plra

Introduction

I’m going to try to do a version of Poisson low rank approximation in R so I can play with it, based on Abhishek Sarkar’s python code (here)[https://github.com/aksarkar/wlra/blob/master/wlra/wlra.py].

Poisson low rank approximation

# return truncated svd (truncated to rank principal components)
lra = function(x, rank){
  x.svd = svd(x,nu=rank,nv=rank)
  
  return(x.svd$u %*% diag(x.svd$d[1:rank],nrow=rank) %*% t(x.svd$v))
}
  
# Return the weighted low rank approximation of x
# Minimize the weighted Frobenius norm between x and the approximation z using EM [SJ03].
#
#' @param x input data (n, p)
#' @param w input weights (n, p)
#' @param rank rank of the approximation (non-negative)
#' @param max_iters - maximum number of EM iterations
#' @param atol - minimum absolute difference in objective function for convergence
#' @param verbose - print objective function updates
#' @return an n by p matrix
wlra = function(x, w, rank, max_iters=1000, atol=1e-3,verbose=FALSE){
  n = nrow(x)
  p = ncol(x)

  # Important: WLRA requires weights 0 <= w <= 1
  w = w/max(w)
  
    # Important: the procedure is deterministic, so initialization
  # matters.
  #
  # Srebro and Jaakkola suggest the best strategy is to initialize
  # from zero, but go from a full rank down to a rank k approximation in
  # the first iterations
  #
  # For now, take the simpler strategy of just initializing to zero. Srebro and
  # Jaakkola suggest this can underfit.
  z = matrix(0,nrow=n,ncol=p)
  obj = mean(w * x^2)
  if(verbose){
    print(paste0("wsvd [0]=",obj))
  }

  for(i in 0:max_iters){
    z1 = lra(w * x + (1 - w) * z, rank)
    update = mean(w * (x - z1)^2)
    if(verbose){print(paste0("wsvd [",i+1,"]=",update))}
    if(update > obj){
      stop("objective increased")
    } else if(max(abs(update-obj))<atol){
      return(z1)
    } else {
      z = z1
      obj = update
    }
  }
  stop("failed to converge")
}

# return log(p(Y|lambda=exp(eta))) for Y \sim Poi(lambda)
pois_llik= function(y, eta){
  sum(dpois(y,exp(eta),log=TRUE))
}

#' @details Assume x_ij ~ Poisson(exp(eta_ij)), eta_ij = L_ik F_kj
#'  Maximize the log likelihood by using Taylor approximation to rewrite the problem as WLRA.
#' @param x: input data (n, p)
#' @param rank: rank of the approximation
#' @param max_outer_iters: maximum number of calls to WLRA
#' @param max_iters: maximum number of EM iterations in WLRA
#' @param verbose: print objective function updates
#' @return low rank approximation (n, p) matrix
pois_lra= function(x, rank, max_outer_iters=50, max_iters=1000, atol=1e-3, verbose=FALSE){
  n = nrow(x)
  p = ncol(x)
  nmf = NNLM::nnmf(x, rank)
  eta = log(nmf$W %*% nmf$H)
  obj = mean(pois_llik(x, eta))
  if(verbose)
    print(paste0("pois_lra [0]:",obj))
  for(i in 0:max_outer_iters){
    lam = exp(eta)
    w = lam
    target = eta + x / lam - 1
    w[is.na(x)]=0 # Mark missing data with weight 0
    
      # Now we can go ahead and fill in the missing values with something
      # computationally convenient, because the WLRA EM update will ignore the
      # value for weight zero.
    target[is.na(x)] = 0
    eta1 = wlra(target, w, rank, max_iters=max_iters, atol=atol, verbose=verbose)
    update = mean(pois_llik(x, eta1))
    if(verbose){
      print(paste0("pois_lra [",i + 1,"]:",update))
    }
    if(max(abs(update-obj))<atol){
      return(list(fit=eta1,w=w,target=target))
    } else {
      eta = eta1
      obj = update
    }
  }
  stop("failed to converge")
}

