Last updated: 2021-06-10

Checks: 7 0

Knit directory: ebpmf_data_analysis/

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    Modified:   topicView-app/app.R

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Introduction

  • I implemented the NMF with L2 loss, with the cone assumption discussed here
  • The derivation is here. I derived it from the HALS idea. Note it only uses information of \(X^T X\)
  • I use simulated data as in here. Note the setting is for multinomial/poisson data. But I still fit with NMF-L2 and convert to multinomial model for comparison. So comparing with the truth does not make much sense.
  • It’s interesting that the fitted \(B,W^T\) are bascially the same! Thus we might make assumption that \(B = W^T\) and make more efficient algorithm. s
  • The algorithm of cone NMF converges on this simulated data.
  • The estimates are slightly worse than regular NMF in terms of objective function. Compared to truth and regular NMF fit, we can see a similar “soft-thresholding”-like behavior on loadings. It’s a bit too strong for the truth. (But note that regular NMF actually needs a bit of softthresholding on loadings).
  • The code can be optimized: it’s slower than nnmf; the initialization is very naive
rm(list = ls())
source("code/smallsim2_functions.R")
source("code/misc.R")
source("code/util.R")

Implementaion of cone NMF

## dim(X) = (p, n)
## init is list(B = B, W = W, R = R)
## K is the number of factors; can be null when init is provided
cone_nmf_l2 <- function(X, K, init = NULL, maxiter = 50, seed = 123){
  set.seed(seed)
  if(is.null(init)){ init <- init_cone_nmf_l2(X, K) }
  M = t(X) %*% X
  fit = cone_nmf_l2_util(M = M, B = init$B, W = init$W, R = init$R, maxiter = maxiter)
  fit[["A"]] = X %*% fit$B
  return(fit)
}


## dim(M) = (n, n); M = t(X) %*% X
## dim(B) = (n, K)
## dim(W) = (K, n); need each row has L2 norm 1
cone_nmf_l2_util <- function(M, B, W, R, maxiter = 50){
  K = ncol(B)
  loss = rep(NaN, maxiter)
  for(i in 1:maxiter){
    #browser()
    for(k in 1:K){
      Rk = R + B[,k] %o% W[k,]
      B[,k] <- threshold(Rk %*% W[k,], 1e-12)
      W[k,] <- threshold(t(Rk) %*% (M %*% B[,k]), 1e-12)
      W[k,] <- W[k,]/(sqrt(sum(W[k,]^2)))
      R <- Rk - B[,k] %o% W[k,]
    }
    loss[i] <- compute_loss_l2(M, R)
  }
  return(list(B = B, W = W, loss = loss))
}

compute_loss_l2 <- function(M, R){
  loss = sum(diag(t(R) %*% M %*% R))
  return(loss)
}

init_cone_nmf_l2 <- function(X, K){
  p = nrow(X)
  n = ncol(X)
  B = matrix(runif(n*K), nrow = n)
  W = matrix(runif(n*K), nrow = K)
  W <- W / sqrt(rowSums(W^2))
  R = diag(replicate(n, 1)) - B %*% W
  return(list(B = B, W = W, R = R))
}

threshold <- function(x, eps){
  x[x < eps] <- eps
  return(x)
}

simulation & fitting

  • I use the same simulation setting as in here. (except I increase the number of features 10 fold). Can see the loss decreases after each iteration and converges rather quickly.
  • The fits on both small and large data are reasonably well. They are slightly worse in MSE compared to regular NMF.
  • Note the “soft-thresholding”-like effect on \(L\) compared to regular NMF!
n <- 300
m <- 10000
k <- 3
n_sample = 60
doc_len = m ## average number of words in each document

S <- 15*diag(k) - 2
F <- simulate_factors(m = m, k = k)
L <- simulate_loadings(n,k,S)
s <- simulate_sizes(n = n, doc_len = doc_len)
X <- simulate_multinom_counts(L,F,s)

Fit with cone NMF and regular NMF below.

## fit with cone NMF
fit_small = cone_nmf_l2(X = t(X[1:n_sample, ]), K = k, maxiter = 50)
fit_big = cone_nmf_l2(X = t(X), K = k, maxiter = 50)
## Fit with regular NMF
fit_small2 <- NNLM::nnmf(A = X[1:n_sample, ], k = k, loss = "mse", method = "scd", trace = 1)
fit_big2 <- NNLM::nnmf(A = X, k = k, loss = "mse", method = "scd", trace = 1)

Compare progress

The red line is the minimum loss attained by regular NMF.

## compare progress
par(mfrow = c(2,2))
loss_ = fit_small$loss/(n_sample*m*2)
plot(loss_, ylim = c(min(fit_small2$target.loss), max(loss_)),
     ylab = "mse (cone NMF)", main = "cone NMF on small data")
abline(h = min(fit_small2$target.loss), col = "red")
plot(fit_small2$target.loss, ylab = "mse (NMF)", main = "NMF on small data")
## MSE on cone NMF vs NMF
print(c(min(loss_), min(fit_small2$target.loss)))
[1] 0.4850875 0.4764580
loss_ = fit_big$loss/(n*m*2)
plot(loss_, ylim = c(min(fit_big2$target.loss),max(loss_)),
     ylab = "mse (cone NMF)", main = "cone NMF on big data")
abline(h = min(fit_big2$target.loss), col = "red")
plot(fit_big2$target.loss, ylab = "mse (NMF)", main = "NMF on big data")

