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File Version Author Date Message
Rmd 4276a57 Matthew Stephens 2026-07-30 Explain objective mismatch between asymmetric and log-cosh for symmetric source
html 4276a57 Matthew Stephens 2026-07-30 Explain objective mismatch between asymmetric and log-cosh for symmetric source
Rmd b8658c2 Matthew Stephens 2026-07-29 Add sorted-objective plots for all examples; fix Gaussian baseline text
html b8658c2 Matthew Stephens 2026-07-29 Add sorted-objective plots for all examples; fix Gaussian baseline text
Rmd 636f7d0 Matthew Stephens 2026-07-29 Add sorted-objective plots to assess objective as a source-selection criterion
html 636f7d0 Matthew Stephens 2026-07-29 Add sorted-objective plots to assess objective as a source-selection criterion
Rmd a875fd7 Matthew Stephens 2026-07-29 Clarify log-cosh local-minimum interpretation for p=0.1 nine-groups case
html a875fd7 Matthew Stephens 2026-07-29 Clarify log-cosh local-minimum interpretation for p=0.1 nine-groups case
Rmd 43a0155 Matthew Stephens 2026-07-29 Add p=0.1 to 9-groups and p=0.2 to single-source k=20 tests
html 43a0155 Matthew Stephens 2026-07-29 Add p=0.1 to 9-groups and p=0.2 to single-source k=20 tests
Rmd 9878c1e Matthew Stephens 2026-07-29 Add log-cosh trace Hessian comparison across all tests
html 9878c1e Matthew Stephens 2026-07-29 Add log-cosh trace Hessian comparison across all tests
Rmd 2f935c7 Matthew Stephens 2026-07-29 Match k=20 p=0.5 simulation exactly to ebproj_newton (set.seed(10), n=200, p=1000)
html 2f935c7 Matthew Stephens 2026-07-29 Match k=20 p=0.5 simulation exactly to ebproj_newton (set.seed(10), n=200, p=1000)
Rmd 5df95fa Matthew Stephens 2026-07-29 Add single-source k=20 tests (p=0.5 and p=0.1)
html 5df95fa Matthew Stephens 2026-07-29 Add single-source k=20 tests (p=0.5 and p=0.1)
Rmd 7686e79 Matthew Stephens 2026-07-29 Add trace Hessian approximation to asymmetric fastICA
html 7686e79 Matthew Stephens 2026-07-29 Add trace Hessian approximation to asymmetric fastICA
Rmd f421e5c Matthew Stephens 2026-07-29 Add warm-start comparison to 9-groups test
html f421e5c Matthew Stephens 2026-07-29 Add warm-start comparison to 9-groups test
Rmd d03ae26 Matthew Stephens 2026-07-29 Add warm-start comparison for symmetric sources
html d03ae26 Matthew Stephens 2026-07-29 Add warm-start comparison for symmetric sources
Rmd 008f9d4 Matthew Stephens 2026-07-29 Add EB fastICA with asymmetric prior
html 008f9d4 Matthew Stephens 2026-07-29 Add EB fastICA with asymmetric prior

Introduction

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:

  1. Updating the projection direction \(w\) via a Newton-like fixed-point step (for fixed \(p\)).
  2. 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:

\[M(x) = \frac{y_1}{cs}\,\pi(x) + \frac{y_0}{cs}\,(1-\pi(x)), \qquad M'(x) = \frac{\pi(x)(1-\pi(x))}{s^2\,p(1-p)}\]

where \(\pi(x) = \sigma(\Delta(x))\) is the logistic sigmoid of

\[\Delta(x) = \frac{1}{\sqrt{p(1-p)}}\!\left(\frac{x}{s} - \frac{c\,(1-2p)}{2s\,\sqrt{p(1-p)}}\right) + \log\frac{p}{1-p}\]

and \(y_1 = c\sqrt{(1-p)/p} > 0\), \(y_0 = -c\sqrt{p/(1-p)} < 0\) are the prior support points.

asym_score = function(x, p, c, s) {
  y1     = c * sqrt((1-p)/p)
  y0     = -c * sqrt(p/(1-p))
  kappa3 = (1 - 2*p) / sqrt(p*(1-p))
  Delta  = (x/s - c*kappa3/(2*s)) / sqrt(p*(1-p)) + log(p/(1-p))
  pi_x   = plogis(Delta)
  list(
    M  = (y1/(c*s)) * pi_x + (y0/(c*s)) * (1 - pi_x),
    Mp = pi_x * (1 - pi_x) / (s^2 * p * (1-p))
  )
}

