biwhitening

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Last updated: 2023-03-28

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Introduction

Let \(Y = X + E\), the biwhitening method proposed by Landa et al 2021 transforms \(Y\) to

\(\tilde Y = D(u) Y D(v) = \tilde X + \tilde E\) such that

\[\mathbb{E}(\tilde E\tilde E^T/n) = I_m,\mathbb{E}(\tilde E^T \tilde E/m) = I_n.\]

That is, the average variance in each row and column of \(\tilde E\) is 1. When \(Y\) is Poisson distributed, this says \[\sum_i^m u_i^2 X_{ij} v_j^2 = m, \sum_j^n u_i^2X_{ij}v_j^2 = n.\]

To find \(u\) and \(v\), the algorithm 2 in the paper presents the Sinkhorn-Knopp algorithm.

Input: Nonnegative matrix \(A\), target row sums \(\boldsymbol r\), and target column sums \(\boldsymbol c\), tolerance \(\delta>0\).

Code below:

#'@param A nonnegative matrix
#'@param rs target row sums, a vector of length m
#'@param cs target col sums, a vector of length n
#'@param tol tol to stop the iterations
Sinkhorn_Knopp = function(A,rs=NULL,cs=NULL,tol=1e-5,maxiter=100){
  
  m = nrow(A)
  n = ncol(A)
  
  if(is.null(rs)){
    rs = rep(n,m)
  }
  if(is.null(cs)){
    cs = rep(m,n)
  }
  
  x = rep(1,m)
  y = rep(1,n)
  
  
  niter = 0
  while(niter<=maxiter) {
    Ax = c(crossprod(A,x))
    if(max(abs(y*Ax))<= tol){
      break
    }
    y = cs/Ax
    Ay = c(A%*%y)
    x = rs/Ay
    #if(max(abs(x*Ay-rs))<=tol & max(abs(y*c(crossprod(A,x))-cs))<= tol){
    #  break
    #}
    niter = niter + 1
  }
  
  return(list(x=x,y=y))
  
}

# example in Appendix C.2
set.seed(12345)
m = 50
n = 100
X = matrix(runif(m*n,1,2),nrow=m,ncol=n)
X = t(t(X*exp(runif(m,-2,2)))*exp(runif(n,-2,2)))
Y = matrix(rpois(m*n,X),nrow=m,ncol=n)

out = Sinkhorn_Knopp(Y,tol=1e-11)
d = diag(c((out$x)))%*%Y%*%diag(c((out$y)))
rowSums(d)

 [1] 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100
[20] 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100
[39] 100 100 100 100 100 100 100 100 100 100 100 100

colSums(d)

  [1] 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50
 [26] 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50
 [51] 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50
 [76] 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

The u, v in the Poisson case would be \(u = sqrt(x), v = sqrt(y)\). The transformed matrix is then

Y_tilde = diag(c(sqrt(out$x)))%*%Y%*%diag(c(sqrt(out$y)))
plot(rowMeans(Y),rowMeans(Y_tilde))

plot(colMeans(Y),colMeans(Y_tilde))

Maybe we can simulate data 100 times, and calculate the variance of each entry, plot the distribution and see how the transforamtion works(on variance and distribution).

set.seed(12345)
n_rep = 1000
Y_tilde = array(dim=c(m,n,n_rep))
Y = array(dim=c(m,n,n_rep))
for(i in 1:n_rep){
  Y[,,i] = matrix(rpois(m*n,X),nrow=m,ncol=n)
  out = Sinkhorn_Knopp(Y[,,i],tol=1e-11)
  Y_tilde[,,i] = diag(c(sqrt(out$x)))%*%Y[,,i]%*%diag(c(sqrt(out$y)))
}

Look at the variance before and after transformation

plot(X,apply(Y_tilde,c(1,2),var),xlab = 'Poisson variance', ylab = 'variance after biwhitening',pch='.',cex=1,col='grey60')

Past versions of unnamed-chunk-5-1.png

Version	Author	Date
35d9064	DongyueXie	2022-04-18

par(mfrow=c(1,2))
hist(apply(Y_tilde,c(1,2),var),breaks = 100,main='',xlab='variance after biwhitening')
boxplot(c(apply(Y_tilde,c(1,2),var)),main='',ylab='variance after biwhitening')

Past versions of unnamed-chunk-5-2.png

Version	Author	Date
35d9064	DongyueXie	2022-04-18

Non of true poisson lambda is 0. Variance after transformation seems to have mean around 1.