# this just does a simple TSE about lam and runs wSVD once without any iteration
pois_lra1 = function(x, rank, lam = ifelse(x>0,x,0.5), max_iters=1000, atol=1e-3, verbose=FALSE){
  target = log(lam) + x / lam - 1
  w = lam
  w[is.na(x)]=0 # Mark missing data with weight 0
    
      # Now we can go ahead and fill in the missing values with something
      # computationally convenient, because the WLRA EM update will ignore the
      # value for weight zero.
    target[is.na(x)] = 0
    eta1 = wlra(target, w, rank, max_iters=max_iters, atol=atol, verbose=verbose)
    return(list(fit=eta1,w=w,target=target))
}

First simulate some data:

set.seed(1)
l = rnorm(100)
f = rnorm(100)
eta = outer(l,f)
lambda = exp(eta)
x = matrix(rpois(length(lambda),lambda),nrow=nrow(lambda))

Now fit various models: the plra, plra1 with the “naive” expansion (around x, or 0.5 for x=0) and around the true value of lambda:

x.plra=pois_lra(x,rank=1,verbose=TRUE)

[1] "pois_lra [0]:-56393.8383721883"
[1] "wsvd [0]=9.11187664142563"
[1] "wsvd [1]=9.09767330084035"
[1] "wsvd [2]=9.0942294282996"
[1] "wsvd [3]=9.09252076154906"
[1] "wsvd [4]=9.09146781601123"
[1] "wsvd [5]=9.09073663864168"
[1] "pois_lra [1]:-20730.5265718132"
[1] "wsvd [0]=0.0254032600526617"
[1] "wsvd [1]=0.018730302153258"
[1] "wsvd [2]=0.0173472213350827"
[1] "wsvd [3]=0.0166205649043371"
[1] "pois_lra [2]:-24476.2310334721"
[1] "wsvd [0]=0.0342958650072393"
[1] "wsvd [1]=0.0302492341697698"
[1] "wsvd [2]=0.0295946545829222"
[1] "pois_lra [3]:-26631.5708456328"
[1] "wsvd [0]=0.0594742391599983"
[1] "wsvd [1]=0.0558114666026009"
[1] "wsvd [2]=0.0551636349735505"
[1] "pois_lra [4]:-26775.5968189102"
[1] "wsvd [0]=0.0654476543467692"
[1] "wsvd [1]=0.0617080003798833"
[1] "wsvd [2]=0.0609634116407461"
[1] "pois_lra [5]:-26663.0720352895"
[1] "wsvd [0]=0.0639419959608682"
[1] "wsvd [1]=0.0601941802731442"
[1] "wsvd [2]=0.0594430040900086"
[1] "pois_lra [6]:-26653.5190878185"
[1] "wsvd [0]=0.0637737656022713"
[1] "wsvd [1]=0.0600262104698666"
[1] "wsvd [2]=0.0592754713652051"
[1] "pois_lra [7]:-26653.9093042342"
[1] "wsvd [0]=0.063778632660418"
[1] "wsvd [1]=0.0600311044827268"
[1] "wsvd [2]=0.0592803862918574"
[1] "pois_lra [8]:-26653.9404182592"
[1] "wsvd [0]=0.0637791754208205"
[1] "wsvd [1]=0.0600316465363189"
[1] "wsvd [2]=0.0592809270276872"
[1] "pois_lra [9]:-26653.9393089239"
[1] "wsvd [0]=0.0637791618254631"
[1] "wsvd [1]=0.0600316328646063"
[1] "wsvd [2]=0.0592809132945037"
[1] "pois_lra [10]:-26653.9392199259"

x.plra1.naive=pois_lra1(x,rank=1,verbose=TRUE)