The marginal log-likelihood in \(p\) for fixed projections \(x = Yw\):

\[J(p) = \frac{1}{n}\sum_{i=1}^n \log\!\left( p\,e^{\,x_i y_1/(cs)\,-\,y_1^2/(2cs)} + (1-p)\,e^{\,x_i y_0/(cs)\,-\,y_0^2/(2cs)}\right)\]

asym_obj_p = function(p, x, c, s) {
  y1  = c * sqrt((1-p)/p)
  y0  = -c * sqrt(p/(1-p))
  a1  = x * y1/(c*s) - y1^2/(2*c*s)
  a0  = x * y0/(c*s) - y0^2/(2*c*s)
  lp  = log(p); l1p = log(1-p)
  m   = pmax(lp + a1, l1p + a0)
  mean(m + log(exp(lp + a1 - m) + exp(l1p + a0 - m)))
}

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)
  objs   = 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)))
    objs[seed]   = mean(log(cosh(as.vector(t(Y) %*% w))))
  }
  structure(maxcor, obj = objs)   # vector; objs accessible via attr(., "obj")
}

run_seeds_asym = function(Y, S_true, anchor, hess = "fastICA", n_seeds = 100) {
  maxcor = numeric(n_seeds); ps = numeric(n_seeds); objs = 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$p)
    objs[seed]   = asym_obj_p(res$p, as.vector(t(Y) %*% res$w), res$c, 1)
  }
  list(maxcor = maxcor, p = ps, obj = objs)
}

# 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); objs = 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$p)
    objs[seed]   = asym_obj_p(res$p, as.vector(t(Y) %*% res$w), res$c, 1)
  }
  list(maxcor = maxcor, p = ps, obj = objs)
}

# Sort by |obj − Gaussian baseline| (most extreme first), colour by success
plot_obj_lc = function(lc_result, title, lc_gauss) {
  objs = attr(lc_result, "obj")
  succ = lc_result > 0.9
  ord  = order(abs(objs - lc_gauss), decreasing = TRUE)
  cols = ifelse(succ[ord], "steelblue", "tomato")
  plot(seq_along(objs), objs[ord],
       pch = 19, cex = 0.7, col = cols,
       xlab = "rank (1 = most extreme log-cosh)",
       ylab = "log-cosh objective",
       main = paste0(title, "\n(blue = found true source, red = not)"))
  abline(h = lc_gauss, lty = 2, col = "grey50")
  legend("topright", c("true source", "other", "Gaussian baseline"),
         pch = c(19, 19, NA), lty = c(NA, NA, 2),
         col = c("steelblue", "tomato", "grey50"), bty = "n", cex = 0.8)
}

plot_obj_asym = function(asym_result, title) {
  objs = asym_result$obj
  succ = asym_result$maxcor > 0.9
  ord  = order(objs, decreasing = TRUE)
  cols = ifelse(succ[ord], "steelblue", "tomato")
  plot(seq_along(objs), objs[ord],
       pch = 19, cex = 0.7, col = cols,
       xlab = "rank (1 = highest asymmetric objective)",
       ylab = "asymmetric log-likelihood",
       main = paste0(title, "\n(blue = found true source, red = not)"))
  legend("topright", c("true source", "other"),
         pch = 19, col = c("steelblue", "tomato"), bty = "n", cex = 0.8)
}

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")

Version Author Date
43a0155 Matthew Stephens 2026-07-29
008f9d4 Matthew Stephens 2026-07-29

The curves are numerically identical.

Theoretical motivation

When does log-cosh fail? For a standardised binary source \((\text{Bernoulli}(p)\), zero mean, unit variance), the expected log-cosh contrast is:

expected_logcosh = function(p) {
  z1 =  sqrt((1-p)/p); z0 = -sqrt(p/(1-p))
  p * log(cosh(z1)) + (1-p) * log(cosh(z0))
}
lc_stable = function(z) abs(z) + log1p(exp(-2*abs(z))) - log(2)
lc_gauss  = integrate(\(z) lc_stable(z) * dnorm(z), -Inf, Inf)$value

pvec = seq(0.01, 0.99, by = 0.01)
lc   = sapply(pvec, expected_logcosh)

plot(pvec, lc, type = "l", col = "steelblue", lwd = 2,
     xlab = "p  (probability of positive state)",
     ylab = "E[log cosh(x)]",
     main = "Log-cosh contrast vs Gaussian baseline")
abline(h = lc_gauss, lty = 2, col = "grey40")
legend("top", c("E[log cosh], binary source", "Gaussian baseline"),
       col = c("steelblue", "grey40"), lty = c(1, 2), lwd = 2, bty = "n")

Version Author Date
43a0155 Matthew Stephens 2026-07-29
008f9d4 Matthew Stephens 2026-07-29

Standard fastICA with log-cosh actively avoids sources with $p < $ about \(0.25\) or \(p > 0.75\): the contrast falls below the Gaussian baseline, so the global maximum of \(E[\log\cosh]\) on the sphere is not at the true source. However, the true source direction can still be a local minimum, and the fixed-point iteration can converge there — especially when \(p\) is small enough to create a deep, distinctive basin (see the \(p=0.1\) nine-groups result below).

