eb_ica
junmingguan
2026-03-03
Last updated: 2026-03-12
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Introduction
We examine an empirical Bayes version of ICA, where the prior \(p_\pi(s_i)=\text{Rad}(s_i;\ \pi)\) is estimated from the data. The steps to find one source are as follows:
M-step: \(\mathbf{w} = Y \mathbf{s}^+ / \| Y \mathbf{s}^+ \|\)
E-step:
\[ \begin{align} (\hat{q}(\mathbf{s}), \hat{\pi}) &=\arg \max_{q, \pi} \mathbb{E}_q\left[\log p(\mathbf{s}, Y \mid \mathbf{w})\right] \\ \\ &= \arg \max_{q, \pi} \mathbb{E}_q\left[ \log p(Y \mid \mathbf{s}, \mathbf{w}) + \sum \log p_\pi(s_i )\right] \\ &= \arg \max_{q, \pi} \left\{-\frac{1}{2\sigma^2} (-2 \text{tr}(\mathbb{E}_q[\mathbf{s}]^\top Y^\top \mathbf{w}) + \mathbb{E}_q\|\mathbf{s}\|^2) -\sum\text{KL}\Big(q(s_i)\ \|\ p_\pi(s_i)\Big)\right\} \end{align} \]
- solved by \(\text{EBNM}(Y^\top\mathbf{w}, \sigma)\)
- \(\mathbf{s}^+=\mathbb{E}_\hat{q}[\mathbf{s}]\).
For a rank-K fit, we proceed as follows:
M-step:
\(W_+ = YS^+\)
Orthogonalization: \(W = U_+ V_+^\top\), where \(W^+=U_+ DV_+^\top\), which is equivalent to \(W= (W_+W_+^\top)^{-1/2} W_+\) proposed in [@Karhunenetal].
E-step: for \(k=1,\dots, K\),
Obtain \(\hat{q}_k, \hat{\pi}_k\) by \(\text{EBNM}(Y^\top\mathbf{w}_k, \sigma)\)
\(\mathbf{s}_k^+=\mathbb{E}_\hat{q}[\mathbf{s}_k]\).
library(flashier)
library(Matrix)
library(fastICA)
library(fastTopics)
library(ebnm)
source('code/ebcd_functions.R')
source('code/ica_functions.R')
source('code/ebnm_rademacher.R')simulate_unbalanaced_rad_groups <- function(
K, n=100, p=1000, noise=1, percent_one=0.2) {
set.seed(1)
L = matrix(-1,nrow=n,ncol=K)
for(i in 1:K){L[sample(1:n,floor(percent_one*n)),i]=1}
FF = matrix(rnorm(p*K), nrow = p, ncol=K)
X = L %*% t(FF) + rnorm(n*p, 0, noise)
return(list(X=X, L=L, FF=FF))
}sim_data <- simulate_unbalanaced_rad_groups(
K=3, n=100, p=1000, percent_one = 0.2, noise = 1
)
X <- sim_data$X
L <- sim_data$L
FF <- sim_data$FF
X1 <- preprocess(X)Unbalanced 3 groups
rank 1 fit
cor_max_ebica <- c() # maximum correlation between the estmate and a true source
cor_max_fastica <- c() # maximum correlation between the estmate and a true source
est_S_ebica <- matrix(0, nrow = nrow(X), ncol = 100) # all 100 solutions
est_S_fastica <- matrix(0, nrow = nrow(X), ncol = 100) # all 100 solutions
for (i in 1:100) {
set.seed(i)
s <- matrix(rnorm(nrow(X)), ncol = 1)
w <- X1 %*% s
ebica_res <- ebica(t(X), K=1,
S_init = s)
fastica_w <- fastica_r1update(w, X1)
fastica_s <- t(X1) %*% fastica_w
cor_max_ebica <- c(cor_max_ebica, max(abs(cor(L, ebica_res$S_plus))))
est_S_ebica[,i] <- ebica_res$S_plus
cor_max_fastica <- c(cor_max_fastica, max(abs(cor(L, fastica_s))))
est_S_fastica[,i] <- fastica_s
}Converged in 3 iterations. Final ELBO: -303290.3
