Last updated: 2021-03-12
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Knit directory: fastTopics-experiments/analysis/
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html | 311659c | Peter Carbonetto | 2021-03-12 | Added LDA results for second “smallsim” example. |
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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 models. This example suggests that the EM updates have difficulty with correlations between topics, even when they are quite modest, The results also suggest that the basic variational EM for LDA also experiences similar difficulties.
Load the packages used in the analysis below, as well as additional functions that we will use to simulate the data.
library(tm)
library(topicmodels)
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)
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:
topic_colors <- c("dodgerblue","darkorange","forestgreen","darkblue",
"gold","skyblue")
p1 <- simdata_structure_plot(L,topic_colors)
print(p1)
Here we compare two different updates for fitting a Poisson NMF model to the simulated counts: EM updates, and sequential coordinate descent (SCD). We perform 100 iterations of each. 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.
control <- list(extrapolate = FALSE,numiter = 4)
fit0 <- fit_poisson_nmf(X,k,numiter = 50,method = "em",control = control)
fit1 <- fit_poisson_nmf(X,fit0=fit0,numiter=100,method="em",control=control)
fit2 <- fit_poisson_nmf(X,fit0=fit0,numiter=100,method="scd",control=control)
fit0 <- poisson2multinom(fit0)
fit1 <- poisson2multinom(fit1)
fit2 <- poisson2multinom(fit2)
This next plot shows the improvement in the solution over time for the EM and SCD updates. The Y axis shows the difference between the current log-likelihood and the best log-likelihood achieved by the two methods.
pdat <- rbind(data.frame(iter = 1:150,
loglik = fit1$progress$loglik.multinom,
res = fit1$progress$res,
method = "em"),
data.frame(iter = 1:150,
loglik = fit2$progress$loglik.multinom,
res = fit2$progress$res,
method = "scd"))
pdat <- subset(pdat,iter >= 50)
pdat <- transform(pdat,
iter = iter - 50,
loglik = max(loglik) - loglik + 0.1)
p2 <- ggplot(pdat,aes(x = iter,y = loglik,color = method)) +
geom_line(size = 0.75) +
scale_y_continuous(trans = "log10") +
scale_color_manual(values = c("dodgerblue","darkorange")) +
labs(x = "iteration",y = "loglik difference") +
theme_cowplot(font_size = 10)
p3 <- ggplot(pdat,aes(x = iter,y = res,color = method)) +
geom_line(size = 0.75) +
scale_color_manual(values = c("dodgerblue","darkorange")) +
labs(x = "iteration",y = "max KKT residual") +
theme_cowplot(font_size = 10)
plot_grid(p2,p3)
Version | Author | Date |
---|---|---|
be1f9a4 | Peter Carbonetto | 2021-03-11 |
daf189c | Peter Carbonetto | 2021-03-09 |
47ed425 | Peter Carbonetto | 2021-03-09 |
907addd | Peter Carbonetto | 2021-03-08 |
1838ec8 | Peter Carbonetto | 2021-03-08 |
50f34a9 | Peter Carbonetto | 2021-03-07 |
1185ab4 | Peter Carbonetto | 2021-03-07 |
913316c | Peter Carbonetto | 2021-03-07 |
51c5321 | Peter Carbonetto | 2021-03-06 |
59a8594 | Peter Carbonetto | 2021-03-05 |
ffad471 | Peter Carbonetto | 2021-03-05 |
Among the two methods compared, the SCD updates progress more rapidly toward a solution. Still, the EM updates recover the same solution after a reasonable number of iterations. Indeed, the EM and SCD estimates of the topic proportions are almost the same:
p4 <- loadings_scatterplot(fit1,fit2,topic_colors,"em","scd")
print(p4)
Next, we turn to the problem of fitting an LDA model to the same data.
We perform 400 iterations of variational EM, initializing the LDA model fits using the MLEs computed above.
lda0 <- run_lda(X,fit0,numiter = 400)
lda1 <- run_lda(X,fit1,numiter = 400)
lda2 <- run_lda(X,fit2,numiter = 400)
Although the LDA fitting problem is different—we are now fitting a (approximate) posterior distribution, and so the estimates are approximate posterior means rather than MLE—initializing the LDA fit using the MLEs greatly improves the LDA model fitting, as we will see from this next plot: even after 400 iterations, variational EM without a good initialization does not approach the quality of the LDA model fits initialized using the EM and SCD estimates.
pdat <- rbind(data.frame(iter = 1:400,elbo = lda0@logLiks,init = "none"),
data.frame(iter = 1:400,elbo = lda1@logLiks,init = "em"),
data.frame(iter = 1:400,elbo = lda2@logLiks,init = "scd"))
pdat <- transform(pdat,elbo = max(elbo) - elbo + 0.1)
p5<-ggplot(pdat,aes(x = iter,y = elbo,color = init)) +
geom_line(size = 0.75) +
scale_y_continuous(trans = "log10") +
scale_color_manual(values=c("darkblue","dodgerblue","darkorange")) +
labs(x = "iteration",y = "ELBO difference") +
theme_cowplot(font_size = 10)
print(p5)
Next we will see an example in which EM updates fail to make reasonable progress.