Last updated: 2020-11-05

Checks: 7 0

Knit directory: ebpmf_data_analysis/

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    Modified:   topicView-app/app.R

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Rmd 1efe752 zihao12 2020-11-05 ebpmf_wbg_simulation_big2_2.Rmd

Introduction

  • What I did:
    • I did experiment with ebpmf-wbg using bigger simulated dataset (\(n = 1100, p = 2100, K = 50\)).
    • The signal from \(L, F\) is low compared to from \(l_0 L, f_0 F\). \(\text{Cor}(L_{k}, L_{k^{'}}) \approx - 0.009\) but \(\text{Cor}(l_0 L_{k}, f_0 L_{k^{'}}) \approx - 0.9\)
    • I compared pmf_bg and ebpmf_wbg on this dataset. (Note pmf_bg has the same objective as pmf, but that I separate \(L\) into \(l_0, L\) and update them separately (same for \(F\)). Or it is just like ebpmf_wbg without the adaptive shrinkage part)
  • What I found:
    • Because the signal from deviation \(L, F\) is too low, both methods have solutions that ignore the signal but achieves better objective value
    • Even when initialized from truth, pmf-bg learns messy structure.
    • When initialized sufficiently close to the truth, ebpmf-wbg recovers the structure well, and the estimate of g also makes sense.
rm(list = ls())
knitr::opts_chunk$set(message = FALSE, warning = FALSE, autodep = TRUE)
library(ggplot2)
library(gridExtra)
library(Matrix)
source("code/misc.R")
source("code/util.R")
data_dir = "output/sim/v0.4.5/exper2"
data_name = "sim_bg_block_n1100_p2100_K50"

Load data and models

## load data
X = read_sim_bag_of_words(sprintf("%s/docword.%s.txt", data_dir, data_name))
truth = readRDS(sprintf("%s/truth.%s.Rds", data_dir, data_name))
n = nrow(X); p = ncol(X); K = ncol(truth$L)

## load wbg models
wbg_from_truth = load_model_ebpmf(data_dir = data_dir, data_name = data_name, 
                                      method_name = "ebpmf_wbg_K50_maxiter5000_from_truth")
wbg_from_pmf_truth = load_model_ebpmf(data_dir = data_dir, data_name = data_name, 
                                  method_name="ebpmf_wbg_K50_maxiter500_pmf_bg_K50_maxiter10_from_truth_scaled0")
wbg_from_pmf_truth_scaled = load_model_ebpmf(data_dir = data_dir, data_name = data_name, 
                                  method_name="ebpmf_wbg_K50_maxiter500_pmf_bg_K50_maxiter10_from_truth_scaled0")

wbg_from_random = load_model_ebpmf(data_dir = data_dir, data_name = data_name, 
                                  method_name="ebpmf_wbg_K50_maxiter100_init_random")

## load pmf models
pmf_bg_from_truth = load_model_pmf(data_dir = data_dir, data_name = data_name, 
                                  method_name = "pmf_bg_K50_maxiter1000_from_truth")

## load pmf models
pmf_bg_from_truth_iter10 = load_model_pmf(data_dir = data_dir, data_name = data_name, 
                                  method_name = "pmf_bg_K50_maxiter10_from_truth")

What does the data look like

  • The data is generated in bigger_simulated_dataset.
  • Data model is \(X \sim \text{Pois}(\Lambda); \Lambda_{ij} = l_{i0} f_{j0} \sum_k l_{ik} f_{jk}\). We call
    • \(l_0, f_0\) background frequency for loading and factor
    • \(L, F\) deviation for loading and factor
    • \(\tilde{\Lambda} := L F^t\) deviation for the mean
  • \(n = 1100, p = 2100, K = 50\)
  • The last 100 words and documents are frequent words/docs
  • For each \(k = 1...K\), it has 20 top words and 10 top documents (100 times more deviation). I arrange them so that \(\tilde{\Lambda}\) has block structures: with \(K\) blocks from \(n = 1...1000, p = 1:2000\), and a block for frequent words/documents at the end.
  • The signal from frequent words is much stronger than those from top words/documents (in \(X\), the block for frequent word has value \(> 10\) times of that in block for top words/documents). Note that this seems to be intrinsic to this type of data: if the amplifying factor for background frequent words, and for top words/documents are the same, the signal from the background is \(K\) times stronger.

