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Standard fastICA maximizes the log-cosh contrast function, which is equivalent to assuming a symmetric binary (Rademacher) prior on the independent components. For sparse or skewed sources (e.g. indicator variables where the “on” fraction \(p \ll 0.5\)), the expected log-cosh contrast falls below the Gaussian baseline, causing fastICA to actively avoid the true source direction.
This analysis generalizes the framework to an asymmetric Rademacher prior parameterized by its up-probability \(p\). We alternate between:
Updating the projection direction \(w\) via a Newton-like fixed-point step (for fixed \(p\)).
Updating \(p\) by maximizing the marginal log-likelihood (for fixed \(w\)).
At \(p = 0.5\) the algorithm exactly reduces to standard log-cosh fastICA. Two scale anchors are compared:
M-estimator (\(c = s\)): robust but slightly over-inflated variance.
Golden-ratio (\(c \approx 0.618\,s\)): satisfies \(c^2 + cs = s^2\), so the assumed generative variance equals the empirical data variance.
Helpers
prewhiten = function(X, n.comp) {
X = X - rowMeans(X)
sqrt(ncol(X)) * t(svd(X)$v[, 1:n.comp])
}
Implementation
With \(\sigma^2 = cs\), the score function and its derivative are:
The Newton-like fixed-point update for \(w\) follows the standard fastICA pattern: \[w \leftarrow \tfrac{1}{n}Y M(x) - \overline{M'(x)}\,w, \qquad
w \leftarrow w / \|w\|\]
fastica_asym_r1 = function(Y, s = 1, anchor = c("M", "golden"),
tol = 1e-6, max_iter = 500, eps = 0.01,
w_init = NULL) {
anchor = match.arg(anchor)
c = if (anchor == "M") s else s * (sqrt(5)-1)/2
m = nrow(Y); n = ncol(Y)
w = if (is.null(w_init)) rnorm(m) else w_init
w = w / sqrt(sum(w^2))
p = 0.5
for (iter in seq_len(max_iter)) {
w_old = w; p_old = p
x = as.vector(t(Y) %*% w)
sc = asym_score(x, p, c, s)
w = as.vector(Y %*% sc$M) / n - mean(sc$Mp) * w
w = w / sqrt(sum(w^2))
x = as.vector(t(Y) %*% w)
opt = optimize(\(pp) asym_obj_p(pp, x, c, s), c(eps, 1-eps), maximum = TRUE)
p = opt$maximum
if (1 - abs(sum(w * w_old)) < tol && abs(p - p_old) < tol) break
}
list(w = w, p = p, iter = iter, c = c)
}
Standard log-cosh fastICA for comparison:
fastica_r1update = function(X, w) {
w = w / sqrt(sum(w^2))
P = t(X) %*% w
G = tanh(P); G2 = 1 - tanh(P)^2
w = X %*% G - mean(G2) * ncol(X) * w
w / sqrt(sum(w^2))
}
run_seeds_lc = function(Y, S_true, n_seeds = 100, n_iter = 200) {
maxcor = numeric(n_seeds)
for (seed in seq_len(n_seeds)) {
set.seed(seed)
w = rnorm(nrow(Y))
for (i in seq_len(n_iter)) w = fastica_r1update(Y, w)
maxcor[seed] = max(abs(cor(t(S_true), t(Y) %*% w)))
}
maxcor
}
run_seeds_asym = function(Y, S_true, anchor, n_seeds = 100) {
maxcor = numeric(n_seeds); ps = numeric(n_seeds)
for (seed in seq_len(n_seeds)) {
set.seed(seed)
res = fastica_asym_r1(Y, anchor = anchor, w_init = rnorm(nrow(Y)))
maxcor[seed] = max(abs(cor(t(S_true), t(Y) %*% res$w)))
ps[seed] = min(res$p, 1 - res$p) # probability of the rare state
}
list(maxcor = maxcor, p = ps)
}
# Warm-start variant: run log-cosh to convergence, then hand off to asymmetric
run_seeds_asym_warm = function(Y, S_true, anchor, n_seeds = 100,
n_iter_lc = 200) {
maxcor = numeric(n_seeds); ps = numeric(n_seeds)
for (seed in seq_len(n_seeds)) {
set.seed(seed)
w = rnorm(nrow(Y))
for (i in seq_len(n_iter_lc)) w = fastica_r1update(Y, w)
res = fastica_asym_r1(Y, anchor = anchor, w_init = w)
maxcor[seed] = max(abs(cor(t(S_true), t(Y) %*% res$w)))
ps[seed] = min(res$p, 1 - res$p)
}
list(maxcor = maxcor, p = ps)
}
Sanity check: \(p = 0.5\) recovers log-cosh
At \(p = 0.5\), \(s = 1\): \(\Delta(x) = 2x\), \(\pi(x) = (1 + \tanh x)/2\), and \(M(x) = \tanh(x)\) — the standard fastICA score.
x_grid = seq(-3, 3, length.out = 300)
sc05 = asym_score(x_grid, p = 0.5, c = 1, s = 1)
plot(x_grid, sc05$M, type = "l", col = "steelblue", lwd = 2,
xlab = "x", ylab = "M(x)",
main = "Score function at p = 0.5 vs tanh(x)")
lines(x_grid, tanh(x_grid), col = "tomato", lty = 2, lwd = 2)
legend("topleft", c("asymmetric M(x), p = 0.5", "tanh(x)"),
col = c("steelblue", "tomato"), lty = c(1, 2), lwd = 2, bty = "n")
Standard fastICA with log-cosh actively avoids sources with $p < $ about \(0.1\) or \(p > 0.9\): the contrast falls below the Gaussian baseline, so the algorithm prefers noise directions over the true source.
