Last updated: 2024-06-23

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Knit directory: fastTopics-experiments/analysis/

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Rmd 4e9dfbe Peter Carbonetto 2024-06-23 I’ve arrived at a good ‘smallsim’ example.
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Rmd 0bb4102 Peter Carbonetto 2024-06-22 Made some improvements to the smallsim_hard example.
Rmd 0b42379 Peter Carbonetto 2024-06-22 Working on new ‘smallsim_hard’ example.

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 basic variational EM algorithm for LDA also struggles with this data set.

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 (i) 180 EM updates (without extrapolation), or (ii) 180 SCD updates (with extrapolation). Both of the fits are initialized by running 20 EM updates.

control <- list(extrapolate = FALSE,numiter = 4)
fit0 <- fit_poisson_nmf(X,k,numiter=20,method="em",control=control)
fit1 <- fit_poisson_nmf(X,fit0=fit0,numiter=180,method="em",control=control)
fit2 <- fit_poisson_nmf(X,fit0=fit0,numiter=80,method="scd",control=control)
control$extrapolate <- TRUE
fit2 <- fit_poisson_nmf(X,fit0=fit2,numiter=100,method="scd",control=control)

EM and SCD produce quite different estimates, and among the two, the SCD estimates are much closer to the 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(fit1)$L,loadings_order,
                             topic_colors[k_set],title = "EM")
p3 <- simdata_structure_plot(poisson2multinom(fit2)$L,loadings_order,
                             topic_colors[k_set],title = "SCD")
plot_grid(p1,p2,p3,nrow = 3,ncol = 1)

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

c("em" = sum(loglik_multinom_topic_model(X,fit1)),
  "scd" = sum(loglik_multinom_topic_model(X,fit2)))
#        em       scd 
# -31465.61 -30874.44

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

c("em" = sum(loglik_multinom_topic_model(X,fit1)),
  "scd" = sum(loglik_multinom_topic_model(X,fit2)))/n
#        em       scd 
# -314.6561 -308.7444

What is reassuring is that if we continue to perform the EM updates, we 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.

control$extrapolate <- FALSE
fit3 <- fit_poisson_nmf(X,fit0=fit1,numiter=600,method="em",control=control)
control$extrapolate <- TRUE
fit2 <- fit_poisson_nmf(X,fit0=fit2,numiter=600,method="scd",control=control)
fit4 <- fit_poisson_nmf(X,fit0=fit1,numiter=600,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)


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_0.6-175
# [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