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Introduction

Consider a matrix factorization problem \[Y = LF+E,\]

where \(Y\) is a \(N\times p\) matrix, \(L\) is a \(N\times K\) matrix, \(F\) is a \(K\times p\) matrix and \(E\) is a \(N\times p\) matrix of residuals.

We assume each row of \(F\) is smooth or spatially-structured. Then each row of \(Y\) is from a smooth function/curve with added noises. Matrix \(L\) is assumed to be sparse.

The question is how to estimate \(L\) and \(F\).

This is very similar to functional principal component analysis, which considers finding weights and principal components of a collection of curves. A common approach is adding a roughness penalty of the weights to obtain smooth estimates.

Here, we consider using a specific basis to represent the smooth curves - wavelet. Let \(W\) be the discrete wavelet transformation(DWT) matrix. We perform wavelet decomnposition on both sides then \[YW=LFW+EW,\] or \[\tilde{Y} = L\tilde{F}+\tilde{E}.\]

Now each row of \(\tilde{F}\) is sparse. We can then apply any penalized matrix factorization algorithm to \(\tilde{Y}\) to obtain sparse estimates of \(L\) and \(\tilde{F}\). Applying inverse DWT gives \(\hat{F}\).

Simulation

In this simulation study, we choose EBMF framework and compare it with the wavelet approach.

library(wavethresh)
library(flashr)
# wavelet-based matrix factorization
#'@ y: observed matrix
#'@ k: number of factors
#'@ filter.number, family: wavelet type

WaveEBMF = function(y,k,filter.number = 1,family = 'DaubExPhase'){
  N=nrow(y)
  p=ncol(y)
  W = GenW(n=p,filter.number = filter.number,family = family)
  y_tilde = y%*%W
  f2 = flash(y_tilde,Kmax=k,var_type = 'constant',backfit = T,verbose=F)
  f2_fitted = flash_get_ldf(f2)
  f_hat = (W%*%f2_fitted$f)
  return(list(f=f_hat,l=f2_fitted$l))
}

A single factor example

Simulate \(N=200\) and \(p=256\) under single-factor model \[l_i\sim \pi_0\delta_0+(1-\pi_0)\sum_{m=1}^5\frac{1}{5}N(0,\sigma^2_m)\]

Step function factor

\(f\) is a step function.

rmse = function(x1,x2){sqrt(mean((x1-x2)^2))}
set.seed(12345)
N = 200
p = 256
pi0 = 0.3
f = c(rep(2,p/4), rep(5, p/4), rep(6, p/4), rep(2, p/4))
l = c()
for (i in 1:N) {
  r = rbinom(1,1,pi0)
  if(r==1){
    l[i]=0
  }else{
      l[i] = mean(c(rnorm(1,0,sqrt(0.25)),rnorm(1,0,sqrt(0.5)),rnorm(1,0,sqrt(1)),
                    rnorm(1,0,sqrt(2)),rnorm(1,0,sqrt(4))))
      }
}
y = l%*%t(f)+matrix(rnorm(N*p,0,1),ncol=p)

# apply flash directly
f1 = flash(y,Kmax=1,var_type = 'constant',backfit = T,verbose=F)
f1 = flash_get_ldf(f1)

# apply wavelet transform

# use Haar wavelet

f2 = WaveEBMF(y,k=1)

paste('RMSE of l flash estimate:',round(rmse(f1$l,l/norm(l,'2')),5))
[1] "RMSE of l flash estimate: 0.00208"
paste('RMSE of l Waveflash estimate:',round(rmse(f2$l,l/norm(l,'2')),5))
[1] "RMSE of l Waveflash estimate: 0.00209"
plot(f2$l,l/norm(l,'2'),xlab='Estimate',ylab='True l')
lines(f2$l,f2$l)

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plot(f1$f,col = 2,type='l',xlab='',ylab='',main='Estimate of factors')
lines(f/norm(f,'2'),col='grey80',type='p')
lines(f2$f,col=4)
legend('topright',c('true f','flash','Waveflash'),col=c('grey80',2,4),lty=c(1,1,1))

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666296b Dongyue Xie 2019-07-23

HeavySine function factor

f=DJ.EX(p,signal = 2)$heavi
y = l%*%t(f)+matrix(rnorm(N*p,0,1),ncol=p)

# apply flash directly
f1 = flash(y,Kmax=1,var_type = 'constant',backfit = F,verbose=F)
f1 = flash_get_ldf(f1)

# apply wavelet transform

# use symmlet10

f2 = WaveEBMF(y,k=1,filter.number = 10,family = 'DaubLeAsymm')

paste('RMSE of l flash estimate:',round(rmse(f1$l,l/norm(l,'2')),5))
[1] "RMSE of l flash estimate: 0.00428"
paste('RMSE of l Waveflash estimate:',round(rmse(f2$l,l/norm(l,'2')),5))
[1] "RMSE of l Waveflash estimate: 0.00421"
plot(f2$l,l/norm(l,'2'),xlab='Estimate',ylab='True l')
lines(f2$l,f2$l)