[1] "wsvd [0]=0.0357370934894206"
[1] "wsvd [1]=0.0216973393395743"
[1] "wsvd [2]=0.0183559836838193"
[1] "wsvd [3]=0.0166671902679346"
[1] "wsvd [4]=0.0156058206162763"
[1] "wsvd [5]=0.0148556454379835"

x.plra1.true = pois_lra1(x,1,lam=lambda) 

pois_llik(x,eta) #log likelihood at true value of eta

[1] -13269.62

pois_llik(x,x.plra$fit)

[1] -26653.94

pois_llik(x,x.plra1.naive$fit)

[1] -20770.27

pois_llik(x,x.plra1.true$fit)

[1] -20830.33

This was weird that expansion around the truth was so bad. So I tried more stringent convergence:

x.plra1.true2 = pois_lra1(x,1,lam=lambda,max_iters = 10000,atol=1e-9,verbose=TRUE) 

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[1] "wsvd [77]=0.00755453841945491"
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[1] "wsvd [640]=0.0038371993915626"
[1] "wsvd [641]=0.00383717178815999"
[1] "wsvd [642]=0.00383714444664052"
[1] "wsvd [643]=0.00383711736441444"
[1] "wsvd [644]=0.00383709053891879"
[1] "wsvd [645]=0.00383706396761709"
[1] "wsvd [646]=0.00383703764799912"
[1] "wsvd [647]=0.00383701157758054"
[1] "wsvd [648]=0.00383698575390267"
[1] "wsvd [649]=0.00383696017453221"
[1] "wsvd [650]=0.00383693483706093"
[1] "wsvd [651]=0.00383690973910542"
[1] "wsvd [652]=0.0038368848783068"
[1] "wsvd [653]=0.00383686025233048"
[1] "wsvd [654]=0.0038368358588659"
[1] "wsvd [655]=0.00383681169562622"
[1] "wsvd [656]=0.00383678776034811"
[1] "wsvd [657]=0.00383676405079149"
[1] "wsvd [658]=0.00383674056473925"
[1] "wsvd [659]=0.00383671729999702"
[1] "wsvd [660]=0.00383669425439295"
[1] "wsvd [661]=0.00383667142577742"
[1] "wsvd [662]=0.00383664881202281"
[1] "wsvd [663]=0.0038366264110233"
[1] "wsvd [664]=0.00383660422069459"
[1] "wsvd [665]=0.00383658223897368"
[1] "wsvd [666]=0.00383656046381867"
[1] "wsvd [667]=0.00383653889320849"
[1] "wsvd [668]=0.0038365175251427"
[1] "wsvd [669]=0.00383649635764127"
[1] "wsvd [670]=0.00383647538874435"
[1] "wsvd [671]=0.00383645461651206"
[1] "wsvd [672]=0.00383643403902427"
[1] "wsvd [673]=0.0038364136543804"
[1] "wsvd [674]=0.0038363934606992"
[1] "wsvd [675]=0.00383637345611853"
[1] "wsvd [676]=0.00383635363879521"
[1] "wsvd [677]=0.00383633400690475"
[1] "wsvd [678]=0.00383631455864119"
[1] "wsvd [679]=0.00383629529221689"
[1] "wsvd [680]=0.00383627620586233"
[1] "wsvd [681]=0.00383625729782595"
[1] "wsvd [682]=0.0038362385663739"
[1] "wsvd [683]=0.0038362200097899"
[1] "wsvd [684]=0.00383620162637504"
[1] "wsvd [685]=0.0038361834144476"
[1] "wsvd [686]=0.00383616537234284"
[1] "wsvd [687]=0.00383614749841286"
[1] "wsvd [688]=0.00383612979102639"
[1] "wsvd [689]=0.00383611224856863"
[1] "wsvd [690]=0.00383609486944107"
[1] "wsvd [691]=0.00383607765206133"
[1] "wsvd [692]=0.00383606059486295"
[1] "wsvd [693]=0.00383604369629528"
[1] "wsvd [694]=0.00383602695482327"
[1] "wsvd [695]=0.00383601036892733"
[1] "wsvd [696]=0.00383599393710313"
[1] "wsvd [697]=0.00383597765786149"
[1] "wsvd [698]=0.00383596152972818"