Asymmetric objective for \(p_\text{true} = 0.2\)

Unlike log-cosh, the asymmetric objective can be evaluated at the optimal test \(p\) matching the true source, placing it above the Gaussian baseline.

set.seed(42)
n_pop  = 100000
c_gr   = (sqrt(5) - 1) / 2   # golden-ratio anchor
z_02   = ifelse(runif(n_pop) < 0.2, sqrt(0.8/0.2), -sqrt(0.2/0.8))
z_N    = rnorm(n_pop)

p_grid      = seq(0.02, 0.98, by = 0.01)
obj_asym_02 = sapply(p_grid, \(p) asym_obj_p(p, z_02, c_gr, 1))
obj_asym_N  = sapply(p_grid, \(p) asym_obj_p(p, z_N,  c_gr, 1))

plot(p_grid, obj_asym_02, type = "l", col = "steelblue", lwd = 2,
     xlab = "test p  (optimization variable)",
     ylab = "E[asymmetric log-likelihood]",
     main = "Asymmetric objective: source p_true = 0.2  (golden-ratio anchor)")
lines(p_grid, obj_asym_N, col = "grey40", lty = 2, lwd = 2)
abline(v = 0.2, lty = 3, col = "tomato", lwd = 1.5)
legend("topright",
       c("source (p_true = 0.2)", "Gaussian source (varying p_test)", "p = 0.2"),
       col = c("steelblue", "grey40", "tomato"),
       lty = c(1, 2, 3), lwd = c(2, 2, 1.5), bty = "n")

Version Author Date
43a0155 Matthew Stephens 2026-07-29
008f9d4 Matthew Stephens 2026-07-29

The grey dashed curve shows the same asymmetric objective evaluated on a Gaussian source as \(p_\text{test}\) varies — it serves as a reference, not a constant horizontal baseline. The blue curve (true \(p = 0.2\) source) peaks near \(p_\text{test} = 0.2\) and lies above the Gaussian reference curve in that region — the optimizer can successfully identify the source by maximising over \(p\). At \(p_\text{test} = 0.5\) the asymmetric objective reduces to the log-cosh contrast, and the blue curve dips below the grey reference, explaining why standard fastICA fails for this 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")

9 overlapping groups (\(k = 9\))

Nine sparse binary sources, each active in some fraction of 100 samples, whitened to \(k = 9\).

\(p \approx 0.2\) (20/100 active)

set.seed(1)
n = 100; p_dim = 1000; K = 9
L  = matrix(0, nrow = n, ncol = K)
for (i in 1:K) L[sample(n, 20), i] = 1
FF = matrix(rnorm(p_dim * K), nrow = p_dim)
X9 = t(L %*% t(FF) + matrix(rnorm(n * p_dim, 0, 0.1), nrow = n))
Z9 = prewhiten(X9, K)
S9 = t(L)
mc9_lc      = run_seeds_lc(Z9,       S9, hess = "fastICA", n_seeds = 100)
mc9_lc_tr   = run_seeds_lc(Z9,       S9, hess = "trace",   n_seeds = 100)
mc9_M       = run_seeds_asym(Z9,      S9, "M",      hess = "fastICA", n_seeds = 100)
mc9_gr      = run_seeds_asym(Z9,      S9, "golden", hess = "fastICA", n_seeds = 100)
mc9_gr_tr   = run_seeds_asym(Z9,      S9, "golden", hess = "trace",   n_seeds = 100)
mc9_grw     = run_seeds_asym_warm(Z9, S9, "golden", hess = "fastICA", n_seeds = 100)
mc9_grw_tr  = run_seeds_asym_warm(Z9, S9, "golden", hess = "trace",   n_seeds = 100)