Version Author Date
14af863 zihao12 2021-06-09
eadae76 zihao12 2021-06-09 
## MSE on cone NMF vs NMF
print(c(min(loss_), min(fit_big2$target.loss)))
[1] 0.5033803 0.4950356
## transform to multinomial model
fit_small.m = get_multinom_from_pnmf(F = fit_small$A, L = t(fit_small$W))
fit_big.m = get_multinom_from_pnmf(F = fit_big$A, L = t(fit_big$W))
fit_small2.m = get_multinom_from_pnmf(F = t(fit_small2$H), L = fit_small2$W)
fit_big2.m = get_multinom_from_pnmf(F = t(fit_big2$H), L = fit_big2$W)

Cone NMF fit on small data VS truth

model = fit_small.m
idx = match_topics(F1 = F, F2 = model$F)
par(mfrow = c(3,2))
compare_truth_fitted(model, idx, L, F)

Version Author Date
14af863 zihao12 2021-06-09
eadae76 zihao12 2021-06-09

Cone NMF fit on small data VS truth

model = fit_small.m
idx = match_topics(F1 = F, F2 = model$F)
par(mfrow = c(3,2))
compare_truth_fitted(model, idx, L, F)

Version Author Date
14af863 zihao12 2021-06-09
eadae76 zihao12 2021-06-09

Regular NMF fit on small data VS truth

model = fit_small2.m
idx = match_topics(F1 = F, F2 = model$F)
par(mfrow = c(3,2))
compare_truth_fitted(model, idx, L, F)

Version Author Date
eadae76 zihao12 2021-06-09

Cone NMF fit on big data VS truth

model = fit_big.m
idx = match_topics(F1 = F, F2 = model$F)
par(mfrow = c(3,2))
compare_truth_fitted(model, idx, L, F)

Version Author Date
14af863 zihao12 2021-06-09
eadae76 zihao12 2021-06-09

Regular NMF fit on big data VS truth

model = fit_big2.m
idx = match_topics(F1 = F, F2 = model$F)
par(mfrow = c(3,2))
compare_truth_fitted(model, idx, L, F)

Version Author Date
eadae76 zihao12 2021-06-09

Cone NMF vs Regular NMF on small data

model1 = fit_small.m
model2 = fit_small2.m
idx1 = match_topics(F1 = F, F2 = model1$F)
idx2 = match_topics(F1 = F, F2 = model2$F)

par(mfrow = c(3,2))
for(i in 1:k){
  plot(model1$F[,idx1[i]], model2$F[,idx2[i]], 
       xlab = "fhat (cone NMF)", ylab = "fhat (NMF)", main = sprintf("factor %d", i))
  plot(model1$L[,idx1[i]], model2$L[,idx2[i]],
       xlab = "lhat (cone NMF)", ylab = "lhat (NMF)", main = sprintf("loading %d", i))
}

Version Author Date
14af863 zihao12 2021-06-09
eadae76 zihao12 2021-06-09

Cone NMF vs Regular NMF on big data

model1 = fit_big.m
model2 = fit_big2.m
idx1 = match_topics(F1 = F, F2 = model1$F)
idx2 = match_topics(F1 = F, F2 = model2$F)

par(mfrow = c(3,2))
for(i in 1:k){
  plot(model1$F[,idx1[i]], model2$F[,idx2[i]], 
       xlab = "fhat (cone NMF)", ylab = "fhat (NMF)", main = sprintf("factor %d", i))
  plot(model1$L[,idx1[i]], model2$L[,idx2[i]],
       xlab = "lhat (cone NMF)", ylab = "lhat (NMF)", main = sprintf("loading %d", i))
}

Version Author Date
14af863 zihao12 2021-06-09
eadae76 zihao12 2021-06-09

Relationship between B & W

In small data

par(mfrow = c(2,2))
model = fit_small
for(i in 1:k){
  plot(model$B[,i], model$W[i,], xlab = "b",ylab = "w",main = sprintf("topic %d", i))  
  abline(a = 0, b = 1, col= "red")
}

## max difference between B & W
max(abs(model$B- t(model$W)))
[1] 0.01307541

In big data

par(mfrow = c(2,2))
model = fit_big
for(i in 1:k){
  plot(model$B[,i], model$W[i,], xlab = "b",ylab = "w",main = sprintf("topic %d", i))  
  abline(a = 0, b = 1, col= "red")
}

## max difference between B & W
max(abs(model$B- t(model$W)))
[1] 0.005823909


sessionInfo()
R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS  10.15.7

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] pheatmap_1.0.12 mvtnorm_1.1-0   workflowr_1.6.2

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.5         knitr_1.28         whisker_0.3-2      magrittr_1.5      
 [5] munsell_0.5.0      colorspace_1.4-1   R6_2.4.1           rlang_0.4.5       
 [9] stringr_1.4.0      tools_3.5.1        grid_3.5.1         gtable_0.3.0      
[13] xfun_0.8           git2r_0.26.1       htmltools_0.5.0    yaml_2.2.0        
[17] digest_0.6.25      rprojroot_1.3-2    lifecycle_0.2.0    RColorBrewer_1.1-2
[21] later_1.1.0.1      promises_1.1.1     fs_1.3.1           NNLM_0.4.2        
[25] glue_1.4.1         evaluate_0.14      rmarkdown_2.1      stringi_1.6.2     
[29] compiler_3.5.1     scales_1.1.1       backports_1.1.7    httpuv_1.5.4