle initial first coefficient exponentially common offset entire still element sort tend draw
motivated circumstance article estimator computationally practice deep natural image patch investigate patch contrary claim qualitative present investigation particularly good image statistical occur constitute tool many learn lee prediction statistical image understand include approach assumption concerned goal important assess compare instance tell everything decide also visible unit boltzmann mass give zero mark boltzmann operate reach conditional unit eq logistic boltzmann see network unit otherwise increasingly magnitude deterministic boltzmann interest build boltzmann partially state state variable ml conceptually term right hand gradient draw distribution follow energy state energy connect hide interpret anti anti evaluate computationally intractable carlo typically slow measure machine feasible replace simple former lead rbms
kolmogorov real theorem close respect kolmogorov r ni nh half ni ni nm utilize know vc arithmetic c imply acknowledgment constructive theorem institute mathematical foundation study space mean minimize kullback type support hyperplane approximation estimator concave density fairly condition us model show respect convergence result independent regression identically distribute concave zero likelihood illustrate real defer proof detail hereafter
expert additionally mistake step back execute drawback impractical different sequentially adopt policy train iteration initially start policy expert interpret remove iteration action stop return expert problem aggregation iterative train deterministic simple proceed use trajectory collect add proceed collect iteration train aggregate collect intuition input likely execution previous experience interpret pick good trajectory well expert learn policy query expert collect mistake visit state irrelevant typically expert behavior could expert decay
involve due restrict turn calculate estimate univariate strictly optimisation also attribute fast pool hence solution monotone decrease function solution permutation index x iy arise naturally apply shift response deal separately observation tie covariate work affect tie covariate general solve dimension covariate extend iteratively word component store marginal fit value pm residual kx k na adjust result necessarily converge estimate converge monotonically locally quadratic converge define th step objective direction great evaluate solution derivative follow involve covariate step exactly check simple note towards drop become facilitate fit consideration corollary great thresholded hence small
svd simulated level monitoring water variable chemical available ambient collect minute compare data stepsize stepsize small stepsize large stepsize sampling decrease sampling near normalize yet comparable reconstruction sensor reconstruct instant section subspace column rank identify incomplete onto examine excellent
fail step relative induce select constant uniformly uniformly inequality constant post estimator step allow derive estimator model post post lasso despite post cn constant ol characterize ols post lasso perform strictly comparison support comparison repeat definition discuss post pt grow lasso post coincide occur fail subset however separate coefficient dominate ol post part perfect case sharp ol strict provide select ol fit condition f provide performance ol post fit lasso sparsity lasso ability ols ol post fitness
nc qx qx lemma fact independent identity observe globally approximation ix r prove follow proof normal claim since cauchy schwarz estimate last follow ix control claim obtain weak replacement control ix pt mn following eq notice wasserstein use distribution bound ix mr n gr ix tr ix z ix dt mr ix ix e ix pt proposal lead scale sde recover identity old denote cdf banach continuous define endowed supremum invariant sde density lebesgue interpretation start rwm take explore proposal mala langevin constant maximize limit sde explore invariant quantifie proposal rwm information develop mathematical metropolis hasting fashion criterion practitioner optimize instance rwm mala yet perspective measure class target measure retain inherent simplicity metropolis valuable key acceptance limit rwm maximize
index match plain interval already sequence e loop discover fix configuration fix fix unfortunately combinatorial adopt therefore alternative iterative every execute uniformly principle two dependency iteration pick index interpret continuous determine queue trick dependence make tractable fig execution ready index expect fraction vertex degree sample observe unfortunately interpret proportional match discover node grow e except special case rewrite q traversal ff subject index apply might significantly discover select sampling traversal select show node k concentrate degree
simulate system frequency collect success driving case balanced truncation suggest dynamical ready present case simulate emphasis empirical broadly applicable apply begin high infinite simultaneously identify direction observable assumption linearity exist become nonlinear closely linear feature reduction state design simple half dynamical behavior problem rkhs balancing adapt balancing propose determine reduction approach system control finite control balancing
curvature local version mcmc may extend riemannian reversible jump extension geometric extend efficiently community geometric
homology group induce topological topological homology background orient module arbitrary unity restrict immediately subset together explicitly unless real say persistence module map otherwise persistence module critical module critical value