Truth (\(l_0, f_0, L, F\) and deviation matrix)

par(mfrow = c(2,2))
plot(truth$l0, log = "y", main = "l0 (truth)")
plot(truth$f0, log = "y", main = "f0 (truth)")

k = 12
plot(truth$L[,k], log = "y", main = sprintf("%dth loading", k))
plot(truth$F[,k], log = "y", main = sprintf("%dth factor", k))

Deviation matrix (block for top words and docs)

image(truth$L[1:50,] %*% t(truth$F[1:100,]), main = "deviation matrix (one block)")

Show some blocks in \(X\)

with topic words/documents, plus neghbors

X[1:15, 1:30]
15 x 30 sparse Matrix of class "dgCMatrix"
                                                                 
 [1,] 1 . . 1 . . . 3 . . 2 1 . . 1 1 . . 1 . . . . . . . . . . .
 [2,] . 1 1 . 2 1 2 . . 1 . . . 2 . 1 . 2 . 1 . . . . . . . . . .
 [3,] . 1 . . . . 1 . 1 1 1 . 3 . . . . . 3 . . . . . . . . . . .
 [4,] . 2 2 1 . 1 . . . 1 1 . . 2 1 . 3 . . 2 . . . . . . . . . .
 [5,] 2 1 1 . 1 . . 1 1 1 . . 1 . . . . 1 . . . . . . . . . . . .
 [6,] 2 . 1 . 2 1 . 1 . 1 1 . 1 . . 1 1 1 2 . . . . . . . . . . .
 [7,] 1 . . 1 . 1 . . . . . . . . . . 1 . . . . . . . . . . . . .
 [8,] 2 3 . . . 1 . 1 . . 1 . . . 1 . . . 1 . . . . . . . . . . .
 [9,] 1 . . 1 . . 1 . 1 1 . . 1 1 . 1 2 . . . . . . . . . . . . .
[10,] 1 . 1 . . 1 . 1 1 . . 2 . . . . 2 . . 1 . . . . . . . . . .
[11,] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
[12,] . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . .
[13,] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
[14,] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
[15,] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

with frequent words/documents

X[(n-10):n, (p-20):p]
11 x 21 sparse Matrix of class "dgCMatrix"
                                                                    
 [1,] 16 16 22 25 16 26 29 15 27 27 22 16 19 19 18 22 26 32 20 24 18
 [2,] 20 16 34 23 28 35 31 22 30 24 21 28 29 19 20 20 29 31 12 20 19
 [3,] 17 14 33 31 17 31 21 22 18 26 28 27 22 29 15 21 30 25 25 29 17
 [4,] 23 22 20 17 19 27 18 12 23 27 34 18 16 14  9 15 24 23 12 13 24
 [5,] 27 26 24 21 22 29 23 19 38 31 25 21 26 20 21 30 16 35 25 18 30
 [6,] 15 19 16 26 11 21 29 16 25 31 23 26 26 19 28 15 37 34 22 27 17
 [7,] 20 17 25 27 18 24 24 34 34 28 22 17 28 29 23 27 26 32 25 22 16
 [8,] 13 14 21 18 14 15  8 13 25 16 12 30 13  8 20 19 26 16 18 16 17
 [9,] 16 10 29 28 24 25 25 16 31 25 13 19 14 18 20 24 24 25 16 16 24
[10,] 20 15 36 30 29 23 26 26 34 25 25 29 16 25 16 19 30 32 25 24 21
[11,] 12 16 25 24 13 25 22 26 23 25 19 15 23 23 12 11 23 29 21 20 14

with frequent words and ordinary documents

X[(n-10):n, 1:20]
11 x 20 sparse Matrix of class "dgCMatrix"
                                             
