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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:
Two diagonal Hessian approximations are supported, following the notation in ebproj_newton:
fastICA: \(\tilde H = \overline{M'(x)}\,\mathbf{I}\) — uniform sample weights.
trace: \(\tilde H = c_\text{trace}\,\mathbf{I}\) where \(c_\text{trace} = \tfrac{1}{k}\sum_i M'(x_i)\,S_{ii}\) and \(S_{ii} = \tfrac{1}{n}\|Y_{:i}\|^2\) is the squared distance of sample \(i\) from the origin (with \(\sum_i S_{ii} = k\) for whitened data).
Both reduce to the same Newton fixed-point update structure: \[w \leftarrow \tfrac{1}{n}Y M(x) - \tilde H\,w, \qquad w \leftarrow w / \|w\|\]
fastica_asym_r1 = function(Y, s = 1, anchor = c("M", "golden"),
hess = c("fastICA", "trace"),
tol = 1e-6, max_iter = 500, eps = 0.01,
w_init = NULL) {
anchor = match.arg(anchor)
hess = match.arg(hess)
c = if (anchor == "M") s else s * (sqrt(5)-1)/2
m = nrow(Y); n = ncol(Y)
S_diag = colSums(Y^2) / n # S_ii = ||Y[:,i]||^2 / n (sum = m)
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)
h = if (hess == "trace") sum(sc$Mp * S_diag) / m else mean(sc$Mp)
w = as.vector(Y %*% sc$M) / n - h * 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$maximum
if (1 - abs(sum(w * w_old)) < tol && abs(p - p_old) < tol) break
}
list(w = w, p = p, iter = iter, c = c)
}
Standard log-cosh fastICA for comparison:
fastica_r1update = function(X, w) {
w = w / sqrt(sum(w^2))
P = t(X) %*% w
G = tanh(P); G2 = 1 - tanh(P)^2
w = X %*% G - mean(G2) * ncol(X) * w
w / sqrt(sum(w^2))
}
fastica_r1update_trace = function(X, w, S_diag) {
w = w / sqrt(sum(w^2))
P = t(X) %*% w
G = tanh(P); G2 = 1 - tanh(P)^2
h = sum(G2 * S_diag) / nrow(X) # trace Hessian: Σ G2_i S_ii / k
w = as.vector(X %*% G) / ncol(X) - h * w
w / sqrt(sum(w^2))
}
run_seeds_lc = function(Y, S_true, hess = "fastICA", n_seeds = 100, n_iter = 200) {
maxcor = numeric(n_seeds)
S_diag = colSums(Y^2) / ncol(Y)
for (seed in seq_len(n_seeds)) {
set.seed(seed)
w = rnorm(nrow(Y))
if (hess == "trace") {
for (i in seq_len(n_iter)) w = fastica_r1update_trace(Y, w, S_diag)
} else {
for (i in seq_len(n_iter)) w = fastica_r1update(Y, w)
}
maxcor[seed] = max(abs(cor(t(S_true), t(Y) %*% w)))
}
maxcor
}
run_seeds_asym = function(Y, S_true, anchor, hess = "fastICA", n_seeds = 100) {
maxcor = numeric(n_seeds); ps = numeric(n_seeds)
for (seed in seq_len(n_seeds)) {
set.seed(seed)
res = fastica_asym_r1(Y, anchor = anchor, hess = hess, w_init = rnorm(nrow(Y)))
maxcor[seed] = max(abs(cor(t(S_true), t(Y) %*% res$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, hess = "fastICA", 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, hess = hess, 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")
Warning! The custom fig.path you set was ignored by workflowr.
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")
Warning! The custom fig.path you set was ignored by workflowr.
Log-cosh fails completely. The asymmetric golden-ratio anchor with fastICA Hessian succeeds from random starts (95%); the trace Hessian is compared directly alongside it.
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")
Warning! The custom fig.path you set was ignored by workflowr.
asym golden (warm, trace) mean = 0.995 frac > 0.9 = 0.99 mean_p = 0.447
Single source, over-complete whitening (\(k = 20\))
In all tests so far the whitening dimension \(k\) matched the number of true sources. Here we use \(k = 20\) whitened components to represent a single source — the “over-complete” regime from ebproj_newton.