hyperparameter k theorems representation nonparametric marginals p family sufficient statistic form projective projective index analogously model back posterior dirichlet process base next instance parametrization detail illustrate symmetric construction projective finite likelihood projective iii concentrate desire interest measure infinite mean conjugacy read marginal conjugate observation draw multinomial distribution note way dirichlet distribution generate set adequate choice countable algebra extension finite partition partial index space dirichlet hence singleton jj j sum probabilitie j hyperparameter space projective space collection random c vx I additive probability measurable ga projective dimensional projective marginal define multinomial affect henceforth omit multinomial family projective g measure eq additive hyperparameter sampling define provide suitable borel embed set v dirichlet instead marginal vi da ig proposition statistic bayesian hyperparameter index statistic dirichlet base parameter sequence group permutation finite element infinite potential model approach motivate permutation datum model censor symmetric task realistic movie movie cycle induce induce partition prominent restaurant permutation projective limit projective limit virtual permutation group condition limit symmetric sequentially inclusion projective system projection mapping mapping intuitively raise consistently natural vector
context default use yet predictor another length bit predictor see refer subsequence history value binary string investigation choose control first attempt predictor predict next single bit subsequence cover time instant two phase initialize prediction candidate score score subsequence decide count cover unchanged repeat decide candidate parameter repeat value increment follow score
tv verify set yes complete yes complete use choose set yes ct see yes next theorem arbitrary give polynomial follow relate let markov graph denote let weighted edge chain stationary vertex protocol let protocol sd protocol completeness protocol let c co hard vertex hypercube edge weight chain let constant edge add add markov walk loop polynomially couple
expensive decomposition thus check iteration alm adopt alm update project subgradient method schmidt comparison meaningful randomly similar al li create sparse hence iid gaussian matlab alm stopping solve dual additional cpu alm c gap gap e e e e e e e alm increase
line leave replicate density segregation alternative kernel symmetric much nan large monte investigation kernel imply nan imply high density skewed segregation dash ht triangle monte power proportional case asymptotic value segregation top bottom middle empirical base use function expansion mc n mc due magnitude severe segregation mild segregation power moderate power however power within right sided small moderate segregation alternative normal yield triangle value segregation alternative column column bottom uniformly segregation triangle corollary approximation monte get empirical maximize mild segregation severe segregation recommend segregation seem recommend segregation triangle monte replicate consider repeat segregation alternative degenerate relative nan segregation line replicate segregation alternative separation kernel alternative imply density segregation leave right separation nan imply skewed solid segregation dash ht triangle carlo segregation middle row column right mc mc cs present carlo central carlo power estimate get severe segregation high estimate power power power recommend mild segregation severe segregation alternative multiple triangle monte carlo estimate triangle value segregation alternative column middle bottom compute formula corollary normal tend increase get segregation power around moderate severe segregation power empirical seem appropriate
sec sec performance summary discussion sec random investigate add random mechanism easy nontrivial instance complex learning I regard perceptron patterns open configuration circle represent configuration flip ise perceptron depict unit perceptron association classification pattern actual perceptron modify configuration pattern space ise perceptron configuration ise perceptron paper binary pattern uniformly long solution quite configuration appear message transform transform
generate certainly moment algorithm quite develop st kind systematic mechanism genetic st bootstrap subset selection simply pre screen st sure screening sis variable subset impose upper st path include design multivariate normal mean simulation zero minimal median simulation signal min median max elastic net sis competitive make signal large reader st sis quite generating lie contrast selection measure average independent measure beneficial attractive many approach decision rank considerably option variable average thresholding rule call look
satisfactory check criterion polynomial candidate primitive apart thus polynomial parameter optimal generator web recommend fail test c easy implement since operation shift unlike primitive characteristic
thin geodesic triangle long organize complex manner loss normalize distance result lipschitz without concern particular prove fashion statement domain exponentially grow confidence every technical observation contradict converse radius concentration around fix constant depending consist statement exist banach regard integer see page respectively function determine lipschitz property preserve exist accord rademacher reference p everywhere regard lebesgue differential denote property though number observation banach banach combine classical constant banach choose distance banach corollary remark scheme exact link histogram lipschitz function get concentrated observation concentration structure dimension indexing scheme curse lipschitz concentration
discount group perform highest discount future algorithmic parameter principled reduction representation carlo select output present input unnecessary analogous input error planning effort prototype module convenience equal output mode reverse backpropagation effect analogous apply model modification need determination encoding mixture permit recurrent map dynamic
ultrametric highlight capable address face provide good tool application face old focused way regularity massive data mining determine express measure reality hierarchical cluster mining finding determination understand express hierarchy intrinsic interest review many ultrametric topology ultrametric discrete focus analyze massive illustrate finance publish study keyword multivariate recognition storage retrieval ultrametric topology economics human science frequently hierarchy scheme mining often instead pointed fundamental complex reality work mathematical complex present make symmetry architecture principle range notation division number field form integer binary define field see practical extensive consider dna encode u offer uniqueness encoding digit expression default digit start section common discuss explore lattice ultrametric distance hierarchy influential paper survey al ultrametric topology ultrametric hausdorff motivation complex give study field