 [1,] 2 1 . 1 2 . 2 . 2 . . . . . . 1 . 2 1 3
 [2,] 1 3 . . 1 . 1 1 . 1 2 . . 1 1 . 1 . 1 .
 [3,] 2 3 1 2 1 . . . 1 . . . . . . 1 1 . 1 1
 [4,] . 1 2 . 2 . 1 1 1 2 1 . . . . 1 2 1 1 1
 [5,] 3 1 1 2 . . . . . . 1 . . 1 . 2 1 . 1 .
 [6,] 1 1 . 1 2 . 3 . 3 . . . 1 . 1 . 1 . . 1
 [7,] . 1 1 1 1 1 1 3 . 1 . 1 . . . . 1 . . 2
 [8,] 1 . . . . . . . 1 . . 1 . 1 1 . . . . .
 [9,] . . 2 2 1 . 1 1 . . 1 . 1 . . . . 1 . 1
[10,] . . . 1 1 . 1 2 1 . . . . 1 2 1 1 . . 1
[11,] . . 2 . 3 2 . 2 . . . 1 4 1 . . . . . 1

with top words and ordinary documents

X[20:30, 1:20]
11 x 20 sparse Matrix of class "dgCMatrix"
                                             
 [1,] . . . . . . . . . . . . . . . . . . . .
 [2,] . . . . . . . . . . . . . . . . . . . .
 [3,] . . . . . . . . . . . . . . . . . . . .
 [4,] . . . . . . . . . . . . . . . . . . . .
 [5,] . . . . . . . . . . . . . . . . . . . .
 [6,] . . . . . . . . . . . . . . . . . . . .
 [7,] . 1 . . . . . . . . . . . . . . . . . .
 [8,] . . . . . . . . . . . . . . . . . . . .
 [9,] . . . . . . . . . . . 1 . . . . . . . .
[10,] . . . . . . . . . . . . . . . . . . . .
[11,] . . . . . . . . . . . . . . . . . . . .

with ordinary documents and ordinary words

X[50:60, 50:70]
11 x 21 sparse Matrix of class "dgCMatrix"
                                               
 [1,] . . . . . . . . . . . . . . . . . . . . .
 [2,] . . . . . . . . . . . . . . . . . . . . .
 [3,] . . . . . . . . . . . . . . . . . . . . .
 [4,] . . . . . . . . . . . . . . . . . . . . .
 [5,] . . . . . . . . . . . . . . . . . . . . .
 [6,] . . . . . . . . . . . . . . . . . . . . .
 [7,] . . . . . . . . . . . . . . . . . . . . .
 [8,] . . . . . . . . . . . . . . . . . . . . .
 [9,] . . . . . . . . . . . . . . . . . . . . .
[10,] . . . . . . . . . . . . . . . . . . . . .
[11,] . . . . . . . . . . . . . . . . . . . . .

pmf_bg from truth

It gets more messy results (why?)

par(mfrow = c(2,2))
k = 13
plot(pmf_bg_from_truth_iter10$L[,k], main = sprintf("10th iter: %dth loading", k), ylab = "loading")
plot(pmf_bg_from_truth$L[,k], main = sprintf("1000th iter: %dth loading", k), ylab = "loading")

plot(pmf_bg_from_truth_iter10$F[,k], main = sprintf("10th iter: %dth factor", k), ylab = "factor")
plot(pmf_bg_from_truth$F[,k], main = sprintf("1000th iter: %dth factor", k), ylab = "factor")

ebpmf_wbg from truth

Posterior mean for \(L, F\) are good

par(mfrow = c(2,2))
k = 13
plot(wbg_from_truth$qg$qls_mean[,k], log = "y", main = sprintf(" %dth loading", k), ylab = "loading")
plot(wbg_from_truth$qg$qfs_mean[,k], log = "y", main = sprintf(" %dth factor", k), ylab = "factor")

k = 29
plot(wbg_from_truth$qg$qls_mean[,k], log = "y", main = sprintf(" %dth loading", k), ylab = "loading")
plot(wbg_from_truth$qg$qfs_mean[,k], log = "y", main = sprintf(" %dth factor", k), ylab = "factor")

The prior g makes sense:
* g has weights on two components, one with small \(\phi\), the other large \(\phi\) * the weights of big \(\phi\) almost equal the proportion of top words/documents for \(L, F\).

