Last updated: 2024-06-28

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Here we perform a small experiment with simulated data to illustrate the behaviour of the EM and SCD algorithms for fitting Poisson NMF (and topic models). The standard variational EM algorithm for LDA has similar struggles in this example.

Load the packages used in the analysis below, as well as some additional functions used to simulate the data and generate the results.

library(tm)
library(topicmodels)
library(fastTopics)
library(mvtnorm)
library(ggplot2)
library(cowplot)
source("../code/smallsim_functions.R")

Simulate a \(100 \times 400\) counts matrix from a multinomial topic model with \(K = 6\) topics.

set.seed(4)
n <- 100
m <- 400
k <- 6
F <- simulate_factors(m,k)
out <- simulate_loadings(n,k)
L <- out$L
major_topic <- out$major_topic
s <- simulate_sizes(n)
X <- simulate_multinom_counts(L,F,s)
cols <- which(colSums(X > 0) > 0)
F <- F[cols,]
X <- X[,cols]

We fit the multinomial topic model by performing 80 EM updates or 80 SCD updates. Both of the fits were first initialized by running 20 EM updates.

control <- list(extrapolate = FALSE,numiter = 4,nc = 2)
fit_init <- init_poisson_nmf(X,k = k)
fit0 <- fit_poisson_nmf(X,fit0=fit_init,numiter=20,method="em",control=control)
fit1 <- fit_poisson_nmf(X,fit0=fit0,numiter=80,method="em",control=control)
fit2 <- fit_poisson_nmf(X,fit0=fit0,numiter=80,method="scd",control=control)

EM and SCD produce quite different estimates, and among the two, the SCD estimates are clearly much closer to the ground truth.

topic_colors <- c("dodgerblue","darkorange","forestgreen","darkblue",
                  "gold","skyblue")
loadings_order <- order(major_topic,L[,1])
k_set <- c(1,3,5,2,4,6)
p1 <- simdata_structure_plot(L,loadings_order,topic_colors,title = "true")
p2 <- simdata_structure_plot(poisson2multinom(fit_init)$L,loadings_order,
                             topic_colors[k_set],title = "initial estimates")
p3 <- simdata_structure_plot(poisson2multinom(fit1)$L,loadings_order,
                             topic_colors[k_set],title = "EM")
p4 <- simdata_structure_plot(poisson2multinom(fit2)$L,loadings_order,
                             topic_colors[k_set],title = "SCD")
plot_grid(p1,p2,p3,p4,nrow = 4,ncol = 1)

Version Author Date
5a2f6a5 Peter Carbonetto 2024-06-26
8759fe2 Peter Carbonetto 2024-06-23

Indeed, the SCD estimates also improve upon the EM estimates in terms of log-likelihood, with a total improvement of about 600 log-likelihood units,

logliks <- c("initial" = sum(loglik_multinom_topic_model(X,fit_init)),
             "em"      = sum(loglik_multinom_topic_model(X,fit1)),
             "scd"     = sum(loglik_multinom_topic_model(X,fit2)))
make_colvec(logliks)
#              [,1]
# initial -38999.28
# em      -31491.97
# scd     -30874.44

or an average of about 6 log-likelihood units per document,

make_colvec(logliks/n)
#              [,1]
# initial -389.9928
# em      -314.9197
# scd     -308.7444

Next we will see that the variational EM (VEM) algorithm for LDA in the topicmodels package similarly has trouble making good progress on this data set, and greatly benefits from a good initialization provided by the SCD algorithm. (Note that the topicmodels package uses the original C implementation of the VEM algorithm.) To show this, we run the EM algorithm in which the parameters are initialized to the estimates obtained by running the EM or SCD updates:

lda0 <- run_lda(X,fit0,numiter = 100)
lda1 <- run_lda(X,fit1,numiter = 100)
lda2 <- run_lda(X,fit2,numiter = 100)

Initalizing to the SCD updates indeed results in a much better fit (as measured by the lower bound to the log-likelihood, i.e., the “ELBO”):

make_colvec(c("default init" = logLik(lda0),
              "100 EM init" = logLik(lda1),
              "20 EM + 80 SCD" = logLik(lda2)))
#                     [,1]
# default init   -356261.1
# 100 EM init    -356180.4
# 20 EM + 80 SCD -355368.5

Note that in this example we have fixed \(\alpha\), which determines the prior on the topic-word frequencies (the F matrix in fastTopics), to 1 rather than estimate it. It is possible to let \(\alpha\) adapt to the data, but doing so complicates the comparison.

