Linear vs. nonlinear embeddings: a thought experiment
Peter Carbonetto
Last updated: 2020-10-15
Checks: 7 0
Knit directory: pcarbo/analysis/
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These are the previous versions of the repository in which changes were made to the R Markdown (analysis/tsne_vs_factors.Rmd) and HTML (docs/tsne_vs_factors.html) files. If you’ve configured a remote Git repository (see ?wflow_git_remote), click on the hyperlinks in the table below to view the files as they were in that past version.
| File | Version | Author | Date | Message |
|---|---|---|---|---|
| Rmd | 54f8c58 | Peter Carbonetto | 2020-10-15 | workflowr::wflow_publish(“tsne_vs_factors.Rmd”) |
| html | 2107d7b | Peter Carbonetto | 2020-10-14 | One more tweak. |
| Rmd | b49f0fa | Peter Carbonetto | 2020-10-14 | workflowr::wflow_publish(“tsne_vs_factors.Rmd”) |
| html | 9a0c13a | Peter Carbonetto | 2020-10-14 | Another few tweaks. |
| Rmd | 6471c5b | Peter Carbonetto | 2020-10-14 | workflowr::wflow_publish(“tsne_vs_factors.Rmd”) |
| html | 8aea435 | Peter Carbonetto | 2020-10-14 | Small tweak to tsne_vs_factors demo. |
| Rmd | 8789475 | Peter Carbonetto | 2020-10-14 | workflowr::wflow_publish(“tsne_vs_factors.Rmd”) |
| html | e378483 | Peter Carbonetto | 2020-10-14 | Added second flash fit to tsne_vs_factors demo. |
| Rmd | 3a07bb9 | Peter Carbonetto | 2020-10-14 | workflowr::wflow_publish(“tsne_vs_factors.Rmd”) |
| html | 8320e44 | Peter Carbonetto | 2020-10-13 | Modified clustering example in tsne_vs_factors demo. |
| Rmd | 23c0081 | Peter Carbonetto | 2020-10-13 | workflowr::wflow_publish(“tsne_vs_factors.Rmd”) |
| html | f7b5c9d | Peter Carbonetto | 2020-10-12 | Fixed dimensions of image in tsne_vs_factors analysis. |
| Rmd | 0aef416 | Peter Carbonetto | 2020-10-12 | workflowr::wflow_publish(“tsne_vs_factors.Rmd”) |
| html | d2ff4e0 | Peter Carbonetto | 2020-10-12 | Fixed the text a bit in tsne_vs_factors. |
| Rmd | 3273db4 | Peter Carbonetto | 2020-10-12 | workflowr::wflow_publish(“tsne_vs_factors.Rmd”) |
| html | 2953d44 | Peter Carbonetto | 2020-10-12 | Added accompanying text to tsne_vs_factors analysis. |
| Rmd | 32402cf | Peter Carbonetto | 2020-10-12 | workflowr::wflow_publish(“tsne_vs_factors.Rmd”) |
| html | 5426424 | Peter Carbonetto | 2020-10-12 | Built page for tsne_vs_factors analysis. |
| Rmd | 82daf35 | Peter Carbonetto | 2020-10-12 | workflowr::wflow_publish(“tsne_vs_factors.Rmd”) |
Load the packages used in the analysis below, and set the seed.
library(mvtnorm)
library(Rtsne)
library(flashier)
library(ggplot2)
library(cowplot)
set.seed(2)Consider the following data set. The 2-d data vectors are simulated so that they cluster into two sets. The second cluster varies much more along the first and second dimensions.
X <- rbind(rmvnorm(300,c(-2,+2),diag(2)/30),
rmvnorm(1000,c(0,0),rbind(c(3,2.9),
c(2.9,3))))
cluster <- factor(c(rep(1,300),rep(2,1000)))
colnames(X) <- c("x1","x2")
p1 <- qplot(X[,1],X[,2],shape = I(21),color = I("white"),fill = cluster) +
scale_fill_manual(values = c("tomato","dodgerblue")) +
xlim(-7,7) +
ylim(-7,7) +
labs(x = "x1",y = "x2") +
theme_cowplot(font_size = 12)
print(p1)
Question: Is there a (linear) matrix factorization method that will produce a 1-d representation that correctly subdivides the points into the two clusters?
