Last updated: 2021-03-22

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Here we show in a small, simulated data set that the SCD and CCD algorithms produce the exact same sequence of iterates.

Load the packages used in the analysis below.

library(R.matlab)
library(NNLM)
library(fastTopics)
library(ggplot2)
library(cowplot)

Set the seed so that the results can be reproduced.

set.seed(1)

Simulate a 100 x 200 counts matrix.

X <- simulate_count_data(100,200,3)$X

“Pre-fit” t he model with the aim of better ensuring that the same local maximum is recovered by all runs below.

fit0 <- fit_poisson_nmf(X,k = 3,numiter = 20,method = "mu",
                        control = list(numiter = 1))
writeMat("../matlab/dat100x200.mat",X = X,L0 = fit0$L,F0 = fit0$F)

Run 40 multiplicative (EM) updates (as implemented in NNLM).

fit1 <- suppressWarnings(nnmf(X,k = 3,init = list(W = fit0$L,H = t(fit0$F)),
                              method = "lee",loss = "mkl",max.iter = 40,
                              trace = 1,rel.tol = 0,inner.max.iter = 1,
                              inner.rel.tol = 0,n.threads = 1))

Run 40 SCD updates (as implemented in NNLM).

fit2 <- suppressWarnings(nnmf(X,k = 3,init = list(W = fit0$L,H = t(fit0$F)),
                              method = "scd",loss = "mkl",max.iter = 40,
                              trace = 1,rel.tol = 0,inner.max.iter = 1,
                              n.threads = 1))

Load the result of running 40 CCD updates on the same data set in MATLAB.

fit3 <- readMat("../matlab/ccd100x200.mat")
fit3$obj <- drop(fit3$obj)

The estimates obtained after running the CCD and SCD updates are nearly the same:

print(range(fit2$W - fit3$W))
print(range(fit2$H - fit3$H))
# [1] -1.116106e-09  1.565456e-09
# [1] -9.941866e-10  1.446030e-09

Finally, I compare against SCD with extrapolation, as implemented in fastTopics.

fit4 <- fit_poisson_nmf(X,fit0 = fit0,numiter = 40,method = "scd",
                        control = list(extrapolate = TRUE,numiter = 1))

This plot shows the improvement in the solutions over time. (Some adjustments need to be made to the fit_poisson_nmf output as it outputs log-likelihoods, whereas the others output generalized KL-divergences.)

y <- fit4$progress[21:60,"loglik"]
y <- sum(X*log(X + 1e-15) - X) - y + sum(fastTopics:::loglik_poisson_const(X))
y <- y/(100*200)
pdat <- rbind(data.frame(iter = 1:40,loss = fit1$mkl,method = "em"),
              data.frame(iter = 1:40,loss = fit2$mkl,method = "scd"),
              data.frame(iter = 1:40,loss = fit3$obj,method = "ccd"),
              data.frame(iter = 1:40,loss = y,method = "scd+ex"))
pdat <- transform(pdat,loss = loss - min(loss))
ggplot(pdat,aes(x = iter,y = loss,color = method,linetype = method)) +
  geom_line(size = 1) +
  scale_color_manual(values = c("darkblue","darkorange","dodgerblue",
                                "tomato")) +
  scale_linetype_manual(values = c("solid","solid","dashed","solid")) +
  labs(x = "iteration",y = "loss") +
  theme_cowplot(font_size = 12)

Version Author Date
32b302f Peter Carbonetto 2021-03-22
323bf3a Peter Carbonetto 2021-03-22

The EM (multiplicative) updates progress much more slowly than the others.

The CCD and SCD updates overlap exactly.

And the extrapolation accelerates convergence of the SCD updates.


sessionInfo()
# R version 3.5.1 (2018-07-02)
# Platform: x86_64-pc-linux-gnu (64-bit)
# Running under: Scientific Linux 7.4 (Nitrogen)
# 
# Matrix products: default
# BLAS/LAPACK: /software/openblas-0.2.19-el7-x86_64/lib/libopenblas_haswellp-r0.2.19.so
# 
# locale:
#  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
#  [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
#  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
#  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
#  [9] LC_ADDRESS=C               LC_TELEPHONE=C            
# [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       
# 
# attached base packages:
# [1] stats     graphics  grDevices utils     datasets  methods   base     
# 
# other attached packages:
# [1] cowplot_1.1.1     ggplot2_3.3.3     fastTopics_0.5-24 NNLM_0.4.4       
# [5] R.matlab_3.6.2   
# 
# loaded via a namespace (and not attached):
#  [1] httr_1.4.2         tidyr_0.8.3        jsonlite_1.6      
#  [4] viridisLite_0.3.0  R.utils_2.7.0      RcppParallel_5.0.1
#  [7] assertthat_0.2.0   mixsqp_0.3-46      yaml_2.2.0        
# [10] progress_1.2.0     ggrepel_0.9.1      pillar_1.5.0      
# [13] backports_1.1.2    lattice_0.20-38    quantreg_5.36     
# [16] glue_1.4.2         quadprog_1.5-5     digest_0.6.18     
# [19] promises_1.0.1     colorspace_1.3-2   htmltools_0.3.6   
# [22] httpuv_1.4.5       Matrix_1.2-15      R.oo_1.22.0       
# [25] pkgconfig_2.0.2    invgamma_1.1       SparseM_1.77      
# [28] purrr_0.3.2        scales_1.0.0       whisker_0.3-2     
# [31] later_0.7.5        Rtsne_0.15         MatrixModels_0.4-1
# [34] git2r_0.26.1       tibble_3.1.0       generics_0.0.2    
# [37] ellipsis_0.2.0.1   withr_2.1.2        ashr_2.2-51       
# [40] lazyeval_0.2.1     magrittr_1.5       crayon_1.3.4      
# [43] mcmc_0.9-7         evaluate_0.12      R.methodsS3_1.7.1 
# [46] fs_1.3.1           fansi_0.4.0        MASS_7.3-51.1     
# [49] truncnorm_1.0-8    tools_3.5.1        data.table_1.12.0 
# [52] prettyunits_1.0.2  hms_0.4.2          lifecycle_1.0.0   
# [55] stringr_1.3.1      MCMCpack_1.4-4     plotly_4.9.3      
# [58] munsell_0.5.0      irlba_2.3.3        compiler_3.5.1    
# [61] rlang_0.4.10       grid_3.5.1         htmlwidgets_1.5.3 
# [64] labeling_0.3       rmarkdown_1.10     gtable_0.2.0      
# [67] DBI_1.0.0          R6_2.3.0           knitr_1.20        
# [70] dplyr_1.0.5        utf8_1.1.4         workflowr_1.6.2   
# [73] rprojroot_1.3-2    stringi_1.2.4      SQUAREM_2017.10-1 
# [76] Rcpp_1.0.4.6       vctrs_0.3.6        tidyselect_1.1.0  
# [79] coda_0.19-2