[1] "wsvd [699]=0.00383594555124378"
[1] "wsvd [700]=0.00383592972096354"
[1] "wsvd [701]=0.00383591403745718"
[1] "wsvd [702]=0.00383589849930878"
[1] "wsvd [703]=0.00383588310511661"
[1] "wsvd [704]=0.003835867853493"
[1] "wsvd [705]=0.00383585274306415"
[1] "wsvd [706]=0.00383583777247004"
[1] "wsvd [707]=0.00383582294036425"
[1] "wsvd [708]=0.00383580824541382"
[1] "wsvd [709]=0.00383579368629912"
[1] "wsvd [710]=0.00383577926171371"
[1] "wsvd [711]=0.00383576497036418"
[1] "wsvd [712]=0.00383575081097007"
[1] "wsvd [713]=0.00383573678226365"
[1] "wsvd [714]=0.00383572288298988"
[1] "wsvd [715]=0.00383570911190619"
[1] "wsvd [716]=0.00383569546778241"
[1] "wsvd [717]=0.00383568194940065"
[1] "wsvd [718]=0.00383566855555509"
[1] "wsvd [719]=0.00383565528505195"
[1] "wsvd [720]=0.00383564213670932"
[1] "wsvd [721]=0.00383562910935702"
[1] "wsvd [722]=0.00383561620183653"
[1] "wsvd [723]=0.00383560341300081"
[1] "wsvd [724]=0.00383559074171422"
[1] "wsvd [725]=0.0038355781868524"
[1] "wsvd [726]=0.00383556574730214"
[1] "wsvd [727]=0.00383555342196125"
[1] "wsvd [728]=0.0038355412097385"
[1] "wsvd [729]=0.00383552910955344"
[1] "wsvd [730]=0.00383551712033633"
[1] "wsvd [731]=0.00383550524102802"
[1] "wsvd [732]=0.00383549347057985"
[1] "wsvd [733]=0.00383548180795352"
[1] "wsvd [734]=0.00383547025212101"
[1] "wsvd [735]=0.00383545880206443"
[1] "wsvd [736]=0.00383544745677597"
[1] "wsvd [737]=0.00383543621525776"
[1] "wsvd [738]=0.00383542507652179"
[1] "wsvd [739]=0.00383541403958978"
[1] "wsvd [740]=0.00383540310349312"
[1] "wsvd [741]=0.00383539226727272"
[1] "wsvd [742]=0.00383538152997897"
[1] "wsvd [743]=0.0038353708906716"
[1] "wsvd [744]=0.00383536034841958"
[1] "wsvd [745]=0.00383534990230106"
[1] "wsvd [746]=0.00383533955140327"
[1] "wsvd [747]=0.00383532929482239"
[1] "wsvd [748]=0.0038353191316635"
[1] "wsvd [749]=0.00383530906104048"
[1] "wsvd [750]=0.00383529908207588"
[1] "wsvd [751]=0.00383528919390089"
[1] "wsvd [752]=0.00383527939565524"
[1] "wsvd [753]=0.00383526968648706"
[1] "wsvd [754]=0.00383526006555287"
[1] "wsvd [755]=0.00383525053201742"
[1] "wsvd [756]=0.00383524108505368"
[1] "wsvd [757]=0.00383523172384271"
[1] "wsvd [758]=0.00383522244757356"
[1] "wsvd [759]=0.00383521325544326"
[1] "wsvd [760]=0.00383520414665666"
[1] "wsvd [761]=0.0038351951204264"
[1] "wsvd [762]=0.00383518617597282"
[1] "wsvd [763]=0.00383517731252387"
[1] "wsvd [764]=0.00383516852931505"
[1] "wsvd [765]=0.00383515982558931"
[1] "wsvd [766]=0.00383515120059699"
[1] "wsvd [767]=0.00383514265359577"
[1] "wsvd [768]=0.00383513418385054"
[1] "wsvd [769]=0.00383512579063336"
[1] "wsvd [770]=0.0038351174732234"
[1] "wsvd [771]=0.00383510923090683"
[1] "wsvd [772]=0.0038351010629768"
[1] "wsvd [773]=0.00383509296873331"
[1] "wsvd [774]=0.00383508494748319"
[1] "wsvd [775]=0.00383507699854"
[1] "wsvd [776]=0.003835069121224"
[1] "wsvd [777]=0.00383506131486201"
[1] "wsvd [778]=0.00383505357878744"