The Structure plots show that the SCD initialization leads to estimates that look much more like the true values than the other initializations:

p1 <- simdata_structure_plot(L,loadings_order,topic_colors,title = "true")
p2 <- simdata_structure_plot(lda0@gamma,loadings_order,
                             topic_colors[k_set],
                             title = "Initialized with 20 EM updates")
p3 <- simdata_structure_plot(lda1@gamma,loadings_order,
                             topic_colors[k_set],
                             title = "Initialized with 100 EM updates")
p4 <- simdata_structure_plot(lda2@gamma,loadings_order,
                             topic_colors[k_set],
                             title = "Initialized with 20 EM + 80 SCD updates")
plot_grid(p1,p2,p3,p4,nrow = 4,ncol = 1)

Version Author Date
574eedb Peter Carbonetto 2024-06-27

Let’s now run the VEM algorithm for longer to see what happens.

lda0 <- run_lda(X,fit0,numiter = 800)
lda1 <- run_lda(X,fit1,numiter = 800)
lda2 <- run_lda(X,fit2,numiter = 800)

The VEM updates continue to approach the estimates obtained from the SCD initialization, but they remain quite far away even after 800 iterations.

These plots show the improvement in the objective (the ELBO) over time.

pdat <- rbind(data.frame(iter = 1:800,elbo = lda0@logLiks,init = "20 EM"),
              data.frame(iter = 1:800,elbo = lda1@logLiks,init = "100 EM"),
              data.frame(iter = 1:800,elbo = lda2@logLiks,init = "20 EM + 80 SCD"))
pdat <- transform(pdat,elbo = max(elbo) - elbo + 0.01)
ggplot(pdat,aes(x = iter,y = elbo,color = init)) +
  geom_line(linewidth = 0.75) +
  scale_x_continuous(breaks = seq(0,800,100)) +
  scale_y_continuous(trans = "log10",breaks = 10^seq(-1,4)) +
  scale_color_manual(values=c("darkblue","dodgerblue","darkorange")) +
  labs(x = "iteration",y = "ELBO difference") +
  theme_cowplot(font_size = 10)

Version Author Date
574eedb Peter Carbonetto 2024-06-27

Returning to the maximum-likelihood estimation problem, it is reassuring is that if we continue to perform the EM updates, they eventually arrive at the same solution as SCD. But SCD is able to “rescue” the EM estimates much more quickly after performing just a few SCD updates:

fit3 <- fit_poisson_nmf(X,fit0=fit1,numiter=700,method="em",control=control)
control$extrapolate <- TRUE
fit2 <- fit_poisson_nmf(X,fit0=fit2,numiter=700,method="scd",control=control)
fit4 <- fit_poisson_nmf(X,fit0=fit1,numiter=700,method="scd",control=control)
fit1 <- poisson2multinom(fit1)
fit2 <- poisson2multinom(fit2)
fit3 <- poisson2multinom(fit3)
fit4 <- poisson2multinom(fit4)
# loadings_scatterplot(F[,k_set],fit1$F,topic_colors,"true","em")
# loadings_scatterplot(F[,k_set],fit2$F,topic_colors,"true","scd")
pdat <- rbind(data.frame(iter   = 1:800,
                         ll     = fit2$progress$loglik.multinom,
                         method = "scd"),
              data.frame(iter   = 1:800,
                         ll     = fit3$progress$loglik.multinom,
                         method = "em"),
              data.frame(iter   = 1:800,
                         ll     = fit4$progress$loglik.multinom,
                         method = "em+scd"))
pdat <- transform(pdat,ll = max(ll) - ll + 0.1)
ggplot(pdat,aes(x = iter,y = ll,color = method)) +
  geom_line(linewidth = 0.75) +
  scale_x_continuous(breaks = seq(0,800,100)) +
  scale_y_continuous(trans = "log10",breaks = 10^seq(-1,4)) +
  scale_color_manual(values = c("dodgerblue","darkorange","magenta")) +
  labs(x = "iteration",y = "loglik difference") +
  theme_cowplot(font_size = 10)