Converged in 3 iterations. Final ELBO: -303042.3
Converged in 2 iterations. Final ELBO: -303042.3
Converged in 2 iterations. Final ELBO: -303042.3
Converged in 2 iterations. Final ELBO: -303259.2
Converged in 4 iterations. Final ELBO: -303042.3
Converged in 3 iterations. Final ELBO: -303042.3
Converged in 2 iterations. Final ELBO: -303042.3
Converged in 2 iterations. Final ELBO: -303279.9
Converged in 4 iterations. Final ELBO: -303279.9
Converged in 3 iterations. Final ELBO: -303290.3
Converged in 2 iterations. Final ELBO: -303290.3
Converged in 3 iterations. Final ELBO: -303279.9
Converged in 4 iterations. Final ELBO: -303042.3
Converged in 2 iterations. Final ELBO: -303259.2
Converged in 3 iterations. Final ELBO: -303279.9
Converged in 4 iterations. Final ELBO: -303042.3
Converged in 4 iterations. Final ELBO: -303042.3
Converged in 5 iterations. Final ELBO: -303042.3
Converged in 4 iterations. Final ELBO: -303042.3
Converged in 2 iterations. Final ELBO: -303259.2
Converged in 2 iterations. Final ELBO: -303259.2
Converged in 4 iterations. Final ELBO: -303042.3
Converged in 3 iterations. Final ELBO: -303259.2
Converged in 3 iterations. Final ELBO: -303042.3
Converged in 2 iterations. Final ELBO: -303290.3
Converged in 5 iterations. Final ELBO: -303042.3
Converged in 2 iterations. Final ELBO: -303279.9
Converged in 3 iterations. Final ELBO: -303042.3
Converged in 2 iterations. Final ELBO: -303290.3
Converged in 3 iterations. Final ELBO: -303279.9
Converged in 2 iterations. Final ELBO: -303042.3
Converged in 3 iterations. Final ELBO: -303279.9
Converged in 4 iterations. Final ELBO: -303279.9
Converged in 2 iterations. Final ELBO: -303042.3
Converged in 4 iterations. Final ELBO: -303042.3
Converged in 2 iterations. Final ELBO: -303279.9
Converged in 4 iterations. Final ELBO: -303042.3
Converged in 2 iterations. Final ELBO: -303042.3
Converged in 4 iterations. Final ELBO: -303259.2
Converged in 2 iterations. Final ELBO: -303042.3
Converged in 4 iterations. Final ELBO: -303279.9
Converged in 4 iterations. Final ELBO: -303042.3
Converged in 3 iterations. Final ELBO: -303259.2
Converged in 4 iterations. Final ELBO: -303290.3
Converged in 4 iterations. Final ELBO: -303042.3
Converged in 2 iterations. Final ELBO: -303259.2
Converged in 2 iterations. Final ELBO: -303290.3
Converged in 2 iterations. Final ELBO: -303259.2
Converged in 2 iterations. Final ELBO: -303279.9
Converged in 2 iterations. Final ELBO: -303279.9
Converged in 4 iterations. Final ELBO: -303259.2
Converged in 2 iterations. Final ELBO: -303290.3
Converged in 3 iterations. Final ELBO: -303042.3
Converged in 4 iterations. Final ELBO: -303042.3
Converged in 2 iterations. Final ELBO: -303259.2
Converged in 2 iterations. Final ELBO: -303259.2