cat("9-groups (p ~ 0.2, k = 9, n_seeds = 100):\n")
9-groups (p ~ 0.2, k = 9, n_seeds = 100):
cat(sprintf("  log-cosh (fastICA)            mean = %.3f   frac > 0.9 = %.2f\n",
    mean(mc9_lc), mean(mc9_lc > 0.9)))
  log-cosh (fastICA)            mean = 0.590   frac > 0.9 = 0.00
cat(sprintf("  log-cosh (trace)              mean = %.3f   frac > 0.9 = %.2f\n",
    mean(mc9_lc_tr), mean(mc9_lc_tr > 0.9)))
  log-cosh (trace)              mean = 0.588   frac > 0.9 = 0.00
cat(sprintf("  asym M-est (random, fastICA)  mean = %.3f   frac > 0.9 = %.2f   mean_p = %.3f\n",
    mean(mc9_M$maxcor),      mean(mc9_M$maxcor      > 0.9), mean(mc9_M$p)))
  asym M-est (random, fastICA)  mean = 0.877   frac > 0.9 = 0.68   mean_p = 0.162
cat(sprintf("  asym golden (random, fastICA) mean = %.3f   frac > 0.9 = %.2f   mean_p = %.3f\n",
    mean(mc9_gr$maxcor),     mean(mc9_gr$maxcor     > 0.9), mean(mc9_gr$p)))
  asym golden (random, fastICA) mean = 0.983   frac > 0.9 = 0.95   mean_p = 0.135
cat(sprintf("  asym golden (random, trace)   mean = %.3f   frac > 0.9 = %.2f   mean_p = %.3f\n",
    mean(mc9_gr_tr$maxcor),  mean(mc9_gr_tr$maxcor  > 0.9), mean(mc9_gr_tr$p)))
  asym golden (random, trace)   mean = 0.974   frac > 0.9 = 0.93   mean_p = 0.142
cat(sprintf("  asym golden (warm, fastICA)   mean = %.3f   frac > 0.9 = %.2f   mean_p = %.3f\n",
    mean(mc9_grw$maxcor),    mean(mc9_grw$maxcor    > 0.9), mean(mc9_grw$p)))
  asym golden (warm, fastICA)   mean = 0.876   frac > 0.9 = 0.65   mean_p = 0.179
cat(sprintf("  asym golden (warm, trace)     mean = %.3f   frac > 0.9 = %.2f   mean_p = %.3f\n",
    mean(mc9_grw_tr$maxcor), mean(mc9_grw_tr$maxcor > 0.9), mean(mc9_grw_tr$p)))
  asym golden (warm, trace)     mean = 0.879   frac > 0.9 = 0.66   mean_p = 0.180

Sorting runs by their objective reveals whether the objective alone can select true sources — making the raw success rate less critical:

par(mfrow = c(1, 2))
plot_obj_lc(mc9_lc,  "log-cosh, 9-groups p≈0.2",  lc_gauss)
plot_obj_asym(mc9_gr, "asymmetric, 9-groups p≈0.2")

Version Author Date
43a0155 Matthew Stephens 2026-07-29
008f9d4 Matthew Stephens 2026-07-29 
par(mfrow = c(1, 1))

\(p \approx 0.1\) (10/100 active)

set.seed(2)
L_01  = matrix(0, nrow = n, ncol = K)
for (i in 1:K) L_01[sample(n, 10), i] = 1
FF_01 = matrix(rnorm(p_dim * K), nrow = p_dim)
X9_01 = t(L_01 %*% t(FF_01) + matrix(rnorm(n * p_dim, 0, 0.1), nrow = n))
Z9_01 = prewhiten(X9_01, K)
S9_01 = t(L_01)
mc9_01_lc     = run_seeds_lc(Z9_01,       S9_01, hess = "fastICA", n_seeds = 100)
mc9_01_lc_tr  = run_seeds_lc(Z9_01,       S9_01, hess = "trace",   n_seeds = 100)
mc9_01_gr     = run_seeds_asym(Z9_01,      S9_01, "golden", hess = "fastICA", n_seeds = 100)
mc9_01_gr_tr  = run_seeds_asym(Z9_01,      S9_01, "golden", hess = "trace",   n_seeds = 100)
mc9_01_grw    = run_seeds_asym_warm(Z9_01, S9_01, "golden", hess = "fastICA", n_seeds = 100)
mc9_01_grw_tr = run_seeds_asym_warm(Z9_01, S9_01, "golden", hess = "trace",   n_seeds = 100)