Version Author Date
5a2f6a5 Peter Carbonetto 2024-06-26
8759fe2 Peter Carbonetto 2024-06-23 

sessionInfo()
# R version 4.3.3 (2024-02-29)
# Platform: aarch64-apple-darwin20 (64-bit)
# Running under: macOS Sonoma 14.5
# 
# Matrix products: default
# BLAS:   /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib 
# LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.11.0
# 
# locale:
# [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
# 
# time zone: America/Chicago
# tzcode source: internal
# 
# attached base packages:
# [1] stats     graphics  grDevices utils     datasets  methods   base     
# 
# other attached packages:
# [1] cowplot_1.1.3      ggplot2_3.5.0      mvtnorm_1.2-4      fastTopics_0.6-184
# [5] topicmodels_0.2-16 tm_0.7-13          NLP_0.2-1         
# 
# loaded via a namespace (and not attached):
#  [1] gtable_0.3.4        xfun_0.42           bslib_0.6.1        
#  [4] htmlwidgets_1.6.4   ggrepel_0.9.5       lattice_0.22-5     
#  [7] quadprog_1.5-8      vctrs_0.6.5         tools_4.3.3        
# [10] generics_0.1.3      stats4_4.3.3        parallel_4.3.3     
# [13] tibble_3.2.1        fansi_1.0.6         highr_0.10         
# [16] pkgconfig_2.0.3     Matrix_1.6-5        data.table_1.15.2  
# [19] SQUAREM_2021.1      RcppParallel_5.1.7  lifecycle_1.0.4    
# [22] truncnorm_1.0-9     farver_2.1.1        compiler_4.3.3     
# [25] stringr_1.5.1       git2r_0.33.0        progress_1.2.3     
# [28] munsell_0.5.0       RhpcBLASctl_0.23-42 httpuv_1.6.14      
# [31] htmltools_0.5.7     sass_0.4.8          lazyeval_0.2.2     
# [34] yaml_2.3.8          plotly_4.10.4       crayon_1.5.2       
# [37] tidyr_1.3.1         later_1.3.2         pillar_1.9.0       
# [40] jquerylib_0.1.4     whisker_0.4.1       uwot_0.1.16        
# [43] cachem_1.0.8        gtools_3.9.5        tidyselect_1.2.1   
# [46] digest_0.6.34       Rtsne_0.17          stringi_1.8.3      
# [49] slam_0.1-50         purrr_1.0.2         dplyr_1.1.4        
# [52] ashr_2.2-66         labeling_0.4.3      rprojroot_2.0.4    
# [55] fastmap_1.1.1       grid_4.3.3          colorspace_2.1-0   
# [58] cli_3.6.2           invgamma_1.1        magrittr_2.0.3     
# [61] utf8_1.2.4          withr_3.0.0         prettyunits_1.2.0  
# [64] scales_1.3.0        promises_1.2.1      httr_1.4.7         
# [67] rmarkdown_2.26      workflowr_1.7.1     hms_1.1.3          
# [70] modeltools_0.2-23   pbapply_1.7-2       evaluate_0.23      
# [73] knitr_1.45          viridisLite_0.4.2   irlba_2.3.5.1      
# [76] rlang_1.1.3         Rcpp_1.0.12         mixsqp_0.3-54      
# [79] glue_1.7.0          xml2_1.3.6          jsonlite_1.8.8     
# [82] R6_2.5.1            fs_1.6.3