Linear vs. nonlinear embeddings: a thought experiment
Peter Carbonetto
Last updated: 2020-10-12
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Knit directory: pcarbo/analysis/
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Add text here.
Load the packages used in the analysis below, and set the seed.
library(mvtnorm)
library(Rtsne)
library(flashier)
library(ggplot2)
library(cowplot)
set.seed(1)Simulate two clusters of 2-d data points. The second cluster varies much more along the second dimension.
X <- rbind(rmvnorm(100,c(-1,0),diag(2)/30),
rmvnorm(900,c(+1,0),diag(c(0.1,3))))
cluster <- factor(c(rep(1,100),rep(2,900)))
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(-6,6) +
ylim(-6,6) +
labs(x = "x1",y = "x2") +
theme_cowplot(font_size = 12)
print(p1)
Unsurprisingly, PCA doesn’t work—he first principal component picks up variation in the second cluster.
pca <- drop(prcomp(X)$x[,1])
pdat <- data.frame(cluster,pca)
p2 <- ggplot(pdat,aes(x = cluster,y = pca)) +
geom_boxplot(width = 0.25) +
theme_cowplot(font_size = 12)
print(p2)
On the other hand, the 1-d embedding generated by t-SNE is able to clearly distinguish the two clusters.
tsne <- drop(Rtsne(X,dims = 1,perplexity = 30,theta = 0.1,max_iter = 1000,
eta = 200,normalize = FALSE,pca = FALSE,verbose = FALSE)$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 one factor does no better than PCA in separating the two clusters.
fl <- flash(X,greedy.Kmax = 1,nullcheck = FALSE)
fl <- drop(fl$loadings.pm[[1]])
pdat <- data.frame(cluster,flash = fl)
p4 <- ggplot(pdat,aes(x = cluster,y = flash)) +
geom_boxplot(width = 0.25) +
theme_cowplot(font_size = 12)
print(p4)
# Adding factor 1 to flash object...
# Wrapping up...
# Done.
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