A small simulation illustrating the trouble with EM
Peter Carbonetto
Last updated: 2021-03-07
Checks: 7 0
Knit directory: fastTopics-experiments/analysis/
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Add text here.
Load the packages used in the analysis below, as well as additional functions that we will use to simulate the data.
library(fastTopics)
library(mvtnorm)
library(ggplot2)
library(cowplot)
source("../code/smallsim_functions.R")Set the seed so that the results can be reproduced.
set.seed(1)Independent topics scenario
In this first example, we simulate a \(100 \times 400\) counts matrix from a multinomial topic model with \(K = 6\) topics.
n <- 100
m <- 400
k <- 6
S <- 13*diag(k) - 2
F <- simulate_factors(m,k)
L <- simulate_loadings(n,k,S)
s <- simulate_sizes(n)
X <- simulate_multinom_counts(L,F,s)
X <- X[,colSums(X > 0) > 0]The topic proportions for each of the 100 samples—that is, a row of the counts matrix—are randomly drawn according to the correlated topic model: \(\eta_i\) for row \(i\) is drawn from the multivariate normal with mean zero and covariance matrix \(S\), such that \(s_{kk} = 11\), \(s_{jk} = -2\) for all \(j \neq k\). Generated in this way, the topic proportions tend to be roughly orthogonal:
y <- apply(L,1,which.max)
y <- rank(y,ties.method = "random")
y <- qqnorm(y,plot.it = FALSE)$x
fit <- list(L = L)
class(fit) <- c("multinom_topic_model_fit","list")
structure_plot(fit,topics = 1:6,perplexity = 30,Y_init = matrix(y),
verbose = FALSE)
| Version | Author | Date |
|---|---|---|
| 913316c | Peter Carbonetto | 2021-03-07 |
Here we compare two different updates for fitting a Poisson NMF model to the simulated counts: EM updates, and sequential coordinate descent (SCD). The model fitting is initialized by first running 50 EM updates, with the aim of better ensuring that the same local maximum is recovered by both runs.
fit0 <- fit_poisson_nmf(X,k,numiter = 50,method = "em",
control = list(extrapolate = FALSE,numiter = 4))
fit1 <- fit_poisson_nmf(X,fit0 = fit0,numiter = 450,method = "em",
control = list(extrapolate = FALSE,numiter = 4))
fit2 <- fit_poisson_nmf(X,fit0 = fit0,numiter = 450,method = "scd",
control = list(extrapolate = FALSE,numiter = 4))
fit1 <- poisson2multinom(fit1)
fit2 <- poisson2multinom(fit2)This next plot shows the improvement in the solution over time for the EM and SCD updates.
pdat<-rbind(data.frame(x=1:500,y=-fit1$progress$loglik.multinom,method="em"),
data.frame(x=1:500,y=-fit2$progress$loglik.multinom,method="scd"))
pdat <- subset(pdat,x >= 50)
ggplot(pdat,aes(x = x,y = y,color = method)) +
geom_line(size = 0.75) +
scale_color_manual(values = c("dodgerblue","darkorange")) +
labs(x = "iteration",y = "negative log-likelihood") +
theme_cowplot(font_size = 10)
Among the two methods compared, the SCD updates progresses more rapidly toward a solution. Still, the EM updates recover the same solution after a reasonable number of iterations. Next we will see an example in which EM updates fail to make reasonable progress.
Correlated topics scenario
The count data in this second example are simulated as before, with one difference: by setting \(s_{56} = s_{65} = 8\), the mixture proportions for topics 5 and 6 are no longer close to orthogonal.
set.seed(1)
S[5,6] <- 8
S[6,5] <- 8
L <- simulate_loadings(n,k,S)
X <- simulate_multinom_counts(L,F,s)
X <- X[,colSums(X > 0) > 0]Compare this Structure plot to the one above—there is more mixing of topics 5 and 6:
y <- apply(L,1,which.max)
y <- rank(y,ties.method = "random")
y <- qqnorm(y,plot.it = FALSE)$x
fit <- list(L = L)
class(fit) <- c("multinom_topic_model_fit","list")
structure_plot(fit,topics = 1:6,perplexity = 30,Y_init = matrix(y),
verbose = FALSE)
As before, we run the EM and SCD updates to fit the multinomial topic model, with a twist that we perform another round of SCD updates after running the EM updates. This will be explained shortly.
