Comparing fastICA and alternating projection for binary group recovery
Matthew Stephens
2026-08-15
Last updated: 2026-08-15
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| File | Version | Author | Date | Message |
|---|---|---|---|---|
| Rmd | e4ed94f | Matthew Stephens | 2026-08-15 | Add fastICA vs alternating projection comparison |
Introduction
We compare two approaches for finding binary group structure in whitened data:
Non-centered fastICA (from
fastica_centered_02.Rmd): maximizes the log-cosh objective over unit-norm weight vectors on whitened data with a row of 1s prepended as an intercept.Alternating projection: iteratively projects a binary vector onto the subspace spanned by the data, then rounds back to ±1. Specifically, each update is
v <- sign(U %*% (t(U) %*% v))where U has orthonormal columns spanning the data subspace.
Both methods receive the same preprocessed data: centered and whitened X1 with a row of 1s prepended to form X1_aug. For the alternating projection, U = t(X1_aug) / sqrt(n), which has orthonormal columns because X1_aug %*% t(X1_aug) = n * I (whitening plus centering makes the intercept row orthogonal to all whitened rows).
For each simulation setting (taken from fastica_centered_02.Rmd) we run each method from 100 random starting points and report the fraction of starts that produce a solution correlating >0.95 with at least one true group column.
Helper functions
# X is (n.comp+1) x n. W is (n.comp+1) x n_starts (one weight vector per column).
# P = t(X) %*% W is n x n_starts (source estimates).
fastica_update = function(X, W) {
P <- t(X) %*% W # n x n_starts: source estimates
G <- tanh(P)
G2 <- 1 - G^2
W <- X %*% G - sweep(W, 2, colSums(G2), "*")
sweep(W, 2, sqrt(colSums(W^2)), "/")
}
# X is (n.comp+1) x n. V is n x n_starts (one binary vector per column).
# P = X %*% V / n is (n.comp+1) x n_starts (subspace coefficients).
alt_proj_update = function(X, V) {
n <- ncol(X)
P <- X %*% V / n # (n.comp+1) x n_starts: subspace coefficients
V <- sign(t(X) %*% P) # n x n_starts: project back and round
V[V == 0] <- 1L
V
}
# Center columns of X, whiten to n.comp dimensions.
preprocess = function(X, n.comp = 10) {
X <- scale(X, scale = FALSE)
sqrt(nrow(X)) * t(svd(X)$u[, 1:n.comp])
}Comparison functions
max_cor_with_truth = function(v, L) max(abs(cor(v, L)))
run_fastica = function(X1_aug, L, n_starts = 100, n_iter = 50, threshold = 0.95) {
set.seed(1)
W <- matrix(rnorm(nrow(X1_aug) * n_starts), nrow(X1_aug), n_starts)
W <- sweep(W, 2, sqrt(colSums(W^2)), "/")
for (i in seq_len(n_iter))
W <- fastica_update(X1_aug, W)
Lhat <- t(X1_aug) %*% W # n x n_starts
success <- sum(apply(Lhat, 2, function(lhat) {
mc <- max_cor_with_truth(lhat, L)
!is.na(mc) && mc > threshold
}))
success / n_starts
}
run_alt_proj = function(X1_aug, L, n_starts = 100, n_iter = 50, threshold = 0.95) {
n <- ncol(X1_aug)
set.seed(1)
V <- sign(matrix(rnorm(n * n_starts), n, n_starts))
V[V == 0] <- 1L
for (i in seq_len(n_iter))
V <- alt_proj_update(X1_aug, V)
success <- sum(apply(V, 2, function(v) {
mc <- max_cor_with_truth(v, L)
!is.na(mc) && mc > threshold
}))
success / n_starts
}
run_comparison = function(X, L, n.comp, label, n_starts = 100, n_iter = 50) {
X1 <- preprocess(X, n.comp = n.comp)
X1_aug <- rbind(rep(1, ncol(X1)), X1)
cat("Running fastICA for:", label, "\n")
ica_time <- system.time(ica_rate <- run_fastica(X1_aug, L, n_starts = n_starts, n_iter = n_iter))
cat("Running alt-proj for:", label, "\n")
ap_time <- system.time(ap_rate <- run_alt_proj(X1_aug, L, n_starts = n_starts, n_iter = n_iter))
cat(sprintf(" fastICA: %.0f%% (%.0fms) AltProj: %.0f%% (%.0fms)\n",
100 * ica_rate, 1000 * ica_time["elapsed"],
100 * ap_rate, 1000 * ap_time["elapsed"]))
c(fastICA = ica_rate, AltProj = ap_rate,
fastICA_ms = 1000 * ica_time["elapsed"], AltProj_ms = 1000 * ap_time["elapsed"])
}Simulation 1: 3 overlapping groups
Three groups, 20 members each out of n=100; groups may overlap. Background is -1, group members are +1.