Score functions for varying \(p\)
As \(p\) decreases below 0.5 the score shifts and steepens, penalising the positive tail more heavily — appropriate for sources that are rarely “on”.
pvec2 = c(0.05, 0.1, 0.2, 0.3, 0.5)
cols = c("purple", "tomato", "darkorange", "steelblue", "black")
plot(NULL, xlim = c(-3, 3), ylim = c(-2.5, 2.5),
xlab = "x", ylab = "M(x)",
main = "Asymmetric score functions (c = s = 1)")
for (i in seq_along(pvec2))
lines(x_grid, asym_score(x_grid, pvec2[i], 1, 1)$M, col = cols[i], lwd = 2)
legend("topleft", paste0("p = ", pvec2), col = cols, lwd = 2, bty = "n")
abline(h = 0, lty = 3, col = "grey60")
Log-cosh fails completely. The asymmetric golden-ratio anchor succeeds from random starts (95%), but the warm-start variant drops to 65%: log-cosh converges to wrong directions on this problem, and the asymmetric method inherits those bad starting points.
Why does the golden-ratio anchor do better?
The two anchors produce different score functions via the \(c\)-dependent bias term in \(\Delta(x)\). The golden-ratio anchor (\(c = 0.618s\)) satisfies \(c^2 + cs = s^2\), so the total assumed generative variance matches the empirical variance \(s^2\). The M-estimator (\(c = s\)) over-inflates the assumed variance to \(2s^2\), shifting the logistic midpoint and softening the asymmetry penalty.
We can visualise this: at \(p = 0.2\), \(s = 1\), the two anchors produce noticeably different score functions:
x_grid2 = seq(-4, 4, length.out = 400)
sc_M = asym_score(x_grid2, p = 0.2, c = 1, s = 1)
sc_gr = asym_score(x_grid2, p = 0.2, c = (sqrt(5)-1)/2, s = 1)
plot(x_grid2, sc_M$M, type = "l", col = "steelblue", lwd = 2,
xlab = "x", ylab = "M(x)",
main = "Score functions at p = 0.2: M-estimator vs golden-ratio")
lines(x_grid2, sc_gr$M, col = "tomato", lwd = 2)
lines(x_grid2, tanh(x_grid2), col = "grey50", lty = 2, lwd = 1.5)
legend("topleft",
c("M-estimator (c = s)", "golden-ratio (c = 0.618s)", "tanh (p = 0.5)"),
col = c("steelblue", "tomato", "grey50"),
lty = c(1, 1, 2), lwd = c(2, 2, 1.5), bty = "n")
cat(sprintf(" asym golden (random) mean = %.3f frac > 0.9 = %.2f mean_p = %.3f\n",
mean(mc_s_gr$maxcor), mean(mc_s_gr$maxcor > 0.9), mean(mc_s_gr$p)))
asym golden (random) mean = 0.888 frac > 0.9 = 0.77 mean_p = 0.377
cat(sprintf(" asym golden (warm) mean = %.3f frac > 0.9 = %.2f mean_p = %.3f\n",
mean(mc_s_grw$maxcor), mean(mc_s_grw$maxcor > 0.9), mean(mc_s_grw$p)))
asym golden (warm) mean = 0.995 frac > 0.9 = 0.99 mean_p = 0.447
Asymmetry parameter recovery (\(k = 1\))
Using \(k = 1\) whitening (which isolates each source exactly), we verify that the estimated \(\hat p\) tracks the true sparse fraction. Because of sign ambiguity in the ICA direction, we report \(\min(\hat p,\, 1-\hat p)\), i.e. the probability of the rare state.
Both anchors track the true sparse fraction closely across \(p \in [0.05, 0.5]\).
Summary
The asymmetric fastICA algorithm alternates between a Newton-like fixed-point update for \(w\) (identical to standard fastICA at \(p = 0.5\)) and 1D optimization of \(p\). Key findings from 100 random seeds each:
Setting
log-cosh
asym golden (random)
asym golden (warm)
Symmetric Rademacher (\(p=0.5\), \(k=9\))
0.99
0.77
0.99
9 overlapping groups (\(p \approx 0.2\), \(k=9\))
0.00
0.95
0.65
Log-cosh fails completely for the 9-groups case because sparse sources have $E[] < $ Gaussian baseline, causing the algorithm to prefer noise directions.
The golden-ratio anchor (\(c \approx 0.618s\)) dramatically outperforms the M-estimator anchor. Its assumed generative variance matches the empirical variance (\(c^2 + cs = s^2\)), giving a better-calibrated asymmetry penalty.
Warm-starting has opposite effects depending on source type:
Symmetric sources: a random start lets \(p\) drift from 0.5 before \(w\) has converged, degrading performance (77%). Warm-starting from log-cosh avoids this and fully recovers 99% success.
Asymmetric sources: log-cosh converges to a wrong direction (noise PC), and the asymmetric method then inherits that bad start. A random start performs better (95% vs 65%) because it can stumble onto the sparse source from a more neutral initial position.
Asymmetry recovery (\(k=1\) whitening): \(\hat p = \min(p, 1-p)\) correctly tracks the true sparse fraction over the range \([0.05, 0.5]\).
Practical guidance: use random starts when sources are expected to be sparse or asymmetric; use warm starts from log-cosh when sources are expected to be symmetric.
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