cat("9-groups (p ~ 0.1, k = 9, n_seeds = 100):\n")
9-groups (p ~ 0.1, k = 9, n_seeds = 100):
cat(sprintf("  log-cosh (fastICA)            mean = %.3f   frac > 0.9 = %.2f\n",
    mean(mc9_01_lc), mean(mc9_01_lc > 0.9)))
  log-cosh (fastICA)            mean = 0.937   frac > 0.9 = 0.82
cat(sprintf("  log-cosh (trace)              mean = %.3f   frac > 0.9 = %.2f\n",
    mean(mc9_01_lc_tr), mean(mc9_01_lc_tr > 0.9)))
  log-cosh (trace)              mean = 0.551   frac > 0.9 = 0.16
cat(sprintf("  asym golden (random, fastICA) mean = %.3f   frac > 0.9 = %.2f   mean_p = %.3f\n",
    mean(mc9_01_gr$maxcor),     mean(mc9_01_gr$maxcor     > 0.9), mean(mc9_01_gr$p)))
  asym golden (random, fastICA) mean = 0.993   frac > 0.9 = 0.98   mean_p = 0.051
cat(sprintf("  asym golden (random, trace)   mean = %.3f   frac > 0.9 = %.2f   mean_p = %.3f\n",
    mean(mc9_01_gr_tr$maxcor),  mean(mc9_01_gr_tr$maxcor  > 0.9), mean(mc9_01_gr_tr$p)))
  asym golden (random, trace)   mean = 0.980   frac > 0.9 = 0.94   mean_p = 0.055
cat(sprintf("  asym golden (warm, fastICA)   mean = %.3f   frac > 0.9 = %.2f   mean_p = %.3f\n",
    mean(mc9_01_grw$maxcor),    mean(mc9_01_grw$maxcor    > 0.9), mean(mc9_01_grw$p)))
  asym golden (warm, fastICA)   mean = 1.000   frac > 0.9 = 1.00   mean_p = 0.050
cat(sprintf("  asym golden (warm, trace)     mean = %.3f   frac > 0.9 = %.2f   mean_p = %.3f\n",
    mean(mc9_01_grw_tr$maxcor), mean(mc9_01_grw_tr$maxcor > 0.9), mean(mc9_01_grw_tr$p)))
  asym golden (warm, trace)     mean = 1.000   frac > 0.9 = 1.00   mean_p = 0.050

Log-cosh fastICA succeeds 82% of the time at \(p \approx 0.1\) despite having $E[(z)] < $ Gaussian baseline. Since the fixed-point iteration converges to any stationary point (maxima or minima) of \(E[G(w^\top x)]\) on the unit sphere, the algorithm may be finding the true source direction as a local minimum. We check this directly.

lc_obj_01 = attr(mc9_01_lc, "obj")   # already computed inside run_seeds_lc
succ = mc9_01_lc > 0.9
cat(sprintf("Gaussian baseline: %.4f\n", lc_gauss))
Gaussian baseline: 0.3746
cat(sprintf("Successful seeds (n=%d): mean log-cosh obj = %.4f  [%s baseline]\n",
    sum(succ),  mean(lc_obj_01[succ]),
    ifelse(mean(lc_obj_01[succ])  < lc_gauss, "BELOW", "ABOVE")))
Successful seeds (n=82): mean log-cosh obj = 0.2801  [BELOW baseline]
cat(sprintf("Failed seeds    (n=%d): mean log-cosh obj = %.4f  [%s baseline]\n",
    sum(!succ), mean(lc_obj_01[!succ]),
    ifelse(mean(lc_obj_01[!succ]) < lc_gauss, "BELOW", "ABOVE")))
Failed seeds    (n=18): mean log-cosh obj = 0.2847  [BELOW baseline]

Both successful and failed seeds land below the Gaussian baseline, confirming that the fixed-point iteration finds local minima in all cases for this dataset. With 9 very sparse sources (\(p = 0.1\)), every direction in the whitened space has $E[] < $ Gaussian baseline — the sparse structure dominates everywhere. The true source directions are the deepest minima because their large active-sample values (\(y_1 \approx 3\)) lower the log-cosh average maximally.

The practical question is whether the objective value alone can select true sources, making the raw success rate less critical:

par(mfrow = c(1, 2))
plot_obj_lc(mc9_01_lc,  "log-cosh, 9-groups p≈0.1",  lc_gauss)
plot_obj_asym(mc9_01_gr, "asymmetric, 9-groups p≈0.1")

par(mfrow = c(1, 1))

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")

Sanity check: symmetric Rademacher (\(p = 0.5\), \(k = 9\))

Both methods should succeed here; the asymmetric algorithm should recover \(\hat p \approx 0.5\) automatically. We compare three variants:

  • log-cosh: standard fastICA.
  • asym golden (random start): alternating optimization from a random \(w\).
  • asym golden (warm start): log-cosh run to convergence first, then hand the resulting \(w\) to the asymmetric optimizer.

The warm start tests whether the degradation seen with random starts is purely an initialization issue.

set.seed(2)
S_sym = matrix(sample(c(-1, 1), K * n, replace = TRUE), nrow = K)
X_sym = t(S_sym) %*% t(FF) + matrix(rnorm(n * p_dim, 0, 0.1), nrow = n)
Z_sym = prewhiten(t(X_sym), K)

mc_s_lc      = run_seeds_lc(Z_sym,       S_sym, hess = "fastICA", n_seeds = 100)
mc_s_lc_tr   = run_seeds_lc(Z_sym,       S_sym, hess = "trace",   n_seeds = 100)
mc_s_gr      = run_seeds_asym(Z_sym,      S_sym, "golden", hess = "fastICA", n_seeds = 100)
mc_s_gr_tr   = run_seeds_asym(Z_sym,      S_sym, "golden", hess = "trace",   n_seeds = 100)
mc_s_grw     = run_seeds_asym_warm(Z_sym, S_sym, "golden", hess = "fastICA", n_seeds = 100)
mc_s_grw_tr  = run_seeds_asym_warm(Z_sym, S_sym, "golden", hess = "trace",   n_seeds = 100)