[1] "wsvd [779]=0.00383504591234015"
[1] "wsvd [780]=0.0038350383148664"
[1] "wsvd [781]=0.00383503078571882"
[1] "wsvd [782]=0.0038350233242563"
[1] "wsvd [783]=0.00383501592984395"
[1] "wsvd [784]=0.00383500860185305"
[1] "wsvd [785]=0.00383500133966096"
[1] "wsvd [786]=0.00383499414265108"
[1] "wsvd [787]=0.00383498701021277"
[1] "wsvd [788]=0.0038349799417413"
[1] "wsvd [789]=0.00383497293663781"
[1] "wsvd [790]=0.00383496599430923"
[1] "wsvd [791]=0.0038349591141682"
[1] "wsvd [792]=0.00383495229563306"
[1] "wsvd [793]=0.00383494553812778"
[1] "wsvd [794]=0.00383493884108187"
[1] "wsvd [795]=0.00383493220393037"
[1] "wsvd [796]=0.00383492562611375"
[1] "wsvd [797]=0.0038349191070779"
[1] "wsvd [798]=0.00383491264627406"
[1] "wsvd [799]=0.00383490624315874"
[1] "wsvd [800]=0.0038348998971937"
[1] "wsvd [801]=0.00383489360784589"
[1] "wsvd [802]=0.0038348873745874"
[1] "wsvd [803]=0.00383488119689538"
[1] "wsvd [804]=0.00383487507425203"
[1] "wsvd [805]=0.00383486900614455"
[1] "wsvd [806]=0.00383486299206503"
[1] "wsvd [807]=0.00383485703151049"
[1] "wsvd [808]=0.00383485112398276"
[1] "wsvd [809]=0.00383484526898847"
[1] "wsvd [810]=0.00383483946603896"
[1] "wsvd [811]=0.0038348337146503"
[1] "wsvd [812]=0.0038348280143432"
[1] "wsvd [813]=0.00383482236464295"
[1] "wsvd [814]=0.0038348167650794"
[1] "wsvd [815]=0.00383481121518693"
[1] "wsvd [816]=0.00383480571450436"
[1] "wsvd [817]=0.00383480026257495"
[1] "wsvd [818]=0.00383479485894632"
[1] "wsvd [819]=0.00383478950317042"
[1] "wsvd [820]=0.00383478419480351"
[1] "wsvd [821]=0.00383477893340609"
[1] "wsvd [822]=0.00383477371854285"
[1] "wsvd [823]=0.00383476854978267"
[1] "wsvd [824]=0.00383476342669853"
[1] "wsvd [825]=0.0038347583488675"
[1] "wsvd [826]=0.0038347533158707"
[1] "wsvd [827]=0.00383474832729324"
[1] "wsvd [828]=0.0038347433827242"
[1] "wsvd [829]=0.00383473848175658"
[1] "wsvd [830]=0.00383473362398727"
[1] "wsvd [831]=0.00383472880901698"
[1] "wsvd [832]=0.00383472403645028"
[1] "wsvd [833]=0.00383471930589545"
[1] "wsvd [834]=0.00383471461696456"
[1] "wsvd [835]=0.00383470996927333"
[1] "wsvd [836]=0.00383470536244117"
[1] "wsvd [837]=0.00383470079609111"
[1] "wsvd [838]=0.00383469626984975"
[1] "wsvd [839]=0.00383469178334728"
[1] "wsvd [840]=0.00383468733621736"
[1] "wsvd [841]=0.00383468292809718"
[1] "wsvd [842]=0.00383467855862736"
[1] "wsvd [843]=0.00383467422745193"
[1] "wsvd [844]=0.0038346699342183"
[1] "wsvd [845]=0.00383466567857725"
[1] "wsvd [846]=0.00383466146018287"
[1] "wsvd [847]=0.00383465727869251"
[1] "wsvd [848]=0.00383465313376681"
[1] "wsvd [849]=0.0038346490250696"
[1] "wsvd [850]=0.00383464495226791"
[1] "wsvd [851]=0.00383464091503192"
[1] "wsvd [852]=0.00383463691303495"
[1] "wsvd [853]=0.00383463294595341"
[1] "wsvd [854]=0.00383462901346676"
[1] "wsvd [855]=0.00383462511525753"
[1] "wsvd [856]=0.00383462125101121"
[1] "wsvd [857]=0.00383461742041631"
[1] "wsvd [858]=0.00383461362316426"
