Last updated: 2021-06-08

Checks: 7 0

Knit directory: ebpmf_data_analysis/

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    Modified:   topicView-app/app.R

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Rmd fa23ffb zihao12 2021-06-08 cone_pmf1.Rmd

Summary

  • Consider the Poisson Matrix Factorization problem: \(X \sim \text{Pois}(\Lambda); \Lambda = L F^T; L, F \geq 0\) with \(\text{dim}(X) = (n, p), n \ll p\). I use the notation \(\text{cone}(A) := \{\sum_k w_k \mathbf{a}_k: w_k \geq 0\} = \{A \mathbf{w}: \mathbf{w} \geq 0\}\) where \(\text{dim}(A) = (n, K)\)
  • In many such applications with \(n \ll p\), I think it’s probably reasonable to assume the factors lie in the convex cone of samples. In other words, each factor is a weighted (non-negative) average of rows of data. This is a generalization of the “anchor sample” assumption (symmetric to the more famous “anchor word” assumption).
  • From numerical experiments with both simulated and real data, I find:
    • first, this assumption probably holds true in many datasets;
    • second, even when we do MLE without using this assumption, the fitted \(F_{\text{mle}}\) can be projected to the \(\text{cone}(X^{t})\) without much loss of information.
    • third, the projection coefficients (\(\mathbf{w}_k\) in \(\text{Proj}_{\text{cone}(X^t)}(\mathbf{f}_k) = X^t \mathbf{w}_k\)) is like applying soft-thresholding to loading \(\mathbf{l}_k\). It’s very sparse and interpretable: for each factor (topic) we can say it’s weighted average of a few “important” documents.
  • Thus this assumption should be useful for better interpretability as well as develop more efficient and stable algorithm. I haven’t figured out a way to implement this, but parameterizing \(f_k\) by \(w_k\) alone is reducing \(p\) parameters to \(n\), so possibly making an easier optimization problem. Also we might impose further assumptions on \(w_k\)… (many possible things to do)
rm(list = ls())
library(NNLM)
library(nnls)
source("code/smallsim2_functions.R")
source("code/misc.R")
source("code/util.R")

set.seed(123)

Simulated Data

  • I find in many of my simulated data, we can easily estimate very high dimensional \(F\) with very few samples.
  • I think it’s because the true Factors lie in convex cone of samples; and the construction of \(L\) makes many “anchor documents”. Therefore, even very few samples contain most of useful information of factors.
  • Below the full data is \(\text{dim}(X) = (300, 1000)\). But in fact we can estimate \(F\) well with only 60 samples. I use functions here
n <- 300
m <- 1000
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)

I initialize with 50 EM runs on small data; then fit on small and large data (NNLM::nnmf supports initializing with only factors).

fit0 = nnmf(A = X[1:n_sample, ], k = k, loss = "mkl", method = "lee", max.iter = 50)

fit_small = nnmf(A = X[1:n_sample, ], k = k, loss = "mkl", method = "scd", init = list(H = fit0$H), max.iter = 200)
fit_big = nnmf(A = X, k = k, loss = "mkl", method = "scd", init = list(H = fit0$H), max.iter = 200)

fit_small.m = get_multinom_from_pnmf(F = t(fit_small$H), L = fit_small$W)
fit_big.m = get_multinom_from_pnmf(F = t(fit_big$H), L = fit_big$W)

idx_small = match_topics(F1 = F, F2 = fit_small.m$F)
idx_big = match_topics(F1 = F, F2 = fit_big.m$F)

Can see the small data recovers \(L, F\) well

model = fit_small.m
idx = idx_small
par(mfrow = c(3,2))
compare_truth_fitted(model, idx, L, F)

  • Below I computed \(\text{Proj}_{\text{cone}(X^t)}(\hat{\mathbf{f}}_k) = X^t \mathbf{w}_k\) by nnls function (solving \(\text{min} \ |\mathbf{f}_k - X^t \mathbf{w}_k|_2^2 \ s.t. \mathbf{w}_k \geq 0\)). It returns both the projected \(f_k\) and weights \(w_k\). Here \(\hat{\mathbf{f}}_k\) is the MLE fit using the small data.
  • We can see:
    • \(\mathbf{f}_k, \text{Proj}_{\text{cone}(X^t)}(\hat{\mathbf{f}}_k)\) are quite similar (at least on important words/genes/features)
    • \(\mathbf{w}_k\) looks like applying soft-thresholding to \(l_k\)!
par(mfrow = c(3, 2))
for(i in 1:k){
  f = t(fit_small$H)[,i]
  l = fit_small.m$L[,i]
  f_proj  = nnls(A = t(X[1:n_sample,]), b = f)
  plot(f, f_proj$fitted, ylab = "f_proj", cex.lab = 1)
  abline(a = 0, b = 1, col = "blue")
  plot(l, f_proj$x, cex = 1, cex.lab = 1,
       xlab = sprintf("loading %d", i), ylab = sprintf("sample weights for factor %d", i))
}

Do the same thing for fit on bigger data. Result is similar.

par(mfrow = c(3, 2))
for(i in 1:k){
  f = t(fit_big$H)[,i]
  l = fit_big.m$L[,i]
  f_proj  = nnls(A = t(X), b = f)
  plot(f, f_proj$fitted, ylab = "f_proj", cex.lab = 1)
  abline(a = 0, b = 1, col = "blue")
  plot(l, f_proj$x, cex = 1, cex.lab = 1,
       xlab = sprintf("loading %d", i), ylab = sprintf("sample weights for factor %d", i))
}

Do the true factors lie in the convex cone of rows of samll subset of \(X\) ? Basically yes

par(mfrow = c(2, 2))
for(i in 1:k){
  f = F[,i]
  f_proj  = nnls(A = t(X[1:n_sample,]), b = f)
  plot(f, f_proj$fitted, ylab = "f_proj", cex.lab = 1)
  abline(a = 0, b = 1, col = "blue")
}

Comment

  • we can see each factor (truth) is basically a weighted sum of a small subset of samples.
  • The small subset contains enough “useful” documents/cells to reconstruct factor. That’s why such a high dimensional problem can be estimated with so few samples.

Real Data

This is a document dataset I fitted before, with \((n, p) = (3430, 6906)\). (I did MLE fit using EM)

data = readRDS("output/kos_coneFactor.Rds")
fit_kos.m = get_multinom_from_pnmf(F = data$pmf$F, L = data$pmf$L)
  • Below I look at the projection operator like in the section above
  • The MLE fits basically lie in \(\text{cone}(X^t)\)
  • The weights vs loading plot is a bit messier than in simulation data, but we can still see similar picture: \(w_k\)’s capture similar information as loadings, and are sparse and interpretable.
par(mfrow = c(20, 2))
for(i in 1:20){
  f = data$pmf$F[,i]
  l = fit_kos.m$L[,i]
  f_proj  = data$f_proj[[i]]
  plot(f, f_proj$fitted, ylab = "f_proj", cex.lab = 1)
  abline(a = 0, b = 1, col = "blue")
  plot(l, f_proj$x, cex = 1, cex.lab = 1,
       xlab = sprintf("loading %d", i), ylab = sprintf("sample weights for factor %d", i))
}


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   nnls_1.4        NNLM_0.4.2     
[5] 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           glue_1.4.1        
[25] evaluate_0.14      rmarkdown_2.1      stringi_1.6.2      compiler_3.5.1    
[29] scales_1.1.1       backports_1.1.7    httpuv_1.5.4