presence constraint high optimization whereas technique coordinate considerable devote make feasible important modern obtain typically differentiable functional chapter analyze functional differential calculus operator study minimization empirical plus squared kernel hilbert reproduce namely unique combination characterize coefficient family equation introduce two general large regularization regard non linear gauss kernel solution symmetric result semi separable separable call dimensional whose follow parametric method characterize problem subdifferential
lemma combine fact fourth combine decomposition guarantee identifiability condition identifiable row column vector consider also recover certain program program constraint entry natural convex surrogate sparsity rank optimize formulation suitable enjoy recovery guarantee work closely sparsity application characterize incoherence sufficient identifiability analysis characterization yield strong favorable
interested simulate evolution spin represent spin obtain discuss spin monte improvement initialization strategy introduction mechanism carlo physics orient concept briefly review quantum simulate skip simply imagine intractable binary value idea contribution improve discuss quantum spin present excellent suppose distribution member dependent implement naive mcmc metropolis heat mixing
gauss z n model mh proposal proposal proposal construct j unity many density suggest sampler generate inverse straightforward piecewise quadratic coefficient z precision compute contribution reverse value current besides mh compute estimate section review new choose high detailed ensure estimate draw acceptance error variance ordinary square laplace sum form truncate yield incomplete gamma whether include nine covariate linear
r x ball disjoint intersect follow decay exponentially state nn ball region pr ok note ht sample disjoint note correspond furthermore kk p x joint power x identical derivation lemma define cx e cx r ok cauchy schwarz pr ok cx tx cx tx cx I tx u mu cx tx ne ok cx r follow pr ok eq note cx rx cx tx le rx also use clear I tx le ok disjoint conjunction pr ok observe schwarz subsection imply ok k imply integer c uniform identical result coverage ball density depend kernel x volume intersection let coverage estimator ball estimator begin establish density give pr q expression coverage neighbor field point definition p iii use cauchy schwarz cx cx cx cx estimate estimate cauchy schwarz obtain term ii iv ex cauchy locate term previously establish ball lemma obtain arbitrary continuous function x ok moment density x nx kx invoke point invoke moment nn density estimate density truncate volume ball ball volume
coherence approximation approximation applicability sample provide far fit corollary definition university california berkeley berkeley approximation algorithm coherence ability extract matrix completion question paper coherence formally new analysis propose whenever coherence across wide coherence estimate excellent become variety method spectral cluster manifold technique ridge algorithm order
update user count serve learn channel channel channel user denote total far transmission slot channel statistic function horizon collect discard policy regret maximum u regret transmission bind policy part asymptotically combine policy regret upon rank channel channel perfect would increment hypothesis test current alarm need decay asymptotically select count first user u n secondary implement bad u none estimate ensure go threshold count trivially occur rank user wrong rank line maximum start user configuration generalize user transition probability define note surely otherwise time spend wrong stochastic go threshold none give main section threshold function condition policy decentralize setting regret user user
information present long specie present sequence reaction hundred amount suffice million enable redundancy extraction mixture sequence two contribute nucleotide enable level comparable measure indicate enough sequence mix reconstruction decay prominent challenge sequence amplitude position peak correction raw context peak incorporate effect complicated peak cumulative calculation overcome sized peak dependent nucleotide position current preprocessing peak position position ordinary tool need problem force remove develop improve cs solver cs greedy pursuit approach might without removal reconstruction consider remove frequency explicitly utilize accurate rip reduce coherence novel dictionary sparse redundant easily sequence reaction sequence increase accuracy case noise sequence easily sequence additionally several sequencing enable enable single universal sequencing region align sequence identical identification via sequencing specie clinical remain species population specie
show trade normalize minimize abstract must deterministic giving limit length follow look practice focus city position certain network large state give bound tractable bad perhaps design network provide main city standardize apart distance apart elaborate outline elsewhere conceptually define network concrete language regard city line segment define applie rule configuration configuration euclidean randomness city euclidean plane though configuration city
message classifier indeed risk conjunction large attribute message string conjunction small string basic compression datum reconstruct small reconstruction return training subset contain need reconstruct compression define set index denote present arbitrary compression reconstruction consist message classifier return message message consist define attribute threshold compression set consist per threshold compression reconstruction map train information equation specify message conjunction specify conjunction attribute threshold value attribute attribute threshold choose subset specifie threshold assign eq complete compression aim obtain sparsity minimal encoding string examine large separate yield pac formulate dependent partly decision gibbs select assign risk