Unsurprisingly, PCA doesn’t work—the first PC picks up variation in the second cluster.
pca <- drop(prcomp(X,scale. = TRUE)$x[,1])
pdat <- data.frame(cluster,pca)
p2 <- ggplot(pdat,aes(x = cluster,y = pca)) +
geom_boxplot(width = 0.25) +
theme_cowplot(font_size = 12)
print(p2)
On the other hand, the 1-d embedding generated by t-SNE is able to clearly distinguish the two clusters.
tsne <- drop(Rtsne(X,dims = 1,perplexity = 30,theta = 0.1,max_iter = 1000,
eta = 200,normalize = FALSE,pca = FALSE,verbose = FALSE)$Y)
pdat <- data.frame(cluster,tsne)
p3 <- ggplot(pdat,aes(x = cluster,y = tsne)) +
geom_boxplot(width = 0.25) +
theme_cowplot(font_size = 12)
print(p3)
FLASH with the default prior choices does no better than PCA in separating the two clusters.
fl1 <- flash(X,greedy.Kmax = 1,backfit = TRUE,nullcheck = FALSE,
verbose.lvl = 0)
y <- drop(fl1$loadings.pm[[1]])
pdat <- data.frame(cluster,flash = y)
p4 <- ggplot(pdat,aes(x = cluster,y = flash)) +
geom_boxplot(width = 0.25) +
theme_cowplot(font_size = 12)
print(p4)
| Version | Author | Date |
|---|---|---|
| e378483 | Peter Carbonetto | 2020-10-14 |
By contrast, FLASH with a unimodal prior on the loadings can separate the two clusters with a random initialization.
fl2 <- flash.init(X)
fl2 <- flash.init.factors(fl2,list(matrix(rnorm(nrow(X))),
matrix(rnorm(ncol(X)))),
prior.family = c(prior.unimodal(),prior.normal()))
fl2 <- flash.backfit(fl2,verbose.lvl = 0)
y <- drop(fl2$loadings.pm[[1]])
pdat <- data.frame(cluster,flash = y)
p5 <- ggplot(pdat,aes(x = cluster,y = flash)) +
geom_boxplot(width = 0.25) +
theme_cowplot(font_size = 12)
print(p5)
| Version | Author | Date |
|---|---|---|
| e378483 | Peter Carbonetto | 2020-10-14 |
sessionInfo()
# R version 3.6.2 (2019-12-12)
# Platform: x86_64-apple-darwin15.6.0 (64-bit)
# Running under: macOS Catalina 10.15.6
#
# 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 flashier_0.2.7 Rtsne_0.15 mvtnorm_1.0-11
#
# loaded via a namespace (and not attached):
# [1] Rcpp_1.0.5 pillar_1.4.3 compiler_3.6.2
# [4] later_1.0.0 git2r_0.26.1 workflowr_1.6.2.9000
# [7] tools_3.6.2 digest_0.6.23 tibble_2.1.3
# [10] evaluate_0.14 lifecycle_0.1.0 gtable_0.3.0
# [13] lattice_0.20-38 pkgconfig_2.0.3 rlang_0.4.5
# [16] Matrix_1.2-18 yaml_2.2.0 parallel_3.6.2
# [19] ebnm_0.1-21 xfun_0.11 invgamma_1.1
# [22] withr_2.1.2 dplyr_0.8.3 stringr_1.4.0
# [25] knitr_1.26 fs_1.3.1 tidyselect_0.2.5
# [28] rprojroot_1.3-2 grid_3.6.2 glue_1.3.1
# [31] R6_2.4.1 rmarkdown_2.3 mixsqp_0.3-44
# [34] irlba_2.3.3 farver_2.0.1 purrr_0.3.3
# [37] ashr_2.2-51 magrittr_1.5 whisker_0.4
# [40] backports_1.1.5 scales_1.1.0 promises_1.1.0
# [43] htmltools_0.4.0 assertthat_0.2.1 colorspace_1.4-1
# [46] httpuv_1.5.2 labeling_0.3 stringi_1.4.3
# [49] munsell_0.5.0 truncnorm_1.0-8 SQUAREM_2017.10-1
# [52] crayon_1.3.4