cat("Symmetric Rademacher (p_true = 0.5, k = 9):\n")
Symmetric Rademacher (p_true = 0.5, k = 9):
cat(sprintf("  log-cosh (fastICA)            mean = %.3f   frac > 0.9 = %.2f\n",
    mean(mc_s_lc), mean(mc_s_lc > 0.9)))
  log-cosh (fastICA)            mean = 0.995   frac > 0.9 = 0.99
cat(sprintf("  log-cosh (trace)              mean = %.3f   frac > 0.9 = %.2f\n",
    mean(mc_s_lc_tr), mean(mc_s_lc_tr > 0.9)))
  log-cosh (trace)              mean = 1.000   frac > 0.9 = 1.00
cat(sprintf("  asym golden (random, fastICA) 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$p)))
  asym golden (random, fastICA) mean = 0.888   frac > 0.9 = 0.77   mean_p = 0.377
cat(sprintf("  asym golden (random, trace)   mean = %.3f   frac > 0.9 = %.2f   mean_p = %.3f\n",
    mean(mc_s_gr_tr$maxcor),  mean(mc_s_gr_tr$maxcor  > 0.9), mean(mc_s_gr_tr$p)))
  asym golden (random, trace)   mean = 0.871   frac > 0.9 = 0.74   mean_p = 0.368
cat(sprintf("  asym golden (warm, fastICA)   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$p)))
  asym golden (warm, fastICA)   mean = 0.995   frac > 0.9 = 0.99   mean_p = 0.447
cat(sprintf("  asym golden (warm, trace)     mean = %.3f   frac > 0.9 = %.2f   mean_p = %.3f\n",
    mean(mc_s_grw_tr$maxcor), mean(mc_s_grw_tr$maxcor > 0.9), mean(mc_s_grw_tr$p)))
  asym golden (warm, trace)     mean = 0.995   frac > 0.9 = 0.99   mean_p = 0.447
par(mfrow = c(1, 2))
plot_obj_lc(mc_s_lc,  "log-cosh, 9-groups p=0.5",  lc_gauss)
plot_obj_asym(mc_s_gr, "asymmetric, 9-groups p=0.5")

par(mfrow = c(1, 1))

Are the two objectives comparable? At \(p = 0.5\) the asymmetric score function reduces to \(\tanh(x/s)/s\), the same as log-cosh, so the fixed-point update for \(w\) is identical to log-cosh. However the objective value differs by a constant offset: \[\text{asym\_obj}(p{=}0.5,\, x,\, c,\, s) = \overline{\log\cosh(x/s)} - \tfrac{c}{2s}\] For the golden-ratio anchor (\(c \approx 0.618\), \(s = 1\)) the offset is \(-c/2 \approx -0.309\).

succ_s   = mc_s_gr$maxcor > 0.9
c_gr     = (sqrt(5) - 1) / 2
lc_obj_s = attr(mc_s_lc, "obj")

cat(sprintf("Successful seeds (n=%d): mean p = %.3f   mean asym_obj = %.4f\n",
    sum(succ_s),  mean(mc_s_gr$p[succ_s]),  mean(mc_s_gr$obj[succ_s])))
Successful seeds (n=77): mean p = 0.445   mean asym_obj = 0.1268
cat(sprintf("Failed seeds    (n=%d): mean p = %.3f   mean asym_obj = %.4f\n",
    sum(!succ_s), mean(mc_s_gr$p[!succ_s]), mean(mc_s_gr$obj[!succ_s])))
Failed seeds    (n=23): mean p = 0.148   mean asym_obj = 0.1216
# Fraction of top-K runs (by asym_obj) that found a true source
for (K_top in c(10, 20, 50)) {
  top_K = order(mc_s_gr$obj, decreasing = TRUE)[1:K_top]
  cat(sprintf("Fraction of top-%d asym_obj runs that are true sources: %.2f\n",
      K_top, mean(succ_s[top_K])))
}
Fraction of top-10 asym_obj runs that are true sources: 0.60
Fraction of top-20 asym_obj runs that are true sources: 0.60
Fraction of top-50 asym_obj runs that are true sources: 0.84
# Theoretical offset at p = 0.5: asym_obj = lc_obj - c/2
cat(sprintf("\nTheoretical offset (asym at p=0.5) - lc_obj = -c/2 = %.4f\n", -c_gr/2))

Theoretical offset (asym at p=0.5) - lc_obj = -c/2 = -0.3090
cat(sprintf("Observed offset for successful seeds: %.4f  (p ≠ 0.5, so not exact)\n",
    mean(mc_s_gr$obj[succ_s] - lc_obj_s[succ_s])))
Observed offset for successful seeds: -0.3055  (p ≠ 0.5, so not exact)