Version Author Date
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plot(f1$f,col = 2,type='l',xlab='',ylab='',main='Estimate of factors')
lines(f/norm(f,'2'),col='grey80',type='p')
lines(f2$f,col=4)
legend('topright',c('true f','flash','Waveflash'),col=c('grey80',2,4),lty=c(1,1,1))

Version Author Date
666296b Dongyue Xie 2019-07-23

Spike function factor

spike.f = function(x) (0.75 * exp(-500 * (x - 0.23)^2) + 1.5 * exp(-2000 * (x - 0.33)^2) + 3 * exp(-8000 * (x - 0.47)^2) + 2.25 * exp(-16000 * 
    (x - 0.69)^2) + 0.5 * exp(-32000 * (x - 0.83)^2))

t = 1:p/p
f = 2*spike.f(t)

y = l%*%t(f)+matrix(rnorm(N*p,0,1),ncol=p)

# apply flash directly
f1 = flash(y,Kmax=1,var_type = 'constant',backfit = F,verbose=F)
f1 = flash_get_ldf(f1)

# apply wavelet transform

# use symmlet10

f2 = WaveEBMF(y,k=1,filter.number = 10,family = 'DaubLeAsymm')

paste('RMSE of l flash estimate:',round(rmse(f1$l,l/norm(l,'2')),5))
[1] "RMSE of l flash estimate: 0.00897"
paste('RMSE of l Waveflash estimate:',round(rmse(f2$l,l/norm(l,'2')),5))
[1] "RMSE of l Waveflash estimate: 0.00905"
plot(f2$l,l/norm(l,'2'),xlab='Estimate',ylab='True l')
lines(f2$l,f2$l)

Version Author Date
666296b Dongyue Xie 2019-07-23 
plot(f1$f,col = 2,type='l',xlab='',ylab='',main='Estimate of factors')
lines(f/norm(f,'2'),col='grey80',type='p')
lines(f2$f,col=4)
legend('topright',c('true f','flash','Waveflash'),col=c('grey80',2,4),lty=c(1,1,1))

Version Author Date
666296b Dongyue Xie 2019-07-23

Three factors example

Simulate \(N=200\) and \(p=256\) under the factor model \[l_i\sim \pi_0\delta_0+(1-\pi_0)\sum_{m=1}^5\frac{1}{5}N(0,\sigma^2_m)\]

We set \(K=3\) and three factors are step function, Heavysine and spike functions.

K=3
set.seed(12345)
L = matrix(nrow=N,ncol=K)
for (i in 1:N) {
  for(j in 1:K){
    
    r = rbinom(1,1,pi0)
    if(r==1){
       L[i,j]=0
    }else{
      L[i,j] = mean(c(rnorm(1,0,sqrt(0.25)),rnorm(1,0,sqrt(0.5)),rnorm(1,0,sqrt(1)),
                    rnorm(1,0,sqrt(2)),rnorm(1,0,sqrt(4))))
    }
  }
}

f_1 = c(rep(2,p/4), rep(5, p/4), rep(6, p/4), rep(2, p/4))
f_2 = DJ.EX(p,signal = 2)$heavi
f_3 = 2*spike.f(t)
FF = rbind(f_1,f_2,f_3)
E = matrix(rnorm(N*p,0,1),ncol=p)
Y = L%*%FF + E

# apply flash directly
f1 = flash(Y,Kmax=3,var_type = 'constant',backfit = F,verbose=F)
f1 = flash_get_ldf(f1)

# apply wavelet transform

# use symmlet10

f2 = WaveEBMF(Y,k=3,filter.number = 10,family = 'DaubLeAsymm')


plot(-f1$f[,1],col = 2,type='l',xlab='',ylab='',main='Estimate of factor 1')
lines(f_1/norm(f_1,'2'),col='grey80',type='p')
lines(-f2$f[,1],col=4)
legend('topright',c('true f','flash','Waveflash'),col=c('grey80',2,4),lty=c(1,1,1))

Version Author Date
666296b Dongyue Xie 2019-07-23 
plot(-f1$f[,2],col = 2,type='l',xlab='',ylab='',main='Estimate of factor 1')
lines(f_2/norm(f_2,'2'),col='grey80',type='p')
lines(-f2$f[,2],col=4)
legend('topright',c('true f','flash','Waveflash'),col=c('grey80',2,4),lty=c(1,1,1))

Version Author Date
666296b Dongyue Xie 2019-07-23 
plot(f1$f[,3],col = 2,type='l',xlab='',ylab='',main='Estimate of factor 1')
lines(f_3/norm(f_3,'2'),col='grey80',type='p')
lines(f2$f[,3],col=4)
legend('topright',c('true f','flash','Waveflash'),col=c('grey80',2,4),lty=c(1,1,1))

Version Author Date
666296b Dongyue Xie 2019-07-23

How about use Haar wavelet?