[1] "wsvd [859]=0.00383460985894942"
[1] "wsvd [860]=0.00383460612746904"
[1] "wsvd [861]=0.00383460242842323"
[1] "wsvd [862]=0.00383459876151492"
[1] "wsvd [863]=0.00383459512644987"
[1] "wsvd [864]=0.00383459152293662"
[1] "wsvd [865]=0.00383458795068643"
[1] "wsvd [866]=0.00383458440941333"
[1] "wsvd [867]=0.003834580898834"
[1] "wsvd [868]=0.00383457741866784"
[1] "wsvd [869]=0.00383457396863686"
[1] "wsvd [870]=0.00383457054846572"
[1] "wsvd [871]=0.00383456715788165"
[1] "wsvd [872]=0.00383456379661446"
[1] "wsvd [873]=0.00383456046439651"
[1] "wsvd [874]=0.00383455716096268"
[1] "wsvd [875]=0.00383455388605033"
[1] "wsvd [876]=0.00383455063939932"
[1] "wsvd [877]=0.00383454742075194"
[1] "wsvd [878]=0.00383454422985289"
[1] "wsvd [879]=0.00383454106644931"
[1] "wsvd [880]=0.00383453793029068"
[1] "wsvd [881]=0.00383453482112885"
[1] "wsvd [882]=0.003834531738718"
[1] "wsvd [883]=0.00383452868281462"
[1] "wsvd [884]=0.00383452565317749"
[1] "wsvd [885]=0.00383452264956765"
[1] "wsvd [886]=0.00383451967174838"
[1] "wsvd [887]=0.00383451671948518"
[1] "wsvd [888]=0.00383451379254576"
[1] "wsvd [889]=0.0038345108907"
[1] "wsvd [890]=0.00383450801371994"
[1] "wsvd [891]=0.00383450516137975"
[1] "wsvd [892]=0.00383450233345572"
[1] "wsvd [893]=0.00383449952972625"
[1] "wsvd [894]=0.00383449674997179"
[1] "wsvd [895]=0.00383449399397485"
[1] "wsvd [896]=0.00383449126152"
[1] "wsvd [897]=0.00383448855239379"
[1] "wsvd [898]=0.00383448586638479"
[1] "wsvd [899]=0.00383448320328355"
[1] "wsvd [900]=0.00383448056288256"
[1] "wsvd [901]=0.00383447794497626"
[1] "wsvd [902]=0.00383447534936102"
[1] "wsvd [903]=0.00383447277583508"
[1] "wsvd [904]=0.00383447022419859"
[1] "wsvd [905]=0.00383446769425358"
[1] "wsvd [906]=0.00383446518580389"
[1] "wsvd [907]=0.00383446269865521"
[1] "wsvd [908]=0.00383446023261504"
[1] "wsvd [909]=0.00383445778749268"
[1] "wsvd [910]=0.00383445536309921"
[1] "wsvd [911]=0.00383445295924746"
[1] "wsvd [912]=0.003834450575752"
[1] "wsvd [913]=0.00383444821242915"
[1] "wsvd [914]=0.00383444586909691"
[1] "wsvd [915]=0.00383444354557501"
[1] "wsvd [916]=0.00383444124168482"
[1] "wsvd [917]=0.00383443895724941"
[1] "wsvd [918]=0.00383443669209346"
[1] "wsvd [919]=0.0038344344460433"
[1] "wsvd [920]=0.00383443221892688"
[1] "wsvd [921]=0.00383443001057374"
[1] "wsvd [922]=0.003834427820815"
[1] "wsvd [923]=0.00383442564948335"
[1] "wsvd [924]=0.00383442349641305"
[1] "wsvd [925]=0.00383442136143988"
[1] "wsvd [926]=0.00383441924440115"
[1] "wsvd [927]=0.00383441714513568"
[1] "wsvd [928]=0.00383441506348379"
[1] "wsvd [929]=0.00383441299928728"
[1] "wsvd [930]=0.00383441095238941"
[1] "wsvd [931]=0.0038344089226349"
[1] "wsvd [932]=0.0038344069098699"
[1] "wsvd [933]=0.00383440491394201"
[1] "wsvd [934]=0.0038344029347002"
[1] "wsvd [935]=0.00383440097199488"
[1] "wsvd [936]=0.00383439902567782"
[1] "wsvd [937]=0.00383439709560217"
[1] "wsvd [938]=0.00383439518162244"
[1] "wsvd [939]=0.00383439328359449"