Converged in 2 iterations. Final ELBO: -303290.3
Converged in 3 iterations. Final ELBO: -303279.9
Converged in 3 iterations. Final ELBO: -303259.2
Converged in 2 iterations. Final ELBO: -303279.9
Converged in 3 iterations. Final ELBO: -303042.3
Converged in 4 iterations. Final ELBO: -303290.3
Converged in 2 iterations. Final ELBO: -303279.9
Converged in 2 iterations. Final ELBO: -303290.3
Converged in 3 iterations. Final ELBO: -303290.3
Converged in 3 iterations. Final ELBO: -303042.3
Converged in 4 iterations. Final ELBO: -303042.3
Converged in 2 iterations. Final ELBO: -303279.9
Converged in 4 iterations. Final ELBO: -303042.3
Converged in 2 iterations. Final ELBO: -303259.2
Converged in 3 iterations. Final ELBO: -303042.3
Converged in 2 iterations. Final ELBO: -303259.2
Converged in 2 iterations. Final ELBO: -303290.3
Converged in 4 iterations. Final ELBO: -303042.3
Converged in 3 iterations. Final ELBO: -303290.3
Converged in 4 iterations. Final ELBO: -303042.3
Converged in 2 iterations. Final ELBO: -303042.3
Converged in 2 iterations. Final ELBO: -303290.3
Converged in 5 iterations. Final ELBO: -303042.3
Converged in 2 iterations. Final ELBO: -303290.3
Converged in 2 iterations. Final ELBO: -303259.2
Converged in 2 iterations. Final ELBO: -303042.3
Converged in 2 iterations. Final ELBO: -303259.2
Converged in 2 iterations. Final ELBO: -303042.3
Converged in 6 iterations. Final ELBO: -303290.3
Converged in 2 iterations. Final ELBO: -303259.2
Converged in 2 iterations. Final ELBO: -303279.9
Converged in 3 iterations. Final ELBO: -303042.3
Converged in 5 iterations. Final ELBO: -303042.3
Converged in 3 iterations. Final ELBO: -303290.3
Converged in 2 iterations. Final ELBO: -303259.2
Converged in 3 iterations. Final ELBO: -303042.3
Converged in 5 iterations. Final ELBO: -303042.3
Converged in 2 iterations. Final ELBO: -303279.9
Converged in 3 iterations. Final ELBO: -303042.3
Converged in 4 iterations. Final ELBO: -303042.3
Converged in 5 iterations. Final ELBO: -303279.9
Converged in 2 iterations. Final ELBO: -303259.2
Converged in 3 iterations. Final ELBO: -303042.3 table(abs(cor_max_ebica) > 0.99)
FALSE TRUE
43 57 table(abs(cor_max_fastica) > 0.99)
FALSE
100 image(t(est_S_ebica[, order((cor_max_ebica), decreasing = T)]))
| Version | Author | Date |
|---|---|---|
| 0cc18da | junmingguan | 2026-03-12 |
image(t(est_S_fastica[, order((cor_max_fastica), decreasing = T)]))
| Version | Author | Date |
|---|---|---|
| 0cc18da | junmingguan | 2026-03-12 |
rank 3 fit
K <- 3
ebica_res <- ebica(t(X), K=K,
S_init = NULL)Converged in 3 iterations. Final ELBO: -297232.8 cor(L, ebica_res$S_plus) [,1] [,2] [,3]
[1,] -5.724587e-17 -6.250000e-02 1.000000e+00
[2,] -1.000000e+00 -1.734723e-17 5.724587e-17