What is happening:

  • Successful seeds do not converge to exactly \(p = 0.5\). With finite data and approximate symmetry, the optimizer finds \(p \approx 0.4\)–\(0.5\). At exactly \(p = 0.5\) the asymmetric score equals log-cosh, but the joint \((w,p)\) iteration drifts \(p\) slightly away.
  • Failed seeds converge to \(p \approx 0.15\)–\(0.2\) — they found a noise direction that the optimizer interpreted as a sparse source.
  • The mean asymmetric objective IS higher for successful seeds, so the objective has selection power, but some failed seeds (with inflated objectives from their small \(p\)) rank among the top runs. This makes the asymmetric objective a slightly noisier selector than log-cosh for a symmetric true source.

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.

# Exactly as in ebproj_newton Test 1
set.seed(10)
n_k20 = 200; p_k20 = 1000; k20 = 20
A_k20    = matrix(rnorm(p_k20), nrow = p_k20)
S_k20_05 = matrix(sample(c(-1, 1), n_k20, replace = TRUE), nrow = 1)
X_k20_05 = A_k20 %*% S_k20_05 + matrix(rnorm(p_k20 * n_k20, 0, 0.1), nrow = p_k20)
Z_k20_05 = prewhiten(X_k20_05, k20)

# Same mixing vector, sparse binary sources (p = 0.2 and p = 0.1)
set.seed(11)
S_k20_02_raw = matrix(as.numeric(runif(n_k20) < 0.2), nrow = 1)
S_k20_02     = (S_k20_02_raw - 0.2) / sqrt(0.2 * 0.8)
X_k20_02 = A_k20 %*% S_k20_02 + matrix(rnorm(p_k20 * n_k20, 0, 0.1), nrow = p_k20)
Z_k20_02 = prewhiten(X_k20_02, k20)

set.seed(12)
S_k20_01_raw = matrix(as.numeric(runif(n_k20) < 0.1), nrow = 1)
S_k20_01     = (S_k20_01_raw - 0.1) / sqrt(0.1 * 0.9)
X_k20_01 = A_k20 %*% S_k20_01 + matrix(rnorm(p_k20 * n_k20, 0, 0.1), nrow = p_k20)
Z_k20_01 = prewhiten(X_k20_01, k20)

\(p = 0.5\), \(k = 20\)

mc_k20_05_lc     = run_seeds_lc(Z_k20_05,       S_k20_05, hess = "fastICA", n_seeds = 100)
mc_k20_05_lc_tr  = run_seeds_lc(Z_k20_05,       S_k20_05, hess = "trace",   n_seeds = 100)
mc_k20_05_gr     = run_seeds_asym(Z_k20_05,      S_k20_05, "golden", hess = "fastICA", n_seeds = 100)
mc_k20_05_gr_tr  = run_seeds_asym(Z_k20_05,      S_k20_05, "golden", hess = "trace",   n_seeds = 100)
mc_k20_05_grw    = run_seeds_asym_warm(Z_k20_05, S_k20_05, "golden", hess = "fastICA", n_seeds = 100)
mc_k20_05_grw_tr = run_seeds_asym_warm(Z_k20_05, S_k20_05, "golden", hess = "trace",   n_seeds = 100)

cat("Single source, k=20 whitening, p=0.5 (n_seeds=100):\n")
Single source, k=20 whitening, p=0.5 (n_seeds=100):
cat(sprintf("  log-cosh (fastICA)            mean = %.3f   frac > 0.9 = %.2f\n",
    mean(mc_k20_05_lc), mean(mc_k20_05_lc > 0.9)))
  log-cosh (fastICA)            mean = 0.178   frac > 0.9 = 0.06
cat(sprintf("  log-cosh (trace)              mean = %.3f   frac > 0.9 = %.2f\n",
    mean(mc_k20_05_lc_tr), mean(mc_k20_05_lc_tr > 0.9)))
  log-cosh (trace)              mean = 0.321   frac > 0.9 = 0.19
cat(sprintf("  asym golden (random, fastICA) mean = %.3f   frac > 0.9 = %.2f   mean_p = %.3f\n",
    mean(mc_k20_05_gr$maxcor),     mean(mc_k20_05_gr$maxcor     > 0.9), mean(mc_k20_05_gr$p)))
  asym golden (random, fastICA) mean = 0.201   frac > 0.9 = 0.03   mean_p = 0.076
cat(sprintf("  asym golden (random, trace)   mean = %.3f   frac > 0.9 = %.2f   mean_p = %.3f\n",
    mean(mc_k20_05_gr_tr$maxcor),  mean(mc_k20_05_gr_tr$maxcor  > 0.9), mean(mc_k20_05_gr_tr$p)))
  asym golden (random, trace)   mean = 0.224   frac > 0.9 = 0.04   mean_p = 0.085
cat(sprintf("  asym golden (warm, fastICA)   mean = %.3f   frac > 0.9 = %.2f   mean_p = %.3f\n",
    mean(mc_k20_05_grw$maxcor),    mean(mc_k20_05_grw$maxcor    > 0.9), mean(mc_k20_05_grw$p)))
  asym golden (warm, fastICA)   mean = 0.169   frac > 0.9 = 0.06   mean_p = 0.075
cat(sprintf("  asym golden (warm, trace)     mean = %.3f   frac > 0.9 = %.2f   mean_p = %.3f\n",
    mean(mc_k20_05_grw_tr$maxcor), mean(mc_k20_05_grw_tr$maxcor > 0.9), mean(mc_k20_05_grw_tr$p)))
  asym golden (warm, trace)     mean = 0.170   frac > 0.9 = 0.06   mean_p = 0.074
par(mfrow = c(1, 2))
plot_obj_lc(mc_k20_05_lc,  "log-cosh, 1-source k=20 p=0.5",  lc_gauss)
plot_obj_asym(mc_k20_05_gr, "asymmetric, 1-source k=20 p=0.5")

