Last updated: 2019-10-18

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    Modified:   analysis/minque.Rmd
    Modified:   analysis/primepca.Rmd

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Rmd 7c973e9 Matthew Stephens 2019-10-18 workflowr::wflow_publish(“softimpute_convergence_problem.Rmd”)

Introduction

Here I investigate an example sent to me by Ziwei Zhu, where softimpute (lambda=0) seems to perform poorly, but primePCA does not. My results suggest that this seems to be an issue with the initialization of softimpute being less good than the initialization used by primePCA.

library(primePCA)
library(softImpute)
Loading required package: Matrix
Loaded softImpute 1.4

Here is the example:

l2norm <- function(x){ return(sqrt(sum(x^2)))}
set.seed(123)
n = 1000
p = 200
missprob = rep(.95,n)
u = rnorm(n)
v = rnorm(p)
X = 2*u %*% t(v) + rnorm(n*p)

for(i in 1:n){
  for(j in 1:p){
    if(runif(1)<missprob[i]){X[i,j]=NA}
  }
}
res.p = primePCA(X, 1,trace.it=FALSE,center=FALSE,thresh_sigma=100)
Convergence threshold is hit.
res.s = softImpute(X,1,maxit=1000)
plot(res.s$v,res.p$V_cur,main="v from primePCA vs softimpute",xlab="softimpute",ylab="primePCA")

We can see that softImpute gives three points that are outlying, which ruins its squared error performance.

Increase convergence tolerance stringency

Increasing stringency of tolerance seems to improve things, but convergence is clearly slow…

res.s = softImpute(X,1,maxit=1000,thresh=1e-8)
Warning in simpute.als(x, J, thresh, lambda, maxit, trace.it, warm.start, :
Convergence not achieved by 1000 iterations
plot(res.s$v,res.p$V_cur,main="v from primePCA vs softimpute",xlab="softimpute",ylab="primePCA")

Changing initialization

I suspect differences in initialization could be responsible. Let’s take a look.

First try softImpute from a different initialization: I use svd of the filled X matrix, filling NA with 0s:

Xfill = X
Xfill[is.na(X)]=0
Xfill.svd=svd(Xfill,1)

res.s.warm = softImpute(X,1,maxit =1000, warm.start = Xfill.svd)
plot(res.s.warm$v,res.p$V_cur,main="v from primePCA vs softimpute with svd-initialization",xlab="softimpute (svd init)",ylab="primePCA")

Now I try running primePCA using the (same random) initialization used by softImpute (Note: softImpute actually initializes u - essentially randomly – and not v; thus by running 1 iteration of softimpute we get effectively obtain its initial value for v.)

Note that to make sure I get the same random initialization as above I have to run the same code again…

set.seed(123)
n = 1000
p = 200
missprob = rep(.95,n)
u = rnorm(n)
v = rnorm(p)
X = 2*u %*% t(v) + rnorm(n*p)

for(i in 1:n){
  for(j in 1:p){
    if(runif(1)<missprob[i]){X[i,j]=NA}
  }
}
res.p = primePCA(X, 1,trace.it=FALSE,center=FALSE,thresh_sigma=100)
Convergence threshold is hit.
res.s.1 = softImpute(X,1,maxit=1)
Warning in simpute.als(x, J, thresh, lambda, maxit, trace.it, warm.start, :
Convergence not achieved by 1 iterations

We call this a “cold start”; from this cold start primePCA runs the full 1000 iterations and produces a result qualitatively similar to softImpute.

V_init = cbind(res.s.1$v)
res.p.cold = primePCA(X,1,trace.it=FALSE,center=FALSE,thresh_sigma=100,V_init = V_init) # runs the full 1000 iterations
Max iteration number is hit.
plot(res.p.cold$V_cur, res.p$V_cur, main="v from primePCA: cold start vs regular",xlab="primePCA (cold start)",ylab="primePCA (default)")


sessionInfo()
R version 3.6.0 (2019-04-26)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Mojave 10.14.4

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.6/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     

other attached packages:
[1] softImpute_1.4 Matrix_1.2-17  primePCA_1.0  

loaded via a namespace (and not attached):
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[13] fs_1.3.1        whisker_0.3-2   rmarkdown_1.14  tools_3.6.0    
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