With \(k = 20\) the true source direction occupies only one of the 20 whitened dimensions. The samples where the source is “on” have larger \(S_{ii} = \|Y_{:i}\|^2/n\) than “off” samples (their projection onto the leading singular vector is large), while those same “on” samples have near-zero \(M'(x_i)\) (the posterior is saturated). The trace Hessian therefore down-weights “on” samples relative to “off” samples, which may give a different curvature estimate than the fastICA (isotropic) version.
The \(p = 0.5\) simulation matches ebproj_newton exactly (set.seed(10), \(n = 200\), \(p = 1000\), single mixing vector, Rademacher source, \(k = 20\) whitened components). The \(p = 0.1\) simulation reuses the same mixing vector and dimensions with a sparse binary source.
asym golden (warm, trace) mean = 0.170 frac > 0.9 = 0.06 mean_p = 0.074
All methods fail on this problem: with \(n = 200\) and \(k = 20\) whitened components, the signal-to-noise ratio in any single direction is low and a random \(w\) starts nearly orthogonal to the true source. The asymmetric optimizer also drifts \(\hat p\) far from 0.5 (mean \(\hat p \approx 0.07\)–\(0.09\)) before \(w\) has converged, so it misidentifies the source as highly sparse even when it is symmetric.
asym golden (warm, trace) mean = 0.865 frac > 0.9 = 0.85 mean_p = 0.045
Log-cosh is the best method here on random starts (83%), while the asymmetric fastICA Hessian trails (71%) and trace Hessian is worst (61%). Warm-starting from log-cosh lifts both asymmetric variants to 85%, matching log-cosh.
The trace Hessian gap arises because \[c_{\text{trace}} = \overline{M'(x)} + \frac{n}{k}\,\mathrm{Cov}(M'(x_i),\, S_{ii})\] The “on” samples (\(p = 0.1\), \(\approx 20\) out of \(n=200\)) have small \(M'(x_i) \approx 0\) (posterior saturated) but large \(S_{ii}\) (they lie far from the origin along the source direction), giving \(\text{Cov}(M', S) < 0\). With \(n/k = 200/20 = 10\) this is amplified 10-fold, making \(c_{\text{trace}}\) substantially smaller than \(\overline{M'(x)}\) and destabilising Newton steps from random starts. Warm-starting largely fixes this.
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.
Warning! The custom fig.path you set was ignored by workflowr.
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 \(p\). Two Hessian approximations are compared: fastICA (isotropic, \(\bar{M'(x)}\,\mathbf{I}\)) and trace (weighted by \(S_{ii} = \|Y_{:i}\|^2/n\)). Key findings from 100 random seeds each:
Fraction of seeds achieving max \(|\text{cor}| > 0.9\) (100 seeds):
Setting
lc-fastICA
lc-trace
asym random fastICA
asym random trace
asym warm fastICA
asym warm trace
Sym (\(p=0.5\), \(k=9\))
0.99
1.00
0.77
0.74
0.99
0.99
9 groups (\(p\approx 0.2\), \(k=9\))
0.00
0.00
0.95
0.93
0.65
0.66
1 source (\(p=0.5\), \(k=20\), \(n=200\))
0.06
0.19
0.03
0.04
0.06
0.06
1 source (\(p=0.1\), \(k=20\), \(n=200\))
0.83
0.19
0.71
0.61
0.85
0.85
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. Warm-starting from log-cosh avoids this and fully recovers high success rates.
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 because it can reach the sparse source from a neutral position.
Trace Hessian: helps symmetric, hurts sparse. For symmetric sources (\(p = 0.5\)) the trace correction consistently helps (100% vs 99% at \(k=9\); 19% vs 6% at \(k=20\)). But for sparse sources (\(p = 0.1\), \(k=20\)) it collapses from 83% to 19%. The mechanism is \(n/k\) amplification: \(c_{\text{trace}} = \overline{M'} + (n/k)\,\text{Cov}(M', S)\). For symmetric sources \(\text{Cov}(M', S) \approx 0\), so trace ≈ fastICA but better reflects local geometry; for sparse sources the strong negative covariance (active samples: small \(M'\), large \(S_{ii}\)) is amplified by \(n/k = 10\), shrinking the correction and destabilising Newton steps.
Asymmetry recovery (\(k=1\) whitening): \(\hat p = \min(p, 1-p)\) correctly tracks the true sparse fraction over the range \([0.05, 0.5]\).
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 glue_1.8.0 workflowr_1.7.2 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 tibble_3.3.1 evaluate_1.0.5 jquerylib_0.1.4
[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 pkgconfig_2.0.3
[29] Rcpp_1.1.1 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