computational expense factor make individual simulator costly generation examine simulator deterministic simulator grid obtain display goal build computer fraction budget simulator coverage constitute contour surface true simulator approach outline take figure less interpolation gp fit well approximate predict power surface point popular maximum predict power obtain maximizer whereas assume underlying computer simulator correlation fit determinant inverse section conduct specifically hypercube design propose new accuracy lemma zero reach important remark worth note methodology also several realization use square theoretical may lead mse sometimes fit recall near design say
threshold table tag snps associations furthermore association detect different also snp snp population population associate pick arrange accord category column collect variable gene category gene category snps majority category quite category snps detect use surprising marker hand gene hmm hmm hmm locate strongly correlation combination report snps rs three snps locate indicate region strong four gene summarize snp furthermore four include snps proximity ambiguity snp correspond gene agree snp include snp snps evidence level future multivariate trait rr rr rr hmm hmm rs rs rs c rs rs rs rs discuss three close actually report characterize snps position kb take report pos fairly think region snps variability within snp
quantify sample size successful note side oracle remain multiply constant al error satisfy probability knowledge furthermore rectangular I consistency detailed rate matrix ball bound spirit general weak usual isometry isometry fix coincide scale restrict scaling remark valid replace positive involve restrict isometry observation brevity assume element difference element exist subset containing imply conditionally leibl q satisfy small
code substantial improvement frame intensity contrast spatially localize translation improve inter video potential compression tradeoff explore distortion beyond natural response infer transformation operator neural observer could change transformation learn could provide extension variable response movie might following decompose generator another diagonal following represent degeneracy degeneracy choose minimize practically every institute california berkeley department
regression sparsity gradient nonsmooth apply investigate suitable choice require short euclidean joint exist mean old condition specify boundary common piecewise smooth terminology treat component necessarily constraint framework describe later rate greatly lie manifold learning exponent euclidean let short denote vector exist satisfie estimate confidence except replace euclidean intrinsic dimension restrictive avoid introduce notation somewhat complicated proof base gradient zero natural specifically variable follow set f empirical covariance large variable directional derivative view datum along represent maximize maximize minimax repeat important effective direction eigenvector outer product provide approximate covariance estimate appear zero entry coordinate identify direction refer
seed interested limit scheme period argument advanced use mc numerical estimator consistent produce statistical advanced offer error argument generally consistent unbiased merely introduce completely eliminate price pay
distinguish fig subsection method consider specify distinguish fig statistic sensitive distribution bin bin recommend asymptotically see goodness limit briefly handle multinomial handle infinitely bin observation furthermore handle weighted root consider separate statistic advantage variation sensible availability computer level monte algorithms efficient easy acknowledgement would like thank discussion
expensive model cpu two essential first initial provide adequate compare type design optimal fit around particularly well suited second evaluating initial optimize validation comparison keyword computer compute investigation code realization code output treat realization center computer code degree polynomial sufficient sometimes capture process characterize spatial correlation stationary focused write one function correlation analytical correlation q wide shape predictor formula denote predictor variance
spectral unlikely due difference within like observational noted label object datum show network class classify case validate test argue matter chance contrary possibility inaccurate enough issue problem machine use logic feature use opposite correctly fail close look look failure occur spectra actual likely acceptable unlikely address recognition identify find datum active good free nn incorrectly label sample correctly remove basis citation alternate instead remove fail pass classification retain opposite scheme clean outlier
martingale previous analogous assumption lipschitz conclude decomposition divide probability appeal f thereby complete proof concentration uncorrelated martingale inequality eq extend n martingale equality uncorrelated q quadratic equality dividing complete last report result dual illustrate excellent agreement behavior versus graph grid expect scaling scale minimization hinge pair dy classifier machine choose minimize hinge associate shorthand hinge associate linear classifier sum setting minimization consider form generate qualitatively similar effect graph topology namely cycle grid range iteration vertical horizontal dimensional size spectral gap dot line corollary error versus grid demonstrate define number obtain function shift goal gain understanding discuss cycle grid follow quantity average panel three panel graph type grid blue dotted exhibit exhibit panel show network paper experiment present incremental currently figure optimal stepsize analysis distribute stepsize therein fit plot connectivity averaging gives reach vertical axis horizontal axis incremental show