f2 = WaveEBMF(Y,k=3,filter.number = 1,family = 'DaubExPhase')


plot(-f1$f[,1],col = 2,type='l',xlab='',ylab='',main='Estimate of factor 1')
lines(f_1/norm(f_1,'2'),col='grey80',type='p')
lines(-f2$f[,1],col=4)
legend('topright',c('true f','flash','Waveflash'),col=c('grey80',2,4),lty=c(1,1,1))

Version Author Date
666296b Dongyue Xie 2019-07-23 
plot(-f1$f[,2],col = 2,type='l',xlab='',ylab='',main='Estimate of factor 1')
lines(f_2/norm(f_2,'2'),col='grey80',type='p')
lines(-f2$f[,2],col=4)
legend('topright',c('true f','flash','Waveflash'),col=c('grey80',2,4),lty=c(1,1,1))

Version Author Date
666296b Dongyue Xie 2019-07-23 
plot(f1$f[,3],col = 2,type='l',xlab='',ylab='',main='Estimate of factor 1')
lines(f_3/norm(f_3,'2'),col='grey80',type='p')
lines(f2$f[,3],col=4)
legend('topright',c('true f','flash','Waveflash'),col=c('grey80',2,4),lty=c(1,1,1))

Version Author Date
666296b Dongyue Xie 2019-07-23

How about change the order of three factors?

FF = rbind(f_3,f_2,f_1)
Y = L%*%FF + E

# apply flash directly
f1 = flash(Y,Kmax=3,var_type = 'constant',backfit = F,verbose=F)
f1 = flash_get_ldf(f1)
f2 = WaveEBMF(Y,k=3,filter.number = 1,family = 'DaubExPhase')


plot(f1$f[,1],col = 2,type='l',xlab='',ylab='',main='Estimate of factor 1')
lines(f_1/norm(f_1,'2'),col='grey80',type='p')
lines(f2$f[,1],col=4)
legend('topright',c('true f','flash','Waveflash'),col=c('grey80',2,4),lty=c(1,1,1))

Version Author Date
666296b Dongyue Xie 2019-07-23 
plot(-f1$f[,2],col = 2,type='l',xlab='',ylab='',main='Estimate of factor 1')
lines(f_2/norm(f_2,'2'),col='grey80',type='p')
lines(-f2$f[,2],col=4)
legend('topright',c('true f','flash','Waveflash'),col=c('grey80',2,4),lty=c(1,1,1))

Version Author Date
666296b Dongyue Xie 2019-07-23 
plot(-f1$f[,3],col = 2,type='l',xlab='',ylab='',main='Estimate of factor 1')
lines(f_3/norm(f_3,'2'),col='grey80',type='p')
lines(-f2$f[,3],col=4)
legend('topright',c('true f','flash','Waveflash'),col=c('grey80',2,4),lty=c(1,1,1))

Version Author Date
666296b Dongyue Xie 2019-07-23

Try PMD

PMD

Summary

  1. The wavelet method gives smoother estimate of factors. The smoothness depends on the wavelet basis chosen.
  2. The signs of loadings and factors are not identifiable.
  3. Smash uses non-decimated DWT to give better estimate. If using NDWT in flash, there would be a lot more computational cost.

sessionInfo()
R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.6

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     

other attached packages:
[1] flashr_0.6-6     wavethresh_4.6.8 MASS_7.3-51.1   

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.1        plyr_1.8.4        bindr_0.1.1      
 [4] pillar_1.4.2      compiler_3.5.1    git2r_0.23.0     
 [7] workflowr_1.4.0   iterators_1.0.10  tools_3.5.1      
[10] digest_0.6.20     evaluate_0.11     tibble_2.1.3     
[13] gtable_0.3.0      lattice_0.20-35   pkgconfig_2.0.2  
[16] rlang_0.4.0       Matrix_1.2-14     foreach_1.4.4    
[19] yaml_2.2.0        parallel_3.5.1    ebnm_0.1-24      
[22] bindrcpp_0.2.2    dplyr_0.7.8       stringr_1.3.1    
[25] knitr_1.20        fs_1.2.6          tidyselect_0.2.5 
[28] rprojroot_1.3-2   grid_3.5.1        glue_1.3.1       
[31] R6_2.2.2          rmarkdown_1.10    mixsqp_0.1-97    
[34] reshape2_1.4.3    purrr_0.2.5       ggplot2_3.2.0    
[37] ashr_2.2-38       magrittr_1.5      whisker_0.3-2    
[40] backports_1.1.4   scales_1.0.0      codetools_0.2-15 
[43] htmltools_0.3.6   assertthat_0.2.1  softImpute_1.4   
[46] colorspace_1.3-2  stringi_1.2.4     lazyeval_0.2.2   
[49] doParallel_1.0.14 pscl_1.5.2        munsell_0.5.0    
[52] truncnorm_1.0-8   SQUAREM_2017.10-1 crayon_1.3.4