K <- 3; p <- 1000; n <- 100
set.seed(1)
L <- matrix(-1, nrow = n, ncol = K)
for (i in 1:K) { L[sample(1:n, 20), i] <- 1 }
FF <- matrix(rnorm(p * K), nrow = p, ncol = K)
X <- L %*% t(FF) + rnorm(n * p, 0, 0.01)
res1 <- run_comparison(X, L, n.comp = 10, label = "3 overlapping groups (n=100, K=3)")Running fastICA for: 3 overlapping groups (n=100, K=3)
Running alt-proj for: 3 overlapping groups (n=100, K=3) Warning in cor(v, L): the standard deviation is zero
Warning in cor(v, L): the standard deviation is zero
Warning in cor(v, L): the standard deviation is zero
Warning in cor(v, L): the standard deviation is zero
Warning in cor(v, L): the standard deviation is zero fastICA: 68% (16ms) AltProj: 13% (10ms)Simulation 2: 9 overlapping groups
Nine groups, 20 members each out of n=100. Background is 0, group members are +1.
K <- 9; p <- 1000; n <- 100
set.seed(1)
L <- matrix(0, nrow = n, ncol = K)
for (i in 1:K) { L[sample(1:n, 20), i] <- 1 }
FF <- matrix(rnorm(p * K), nrow = p, ncol = K)
X <- L %*% t(FF) + rnorm(n * p, 0, 0.01)
res2 <- run_comparison(X, L, n.comp = 10, label = "9 overlapping groups (n=100, K=9)")Running fastICA for: 9 overlapping groups (n=100, K=9)
Running alt-proj for: 9 overlapping groups (n=100, K=9)
fastICA: 84% (8ms) AltProj: 17% (5ms)Simulation 3: 4 non-overlapping groups
Four equal non-overlapping groups of 25 (n=100). Using n.comp=10.
n <- 100; p <- 1000
set.seed(1)
L <- matrix(0, nrow = n, ncol = 4)
L[1:25, 1] <- 1
L[26:50, 2] <- 1
L[51:75, 3] <- 1
L[76:100, 4] <- 1
FF <- matrix(rnorm(p * 4), nrow = p)
X <- L %*% t(FF) + rnorm(n * p, 0, 0.01)
res3 <- run_comparison(X, L, n.comp = 10, label = "4 non-overlapping groups (n=100)")Running fastICA for: 4 non-overlapping groups (n=100)
Running alt-proj for: 4 non-overlapping groups (n=100) Warning in cor(v, L): the standard deviation is zero
Warning in cor(v, L): the standard deviation is zero
Warning in cor(v, L): the standard deviation is zero fastICA: 41% (9ms) AltProj: 9% (5ms)Simulation 4: 9 non-overlapping groups
Nine equal non-overlapping groups of 25 (n=225). Using n.comp=20.
n <- 225; p <- 1000
set.seed(1)
L <- matrix(0, nrow = n, ncol = 9)
for (i in 1:9) { L[((i - 1) * 25 + 1):(i * 25), i] <- 1 }
FF <- matrix(rnorm(p * 9), nrow = p)
X <- L %*% t(FF) + rnorm(n * p, 0, 0.01)
res4 <- run_comparison(X, L, n.comp = 20, label = "9 non-overlapping groups (n=225)")Running fastICA for: 9 non-overlapping groups (n=225)
Running alt-proj for: 9 non-overlapping groups (n=225) Warning in cor(v, L): the standard deviation is zero fastICA: 24% (16ms) AltProj: 2% (11ms)Simulation 5: Bifurcating tree
A 4-leaf bifurcating tree with 6 binary branches (n=100). Using n.comp=10 (fewer PCs reduces noise, as found in fastica_centered_02.Rmd).