Look at some distribution.

par(mfrow=c(3,3))
for(i in 1:9){
  print(X[i,i])
  hist(Y_tilde[i,i,],breaks = 50,freq = FALSE,main=paste(i))
  x = seq(range(Y_tilde[i,i,])[1],range(Y_tilde[i,i,])[2],length.out = 100)
  curve(dnorm(x,mean=mean(Y_tilde[i,i,]),sd=sd(Y_tilde[i,i,])), add=TRUE)
}

[1] 2.328184

[1] 0.2150417

[1] 20.36863

[1] 0.0281371

[1] 8.214323

[1] 26.26458

[1] 0.07391752

[1] 4.298053

[1] 0.7949581

for(i in 1:9){
  r = 30
  print(X[r,i])
  hist(Y_tilde[r,i,],breaks = 20,freq = FALSE,main=paste(i))
  x = seq(range(Y_tilde[r,i,])[1],range(Y_tilde[r,i,])[2],length.out = 100)
  curve(dnorm(x,mean=mean(Y_tilde[r,i,]),sd=sd(Y_tilde[r,i,])), add=TRUE)
}

[1] 11.09687

[1] 0.7245544

[1] 11.38213

[1] 0.383629

[1] 8.670353

[1] 14.99803

[1] 0.6120885

[1] 2.394462

[1] 3.502384

for(i in 1:9){
  r = 40
  print(X[r,i])
  hist(Y_tilde[r,i,],breaks = 20,freq = FALSE,main=paste(i))
  x = seq(range(Y_tilde[r,i,])[1],range(Y_tilde[r,i,])[2],length.out = 100)
  curve(dnorm(x,mean=mean(Y_tilde[r,i,]),sd=sd(Y_tilde[r,i,])), add=TRUE)
}

[1] 3.905036

[1] 0.3078226

[1] 5.342475

[1] 0.1117732

[1] 3.450644

[1] 5.646072

[1] 0.4314669

[1] 0.7812951

[1] 1.102982

idx = rbind(c(20,55),c(7,10),c(8,4),c(44,100),c(2,6),c(10,6),c(8,5),c(39,71))
par(mfrow=c(4,2))
for(i in 1:8){
  ii = idx[i,]
  if(i%in%c(7,8)){
    hist(Y_tilde[ii[1],ii[2],],breaks = 50,freq = FALSE,main=paste('x =',round(X[ii[1],ii[2]],2)),xlab=expression(tilde(y)))
  }else{
    hist(Y_tilde[ii[1],ii[2],],breaks = 50,freq = FALSE,main=paste('x =',round(X[ii[1],ii[2]],2)),xlab='')
  }
  x = seq(range(Y_tilde[ii[1],ii[2],])[1],range(Y_tilde[ii[1],ii[2],])[2],length.out = 100)
  curve(dnorm(x,mean=mean(Y_tilde[ii[1],ii[2],]),sd=sd(Y_tilde[ii[1],ii[2],])), add=TRUE)
}

Session information

sessionInfo()

R version 4.1.0 (2021-05-18)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: CentOS Linux 7 (Core)

Matrix products: default
BLAS:   /software/R-4.1.0-no-openblas-el7-x86_64/lib64/R/lib/libRblas.so
LAPACK: /software/R-4.1.0-no-openblas-el7-x86_64/lib64/R/lib/libRlapack.so

locale:
 [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C         LC_TIME=C           
 [4] LC_COLLATE=C         LC_MONETARY=C        LC_MESSAGES=C       
 [7] LC_PAPER=C           LC_NAME=C            LC_ADDRESS=C        
[10] LC_TELEPHONE=C       LC_MEASUREMENT=C     LC_IDENTIFICATION=C 

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

other attached packages:
[1] workflowr_1.6.2

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.10      rstudioapi_0.13  whisker_0.4      knitr_1.33      
 [5] magrittr_2.0.3   R6_2.5.1         rlang_1.0.6      fastmap_1.1.0   
 [9] fansi_1.0.4      highr_0.9        stringr_1.5.0    tools_4.1.0     
[13] xfun_0.24        utf8_1.2.3       cli_3.6.0        git2r_0.28.0    
[17] jquerylib_0.1.4  htmltools_0.5.4  rprojroot_2.0.2  yaml_2.3.7      
[21] digest_0.6.31    tibble_3.1.8     lifecycle_1.0.3  later_1.3.0     
[25] sass_0.4.0       vctrs_0.5.2      promises_1.2.0.1 fs_1.5.0        
[29] glue_1.6.2       evaluate_0.14    rmarkdown_2.9    stringi_1.6.2   
[33] bslib_0.2.5.1    compiler_4.1.0   pillar_1.8.1     jsonlite_1.8.4  
[37] httpuv_1.6.1     pkgconfig_2.0.3