[3,] 1.734723e-17 1.000000e+00 -6.250000e-02fastica_res <- fastICA(X, n.comp = K, alg.typ = "parallel")
cor(L, fastica_res$S) [,1] [,2] [,3]
[1,] 0.5610073 -0.2071364 0.8004757
[2,] 0.5797838 -0.5914011 -0.5590788
[3,] 0.5534336 0.7897468 -0.2615436rank 4 fit
K <- 4
ebica_res <- ebica(t(X), K=K,
S_init = NULL)Converged in 4 iterations. Final ELBO: -296086.9 cor(L, ebica_res$S_plus) [,1] [,2] [,3] [,4]
[1,] -5.724587e-17 0.5837923 6.250000e-02 -1.000000e+00
[2,] -1.000000e+00 -0.2810852 1.734723e-17 -5.724587e-17
[3,] 1.734723e-17 0.5837923 -1.000000e+00 6.250000e-02fastica_res <- fastICA(X, n.comp = K, alg.typ = "parallel")
cor(L, fastica_res$S) [,1] [,2] [,3] [,4]
[1,] 0.71950860 -0.08927415 0.12029707 -0.67695948
[2,] -0.04814472 -0.94855271 -0.30994530 0.01850681
[3,] -0.72395529 0.03350308 -0.03122539 -0.68715783Unbalanced 9 groups
sim_data <- simulate_unbalanaced_rad_groups(
K=9, n=100, p=1000, percent_one = 0.2, noise = 1
)
X <- sim_data$X
L <- sim_data$L
FF <- sim_data$FF
X1 <- preprocess(X)set.seed(1)
K <- 9
ebica_res <- ebica(t(X), K=K,
S_init = NULL)Converged in 8 iterations. Final ELBO: -571230.3 cor(L, ebica_res$S_plus) [,1] [,2] [,3] [,4] [,5] [,6]
[1,] 0.0625 5.377643e-17 0.0625 -3.469447e-17 -6.250000e-02 5.724587e-17
[2,] 0.0625 -6.250000e-02 0.0625 2.341877e-17 -1.734723e-17 1.000000e+00
[3,] 0.2500 -5.030698e-17 -0.1875 -6.250000e-02 1.000000e+00 -1.734723e-17
[4,] -0.0625 -6.250000e-02 -0.0625 -1.000000e+00 6.250000e-02 -2.341877e-17
[5,] 0.0625 6.250000e-02 -0.0625 6.250000e-02 5.898060e-17 6.250000e-02
[6,] -0.0625 -1.000000e+00 0.0625 -6.250000e-02 5.030698e-17 6.250000e-02
[7,] 0.0625 6.250000e-02 -1.0000 -6.250000e-02 1.875000e-01 -6.250000e-02
[8,] 0.1250 6.250000e-02 -0.0625 1.250000e-01 1.875000e-01 1.214306e-16
[9,] -1.0000 -6.250000e-02 0.0625 -6.250000e-02 -2.500000e-01 -6.250000e-02
[,7] [,8] [,9]
[1,] -6.250000e-02 6.25000e-02 -1.000000e+00
[2,] -1.214306e-16 6.25000e-02 -5.724587e-17
[3,] -1.875000e-01 5.89806e-17 6.250000e-02
[4,] 1.250000e-01 -6.25000e-02 -3.469447e-17
[5,] -6.250000e-02 1.00000e+00 -6.250000e-02
[6,] 6.250000e-02 -6.25000e-02 5.377643e-17
[7,] -6.250000e-02 6.25000e-02 6.250000e-02
[8,] -1.000000e+00 6.25000e-02 -6.250000e-02
[9,] 1.250000e-01 -6.25000e-02 6.250000e-02apply(abs(cor(L, ebica_res$S_plus)), 1, max)[1] 1 1 1 1 1 1 1 1 1fastica_res <- fastICA(X, n.comp = K, alg.typ = "parallel")
cor(L, fastica_res$S) [,1] [,2] [,3] [,4] [,5] [,6]
[1,] -0.09068524 -0.12656481 0.30849779 0.52273147 -0.008850283 -0.097420560
[2,] -0.25896415 0.17633223 -0.47914041 -0.32716737 -0.308085791 -0.075623431
[3,] 0.53904316 0.10740969 -0.33597685 -0.19461556 -0.028993764 0.533888523
[4,] -0.33344869 0.16680122 0.54790818 -0.56856914 -0.212175323 0.197115772