## pi = 0.01 for phi = 100, and 0.99 for phi = 0.001 (truth: around 0.01 are top doc)
g = wbg_from_truth$qg$gls
Pi_L = get_prior_summary(g, log10 = TRUE, return_matrix = TRUE)

## pi around 0.01 for phi = 100, and 0.99 for phi = 0.001 (truth: around 0.01 are top words)
g = wbg_from_truth$qg$gfs
Pi_F = get_prior_summary(g, log10 = TRUE, return_matrix = TRUE)

ebpmf_wbg from close to truth

Above we see pmf_bg gets messy when initialized from the truth. I use that pmf_bg of 10 iterations as initialization for ebpmf-wbg (also tried 1000 iteration pmf_bg for initialization but not very good)

Posterior mean for \(L, F\): make a few mistakes, due to initialization

par(mfrow = c(2,2))
k = 19
plot(pmf_bg_from_truth_iter10$L[,k], main = sprintf("init: %dth loading", k), ylab = "loading")
plot(wbg_from_pmf_truth$qg$qls_mean[,k], main = sprintf("wbg: %dth loading", k), ylab = "loading")

plot(pmf_bg_from_truth_iter10$F[,k], main = sprintf("init iter: %dth factor", k), ylab = "factor")
plot(wbg_from_pmf_truth$qg$qfs_mean[,k], main = sprintf("wbg: %dth factor", k), ylab = "factor")

The prior g still mostly makes sense. The proportions are mostly good.
(Note topic 15 and 29 are down-weighted. Their g_F are slightly different than others)

g = wbg_from_pmf_truth$qg$gls
Pi_L = get_prior_summary(g, log10 = TRUE, return_matrix = TRUE)

g = wbg_from_pmf_truth$qg$gfs
Pi_F = get_prior_summary(g, log10 = TRUE, return_matrix = TRUE)

Whereebpmf-wbg can go wrong

  • When the initialization is not good enough, ebpmf-wbg can go wrong.

  • When initialized with \(l_0\), \(f_0\) from rank-1 model (scaled properly), and \(L, F\) uniform with mean 1, the model ignores the signal from \(L,F\) and g puts all mass on very small \(\phi\). As a result, the E-loglik is slightly worse, but the KL divergence is much smaller. This gives the model a higher ELBO than starting the truth. It converges after a couple of iterations.

  • When initialized from pmf which completely misses the structure, the final model also does not make sense, and has low ELBO (didn’t show).

compare_df = data.frame(cbind(as.numeric(wbg_from_random$summary), 
                              as.numeric(wbg_from_truth$summary),
                              as.numeric(wbg_from_pmf_truth$summary)), 
                        row.names = names(wbg_from_random$summary))
colnames(compare_df) <- c("from_random", "from_truth", "from_pmf")
round(compare_df)
             from_random from_truth from_pmf
ELBO             -364607    -370581  -371185
KL                     7      16476    17404
E_loglik         -364600    -354105  -353781
runtime_iter           9          7        8

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 Matrix_1.2-17   gridExtra_2.3   ggplot2_3.3.0  
[5] workflowr_1.6.2

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.5         RColorBrewer_1.1-2 compiler_3.5.1     pillar_1.4.4      
 [5] later_1.1.0.1      git2r_0.26.1       tools_3.5.1        digest_0.6.25     
 [9] lattice_0.20-38    evaluate_0.14      lifecycle_0.2.0    tibble_3.0.1      
[13] gtable_0.3.0       pkgconfig_2.0.3    rlang_0.4.6        yaml_2.2.0        
[17] xfun_0.8           withr_2.2.0        stringr_1.4.0      dplyr_0.8.1       
[21] knitr_1.28         fs_1.3.1           vctrs_0.3.0        rprojroot_1.3-2   
[25] grid_3.5.1         tidyselect_0.2.5   glue_1.4.1         R6_2.4.1          
[29] rmarkdown_2.1      purrr_0.3.4        magrittr_1.5       whisker_0.3-2     
[33] backports_1.1.7    scales_1.1.1       promises_1.1.1     htmltools_0.5.0   
[37] ellipsis_0.3.1     assertthat_0.2.1   colorspace_1.4-1   httpuv_1.5.4      
[41] stringi_1.4.3      munsell_0.5.0      crayon_1.3.4