[1] "wsvd [940]=0.00383439140137549"
[1] "wsvd [941]=0.00383438953482397"
[1] "wsvd [942]=0.00383438768379973"
[1] "wsvd [943]=0.00383438584816388"
[1] "wsvd [944]=0.00383438402777883"
[1] "wsvd [945]=0.00383438222250823"
[1] "wsvd [946]=0.00383438043221702"
[1] "wsvd [947]=0.00383437865677135"
[1] "wsvd [948]=0.00383437689603865"
[1] "wsvd [949]=0.00383437514988753"
[1] "wsvd [950]=0.00383437341818785"
[1] "wsvd [951]=0.00383437170081063"
[1] "wsvd [952]=0.00383436999762813"
[1] "wsvd [953]=0.00383436830851373"
[1] "wsvd [954]=0.00383436663334203"
[1] "wsvd [955]=0.00383436497198874"
[1] "wsvd [956]=0.00383436332433075"
[1] "wsvd [957]=0.00383436169024607"
[1] "wsvd [958]=0.00383436006961383"
[1] "wsvd [959]=0.00383435846231427"
[1] "wsvd [960]=0.00383435686822873"
[1] "wsvd [961]=0.00383435528723967"
[1] "wsvd [962]=0.0038343537192306"
[1] "wsvd [963]=0.0038343521640861"
[1] "wsvd [964]=0.00383435062169183"
[1] "wsvd [965]=0.0038343490919345"
[1] "wsvd [966]=0.00383434757470184"
[1] "wsvd [967]=0.00383434606988263"
[1] "wsvd [968]=0.00383434457736666"
[1] "wsvd [969]=0.00383434309704475"
[1] "wsvd [970]=0.00383434162880869"
[1] "wsvd [971]=0.0038343401725513"
[1] "wsvd [972]=0.00383433872816636"
[1] "wsvd [973]=0.00383433729554862"
[1] "wsvd [974]=0.00383433587459382"
[1] "wsvd [975]=0.00383433446519863"
[1] "wsvd [976]=0.00383433306726069"
[1] "wsvd [977]=0.00383433168067854"
[1] "wsvd [978]=0.0038343303053517"
[1] "wsvd [979]=0.00383432894118057"
[1] "wsvd [980]=0.00383432758806648"
[1] "wsvd [981]=0.00383432624591166"
[1] "wsvd [982]=0.00383432491461924"
[1] "wsvd [983]=0.00383432359409322"
[1] "wsvd [984]=0.0038343222842385"
[1] "wsvd [985]=0.00383432098496082"
[1] "wsvd [986]=0.00383431969616682"
[1] "wsvd [987]=0.00383431841776396"
[1] "wsvd [988]=0.00383431714966057"
[1] "wsvd [989]=0.00383431589176579"
[1] "wsvd [990]=0.00383431464398962"
[1] "wsvd [991]=0.00383431340624286"
[1] "wsvd [992]=0.00383431217843713"
[1] "wsvd [993]=0.00383431096048486"
[1] "wsvd [994]=0.00383430975229929"
[1] "wsvd [995]=0.00383430855379441"
[1] "wsvd [996]=0.00383430736488505"
[1] "wsvd [997]=0.00383430618548678"
[1] "wsvd [998]=0.00383430501551595"
[1] "wsvd [999]=0.00383430385488966"
[1] "wsvd [1000]=0.0038343027035258"
[1] "wsvd [1001]=0.00383430156134296"
[1] "wsvd [1002]=0.0038343004282605"
[1] "wsvd [1003]=0.00383429930419853"
[1] "wsvd [1004]=0.00383429818907784"
[1] "wsvd [1005]=0.00383429708281998"
[1] "wsvd [1006]=0.00383429598534721"
[1] "wsvd [1007]=0.00383429489658247"
[1] "wsvd [1008]=0.00383429381644943"
[1] "wsvd [1009]=0.00383429274487244"
[1] "wsvd [1010]=0.00383429168177654"
[1] "wsvd [1011]=0.00383429062708745"
[1] "wsvd [1012]=0.00383428958073157"
[1] "wsvd [1013]=0.00383428854263597"
[1] "wsvd [1014]=0.00383428751272836"
[1] "wsvd [1015]=0.00383428649093713"
[1] "wsvd [1016]=0.00383428547719132"
[1] "wsvd [1017]=0.0038342844714206"
[1] "wsvd [1018]=0.00383428347355528"