n <- 100; p <- 1000
set.seed(1)
L <- matrix(0, nrow = n, ncol = 6)
L[1:50, 1] <- 1
L[51:100, 2] <- 1
L[1:25, 3] <- 1
L[26:50, 4] <- 1
L[51:75, 5] <- 1
L[76:100, 6] <- 1
FF <- matrix(rnorm(p * 6), nrow = p)
X <- L %*% t(FF) + rnorm(n * p, 0, 0.01)
res5 <- run_comparison(X, L, n.comp = 10, label = "Bifurcating tree (n=100, 6 branches)")Running fastICA for: Bifurcating tree (n=100, 6 branches)
Running alt-proj for: Bifurcating tree (n=100, 6 branches) Warning in cor(v, L): the standard deviation is zero
Warning in cor(v, L): the standard deviation is zero
Warning in cor(v, L): the standard deviation is zero
Warning in cor(v, L): the standard deviation is zero
Warning in cor(v, L): the standard deviation is zero fastICA: 63% (9ms) AltProj: 11% (6ms)Diagnosing the 9 non-overlapping groups case
Is alternating projection finding combinations of multiple groups rather than single groups? We re-run both methods on sim 4, store all 100 result vectors, and examine the absolute correlation of each result with each of the 9 true group columns.
n <- 225; p <- 1000; n_starts <- 100; n_iter <- 50
set.seed(1)
L9 <- matrix(0, nrow = n, ncol = 9)
for (i in 1:9) L9[((i-1)*25+1):(i*25), i] <- 1
FF <- matrix(rnorm(p * 9), nrow = p)
X9 <- L9 %*% t(FF) + rnorm(n * p, 0, 0.01)
X1 <- preprocess(X9, n.comp = 20)
X1_aug <- rbind(rep(1, ncol(X1)), X1)
# fastICA: store all weight vectors
set.seed(1)
W <- matrix(rnorm(nrow(X1_aug) * n_starts), nrow(X1_aug), n_starts)
W <- sweep(W, 2, sqrt(colSums(W^2)), "/")
for (i in seq_len(n_iter)) W <- fastica_update(X1_aug, W)
Lhat_ica <- t(X1_aug) %*% W # n x n_starts
# alt_proj: store all binary vectors
set.seed(1)
V <- sign(matrix(rnorm(n * n_starts), n, n_starts))
V[V == 0] <- 1L
for (i in seq_len(n_iter)) V <- alt_proj_update(X1_aug, V)
Lhat_ap <- V # n x n_starts
# Absolute correlation of each result with each true group: n_starts x 9
cor_ica <- abs(cor(Lhat_ica, L9))
cor_ap <- abs(cor(Lhat_ap, L9))Warning in cor(Lhat_ap, L9): the standard deviation is zeroFor each start, we look at how many groups it correlates with above 0.5 — a single-group solution scores 1, a combination scores 2 or more.
n_groups_hit = function(cor_mat, thresh = 0.5) rowSums(cor_mat > thresh)
tab_ica <- table(n_groups_hit(cor_ica))
tab_ap <- table(n_groups_hit(cor_ap))
cat("fastICA — number of groups with |cor| > 0.5:\n"); print(tab_ica)fastICA <U+2014> number of groups with |cor| > 0.5:
0 1 2
67 24 9 cat("AltProj — number of groups with |cor| > 0.5:\n"); print(tab_ap)AltProj <U+2014> number of groups with |cor| > 0.5:
0 1 2 3
82 2 12 3 We can also check whether the fastICA objective is higher for single-group solutions than for combos, using the weight vectors W already computed.