[5,] -0.17075576 0.33437216 -0.22897974 0.40928402 -0.177834999 -0.051334450
[6,] -0.41640315 -0.33932403 -0.32139446 0.05584344 -0.509918586 0.009837451
[7,] 0.46129730 0.57313013 0.22038054 0.13076840 -0.572907156 0.005648480
[8,] 0.50392918 -0.58040328 0.06660077 0.01164317 -0.426986098 0.071580896
[9,] -0.43888226 -0.03375912 0.21443129 0.25361005 -0.021940185 0.670173376
[,7] [,8] [,9]
[1,] 0.1545986 -0.54223035 0.52738156
[2,] 0.5788604 -0.35497708 0.04864275
[3,] -0.2316659 -0.11287418 0.43973454
[4,] -0.1551914 0.04816204 0.35425421
[5,] 0.2024973 0.60297828 0.44565564
[6,] -0.5676895 -0.14499598 -0.01000411
[7,] -0.1478544 -0.06522789 -0.19243760
[8,] 0.3749530 0.23026102 0.14794953
[9,] 0.1654524 0.03793201 -0.46457058apply(abs(cor(L, fastica_res$S)), 1, max)[1] 0.5422304 0.5788604 0.5390432 0.5685691 0.6029783 0.5676895 0.5731301
[8] 0.5804033 0.6701734
sessionInfo()R version 4.3.1 (2023-06-16)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS 26.2
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.11.0
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
time zone: America/Los_Angeles
tzcode source: internal
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] fastTopics_0.6-192 fastICA_1.2-7 Matrix_1.6-4 flashier_1.0.54
[5] ebnm_1.1-34 workflowr_1.7.1
loaded via a namespace (and not attached):
[1] tidyselect_1.2.1 viridisLite_0.4.2 dplyr_1.1.4
[4] fastmap_1.2.0 lazyeval_0.2.2 promises_1.3.0
[7] digest_0.6.37 lifecycle_1.0.4 processx_3.8.2
[10] invgamma_1.1 magrittr_2.0.3 compiler_4.3.1
[13] rlang_1.1.4 sass_0.4.9 progress_1.2.3
[16] tools_4.3.1 utf8_1.2.4 yaml_2.3.10
[19] data.table_1.16.2 knitr_1.48 prettyunits_1.2.0
[22] htmlwidgets_1.6.4 scatterplot3d_0.3-44 RColorBrewer_1.1-3
[25] Rtsne_0.17 purrr_1.0.2 grid_4.3.1
[28] fansi_1.0.6 git2r_0.35.0 colorspace_2.1-1
[31] ggplot2_3.5.1 scales_1.3.0 gtools_3.9.5
[34] cli_3.6.3 rmarkdown_2.28 crayon_1.5.3
[37] generics_0.1.3 RcppParallel_5.1.9 rstudioapi_0.15.0
[40] httr_1.4.7 pbapply_1.7-2 cachem_1.1.0
[43] stringr_1.5.1 splines_4.3.1 parallel_4.3.1
[46] softImpute_1.4-1 matrixStats_1.3.0 vctrs_0.6.5
[49] jsonlite_1.8.9 callr_3.7.3 hms_1.1.3
[52] mixsqp_0.3-54 ggrepel_0.9.6 irlba_2.3.5.1
[55] horseshoe_0.2.0 trust_0.1-8 plotly_4.10.4
[58] jquerylib_0.1.4 tidyr_1.3.1 glue_1.8.0
[61] ps_1.7.5 uwot_0.1.16 cowplot_1.1.3
[64] stringi_1.8.4 Polychrome_1.5.1 gtable_0.3.6
[67] later_1.3.2 quadprog_1.5-8 munsell_0.5.1
[70] tibble_3.2.1 pillar_1.9.0 htmltools_0.5.8.1
[73] truncnorm_1.0-9 R6_2.5.1 rprojroot_2.0.3
[76] evaluate_1.0.1 lattice_0.21-8 highr_0.11
[79] RhpcBLASctl_0.23-42 SQUAREM_2021.1 ashr_2.2-63
[82] httpuv_1.6.14 bslib_0.8.0 Rcpp_1.0.13
[85] deconvolveR_1.2-1 whisker_0.4.1 xfun_0.48
[88] fs_1.6.4 getPass_0.2-4 pkgconfig_2.0.3