x.plra1.naive2 = pois_lra1(x,1,max_iters = 10000,atol=1e-9,verbose=TRUE) 

[1] "wsvd [0]=0.0357370934894206"
[1] "wsvd [1]=0.0216973393395743"
[1] "wsvd [2]=0.0183559836838193"
[1] "wsvd [3]=0.0166671902679346"
[1] "wsvd [4]=0.0156058206162763"
[1] "wsvd [5]=0.0148556454379835"
[1] "wsvd [6]=0.0142861298681759"
[1] "wsvd [7]=0.013833147538475"
[1] "wsvd [8]=0.0134611235799792"
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[1] "wsvd [944]=0.00396211429205694"
[1] "wsvd [945]=0.00396211329776132"

pois_llik(x,x.plra1.naive2$fit)

[1] -13163.59

pois_llik(x,x.plra1.true2$fit)

[1] -13145.19

Session information

sessionInfo()

R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: OS X El Capitan 10.11.6

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

loaded via a namespace (and not attached):
 [1] workflowr_1.1.1   Rcpp_0.12.19      NNLM_0.4.2       
 [4] digest_0.6.17     rprojroot_1.3-2   R.methodsS3_1.7.1
 [7] backports_1.1.2   git2r_0.23.0      magrittr_1.5     
[10] evaluate_0.11     stringi_1.2.4     whisker_0.3-2    
[13] R.oo_1.22.0       R.utils_2.7.0     rmarkdown_1.10   
[16] tools_3.5.1       stringr_1.3.1     yaml_2.2.0       
[19] compiler_3.5.1    htmltools_0.3.6   knitr_1.20