fit0 <- fit_poisson_nmf(X,k,numiter = 50,method = "em",
control = list(extrapolate = FALSE,numiter = 4))
fit1 <- fit_poisson_nmf(X,fit0 = fit0,numiter = 800,method = "em",
control = list(extrapolate = FALSE,numiter = 4))
fit2 <- fit_poisson_nmf(X,fit0 = fit0,numiter = 1000,method = "scd",
control = list(extrapolate = FALSE,numiter = 4))
fit3 <- fit_poisson_nmf(X,fit0 = fit1,numiter = 200,method = "scd",
control = list(extrapolate = FALSE,numiter = 4))
fit1 <- fit_poisson_nmf(X,fit0 = fit1,numiter = 200,method = "em",
control = list(extrapolate = FALSE,numiter = 4))
fit1 <- poisson2multinom(fit1)
fit2 <- poisson2multinom(fit2)
fit3 <- poisson2multinom(fit3)In this second example, after initially making good progress, the EM estimates remain very far from the solution achieved by SCD even after hundreds of EM updates. This isn’t a case where the EM updates have settled on a different local solution—the SCD updates quickly “rescue” the EM estimates.
pdat<-rbind(data.frame(x=1:1050,y=-fit1$progress$loglik.multinom,method="em"),
data.frame(x=1:1050,y=-fit2$progress$loglik.multinom,method="scd"),
data.frame(x=851:1050,y=-fit3$progress[851:1050,"loglik.multinom"],
method = "em+scd"))
pdat <- subset(pdat,x >= 50)
ggplot(pdat,aes(x = x,y = y,color = method)) +
geom_line(size = 0.75) +
scale_color_manual(values = c("dodgerblue","darkorange","magenta")) +
labs(x = "iteration",y = "negative log-likelihood") +
theme_cowplot(font_size = 10)
This suggests that the EM algorithm may perform poorly in settings where the topics are correlated, even if the correlations are quite modest.
sessionInfo()
# R version 3.6.2 (2019-12-12)
# Platform: x86_64-apple-darwin15.6.0 (64-bit)
# Running under: macOS Catalina 10.15.7
#
# Matrix products: default
# BLAS: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
# LAPACK: /Library/Frameworks/R.framework/Versions/3.6/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] cowplot_1.0.0 ggplot2_3.3.0 mvtnorm_1.0-11 fastTopics_0.5-24
#
# loaded via a namespace (and not attached):
# [1] httr_1.4.2 tidyr_1.0.0 jsonlite_1.6
# [4] viridisLite_0.3.0 RcppParallel_4.4.2 assertthat_0.2.1
# [7] mixsqp_0.3-44 yaml_2.2.0 progress_1.2.2
# [10] ggrepel_0.9.0 pillar_1.4.3 backports_1.1.5
# [13] lattice_0.20-38 quantreg_5.54 glue_1.3.1
# [16] quadprog_1.5-8 digest_0.6.23 promises_1.1.0
# [19] colorspace_1.4-1 htmltools_0.4.0 httpuv_1.5.2
# [22] Matrix_1.2-18 pkgconfig_2.0.3 invgamma_1.1
# [25] SparseM_1.78 purrr_0.3.3 scales_1.1.0
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# [31] MatrixModels_0.4-1 git2r_0.26.1 tibble_2.1.3
# [34] farver_2.0.1 withr_2.1.2 ashr_2.2-51
# [37] lazyeval_0.2.2 magrittr_1.5 crayon_1.3.4
# [40] mcmc_0.9-6 evaluate_0.14 fs_1.3.1
# [43] MASS_7.3-51.4 truncnorm_1.0-8 tools_3.6.2
# [46] data.table_1.12.8 prettyunits_1.1.1 hms_0.5.2
# [49] lifecycle_0.1.0 stringr_1.4.0 MCMCpack_1.4-5
# [52] plotly_4.9.2 munsell_0.5.0 irlba_2.3.3
# [55] compiler_3.6.2 rlang_0.4.5 grid_3.6.2
# [58] htmlwidgets_1.5.1 labeling_0.3 rmarkdown_2.3
# [61] gtable_0.3.0 R6_2.4.1 knitr_1.26
# [64] dplyr_0.8.3 zeallot_0.1.0 workflowr_1.6.2.9000
# [67] rprojroot_1.3-2 stringi_1.4.3 SQUAREM_2017.10-1
# [70] Rcpp_1.0.5 vctrs_0.2.1 tidyselect_0.2.5
# [73] xfun_0.11 coda_0.19-3