test hardware traditional processor compare processor implementation old fast appear clear landscape change publish implementation intend efficient implement recent implementation three method think processor generator uniform apply number normally apply linear avoid consume generator
show unique partition triplet independent strongly let use necessary respect certain distribution even belong mutual case complement multiple equally design complexity example objective recursively partition meet leaf simply stop split search subset time partition item evaluate independent use thin chain structure thin discover thin chain partitioning item thin item n optimize exhaustive observation could advance nonzero second sort know unknown threshold infer item advance fix repeat candidate setting method search exhaustive third strongly remark method somewhat conceptually connect anchor exhaustive path nonetheless effective necessary information triplet sample triplet exhaustive minimizes subset evaluate triangle cross triangle total partition reality large element partition paragraph evaluate optimization big time require cache hierarchy discuss practically confident amount estimate adequate learn would bootstrappe offer approach replacement estimate resample typical bootstrapping structure lie discrete summarize compactly summarize hierarchical fraction set hierarchy confident small tree summarize large size final time run plot solid agree force set partition item structure dataset even hierarchy agree hierarchy tree rarely case entire structure agree make ask agree count correctly leaf meaningful identify hierarchy uniform consistent concentrate even case simple structure much fouri theoretic machine community graphical lie exploit inference operation bias drop formalize recurrence transform dynamic programming branching use recurrence compute
topic work typically far perfect probably combinatorial bi mean objective benefit although exactly combinatorial eqn intractable exist wide range heuristic strength understand approximately optimize method complementary bad flow method spectral bad method piece cut difference flow road might scientific easily well cut bias social property analogous piece regularize suggest novel partition apart insight perspective define perform instead couple noise particularly problem instead much reveal look ensemble cluster property heuristic interested analogy intractable partition less non determine dna look etc structure put x ray one protein physics input hard visualize protein large numerous large piece procedure employ reconstruct visualize adopt interpretable community principle let small scale social function minimum cardinality intuitively surface area aspect capture community community size illustration hard order compute different flow follow processing provide strong heuristic good piece find tight community like surprisingly compare behave graph road datum non manifold identification flat connect moderately graph perhaps surprisingly common model reproduce community flat thus view perspective meaningful piece moderately like graph size fact advantage analysis information network
agent exist question describe researcher ask exist amount sketch represent substitute representation substitute admit perspective structural submodular bound submodular distributional implication theory economics combinatorial present approximation achievable example simple achieve product behave fairly submodular behave technical question economics immediate use complicated approach technical precise achievable submodular upper improve function proof description central submodular central issue distributional bring usual model structural provide understand natural technical gap improve improve likely work real would natural class learn well perhaps distributional assumption learn trivially extend lipschitz concrete value satisfie integer normalize monotone submodular converse also rank property normalize monotone submodular edge monotone rank formal value lattice subset whereas large close na set follow property
pa proof w use w w hand complete definite inactive c small affect onto dimensional clear mapping minimal c j denote topology natural subspace topology fix definition c c c sn j sp sc p ps cp j j j cd ccc cc without spectra show rule setting demonstrate use noise accumulation centroid fair select feature report microarray ignore gene centroid employ microarray essential datum correlation information case whether covariance significant accumulation setup consideration precise suppose normal p setup performance rate pseudo classifier fisher discriminant difficulty discriminant whose first arise noise accumulation centroid challenge
unnecessary sometimes advantage simulation established investigate kind signal arise test spaced test signal scale root ratio replication posterior rule deviation parameter practice estimating significant trial dataset several establish wavelet reconstruct signal ordinary discovery wavelet block signal haar signal analysis package perform wavelet mean interaction ss bt false fdr signal replicate error rr rr ss cv fdr ex ss ss bt replication table present extremely estimator thresholde naturally cluster transform perform moderate reasonably
state markov nucleotide index represent instantaneous substitution nucleotide nucleotide transition change state substitution homogeneous substitution site reversible proportion amount flow flow opposite follow notation nucleotide nucleotide specification substitution distinction substitution rate evolutionary reversible substitution substitution simplify illustrative look wide range substitution cardinality specie nucleotide substitution propose maximize adopt endow prior update follow mx suppose competing substitution eq favor really year monte carry ad hoc currently depending
state current lead current agent receive reach offer mdp require procedure valuable environment seven among leave agent reward otherwise receive monte replication constant ensure hold reward deterministic slightly algorithm benchmark environment agent environment four environment generator environment environment environment average five state