# fastICA objective for each start
P <- t(X1_aug) %*% W # n x n_starts
obj_ica <- colMeans(log(cosh(P)))
n_hit <- n_groups_hit(cor_ica)
boxplot(obj_ica ~ n_hit,
xlab = "Number of groups with |cor| > 0.5",
ylab = "fastICA objective mean(log cosh(Xw))",
main = "fastICA objective by solution type (9 non-overlapping groups)")
Heatmap of absolute correlations for each start (rows = starts sorted by best single-group correlation, columns = true groups):
par(mfrow = c(1, 2))
order_ica <- order(apply(cor_ica, 1, max), decreasing = TRUE)
image(t(cor_ica[order_ica, ]), zlim = c(0, 1), axes = FALSE,
main = "fastICA", xlab = "True group", ylab = "Start (sorted)")
axis(1, at = seq(0, 1, length.out = 9), labels = 1:9)
order_ap <- order(apply(cor_ap, 1, max), decreasing = TRUE)
image(t(cor_ap[order_ap, ]), zlim = c(0, 1), axes = FALSE,
main = "AltProj", xlab = "True group", ylab = "Start (sorted)")
axis(1, at = seq(0, 1, length.out = 9), labels = 1:9)
par(mfrow = c(1, 1))Summary
results <- data.frame(
Simulation = c(
"3 overlapping groups (n=100)",
"9 overlapping groups (n=100)",
"4 non-overlapping groups (n=100)",
"9 non-overlapping groups (n=225)",
"Bifurcating tree (n=100, 6 branches)"
),
n.comp = c(10, 10, 10, 20, 10),
fastICA_pct = round(100 * c(res1["fastICA"], res2["fastICA"], res3["fastICA"],
res4["fastICA"], res5["fastICA"])),
fastICA_ms = round(c(res1["fastICA_ms.elapsed"], res2["fastICA_ms.elapsed"],
res3["fastICA_ms.elapsed"], res4["fastICA_ms.elapsed"],
res5["fastICA_ms.elapsed"])),
AltProj_pct = round(100 * c(res1["AltProj"], res2["AltProj"], res3["AltProj"],
res4["AltProj"], res5["AltProj"])),
AltProj_ms = round(c(res1["AltProj_ms.elapsed"], res2["AltProj_ms.elapsed"],
res3["AltProj_ms.elapsed"], res4["AltProj_ms.elapsed"],
res5["AltProj_ms.elapsed"]))
)
knitr::kable(results,
col.names = c("Simulation", "n.comp",
"fastICA (% success)", "fastICA (ms)",
"AltProj (% success)", "AltProj (ms)"),
caption = "Fraction of 100 random starts recovering a solution with max correlation > 0.95 with any true group column, with wall-clock time in milliseconds."
)| Simulation | n.comp | fastICA (% success) | fastICA (ms) | AltProj (% success) | AltProj (ms) |
|---|---|---|---|---|---|
| 3 overlapping groups (n=100) | 10 | 68 | 16 | 13 | 10 |
| 9 overlapping groups (n=100) | 10 | 84 | 8 | 17 | 5 |
| 4 non-overlapping groups (n=100) | 10 | 41 | 9 | 9 | 5 |
| 9 non-overlapping groups (n=225) | 20 | 24 | 16 | 2 | 11 |
| Bifurcating tree (n=100, 6 branches) | 10 | 63 | 9 | 11 | 6 |
sessionInfo()R version 4.4.2 (2024-10-31)
Platform: aarch64-apple-darwin20
Running under: macOS 26.5.2
Matrix products: default
BLAS: /System/Library/Frameworks/Accelerate.framework/Versions/A/Frameworks/vecLib.framework/Versions/A/libBLAS.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.12.0
locale:
[1] C
time zone: America/Chicago
tzcode source: internal
attached base packages:
[1] stats graphics grDevices utils datasets methods base
loaded via a namespace (and not attached):
[1] vctrs_0.7.2 cli_3.6.5 knitr_1.51 rlang_1.1.7
[5] xfun_0.56 stringi_1.8.7 otel_0.2.0 promises_1.5.0
[9] jsonlite_2.0.0 workflowr_1.7.2 glue_1.8.0 rprojroot_2.1.1
[13] git2r_0.36.2 htmltools_0.5.9 httpuv_1.6.16 sass_0.4.10
[17] rmarkdown_2.30 evaluate_1.0.5 jquerylib_0.1.4 tibble_3.3.1
[21] fastmap_1.2.0 yaml_2.3.12 lifecycle_1.0.5 whisker_0.4.1
[25] stringr_1.6.0 compiler_4.4.2 fs_1.6.6 Rcpp_1.1.1
[29] pkgconfig_2.0.3 later_1.4.6 digest_0.6.39 R6_2.6.1
[33] pillar_1.11.1 magrittr_2.0.4 bslib_0.10.0 tools_4.4.2
[37] cachem_1.1.0