par(mfrow = c(1, 1))

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.

\(p = 0.2\), \(k = 20\)

mc_k20_02_lc     = run_seeds_lc(Z_k20_02,       S_k20_02, hess = "fastICA", n_seeds = 100)
mc_k20_02_lc_tr  = run_seeds_lc(Z_k20_02,       S_k20_02, hess = "trace",   n_seeds = 100)
mc_k20_02_gr     = run_seeds_asym(Z_k20_02,      S_k20_02, "golden", hess = "fastICA", n_seeds = 100)
mc_k20_02_gr_tr  = run_seeds_asym(Z_k20_02,      S_k20_02, "golden", hess = "trace",   n_seeds = 100)
mc_k20_02_grw    = run_seeds_asym_warm(Z_k20_02, S_k20_02, "golden", hess = "fastICA", n_seeds = 100)
mc_k20_02_grw_tr = run_seeds_asym_warm(Z_k20_02, S_k20_02, "golden", hess = "trace",   n_seeds = 100)

cat("Single source, k=20 whitening, p=0.2 (n_seeds=100):\n")
Single source, k=20 whitening, p=0.2 (n_seeds=100):
cat(sprintf("  log-cosh (fastICA)            mean = %.3f   frac > 0.9 = %.2f\n",
    mean(mc_k20_02_lc), mean(mc_k20_02_lc > 0.9)))
  log-cosh (fastICA)            mean = 0.202   frac > 0.9 = 0.00
cat(sprintf("  log-cosh (trace)              mean = %.3f   frac > 0.9 = %.2f\n",
    mean(mc_k20_02_lc_tr), mean(mc_k20_02_lc_tr > 0.9)))
  log-cosh (trace)              mean = 0.164   frac > 0.9 = 0.00
cat(sprintf("  asym golden (random, fastICA) mean = %.3f   frac > 0.9 = %.2f   mean_p = %.3f\n",
    mean(mc_k20_02_gr$maxcor),     mean(mc_k20_02_gr$maxcor     > 0.9), mean(mc_k20_02_gr$p)))
  asym golden (random, fastICA) mean = 0.417   frac > 0.9 = 0.31   mean_p = 0.086
cat(sprintf("  asym golden (random, trace)   mean = %.3f   frac > 0.9 = %.2f   mean_p = %.3f\n",
    mean(mc_k20_02_gr_tr$maxcor),  mean(mc_k20_02_gr_tr$maxcor  > 0.9), mean(mc_k20_02_gr_tr$p)))
  asym golden (random, trace)   mean = 0.350   frac > 0.9 = 0.24   mean_p = 0.091
cat(sprintf("  asym golden (warm, fastICA)   mean = %.3f   frac > 0.9 = %.2f   mean_p = %.3f\n",
    mean(mc_k20_02_grw$maxcor),    mean(mc_k20_02_grw$maxcor    > 0.9), mean(mc_k20_02_grw$p)))
  asym golden (warm, fastICA)   mean = 0.333   frac > 0.9 = 0.17   mean_p = 0.073
cat(sprintf("  asym golden (warm, trace)     mean = %.3f   frac > 0.9 = %.2f   mean_p = %.3f\n",
    mean(mc_k20_02_grw_tr$maxcor), mean(mc_k20_02_grw_tr$maxcor > 0.9), mean(mc_k20_02_grw_tr$p)))
  asym golden (warm, trace)     mean = 0.328   frac > 0.9 = 0.17   mean_p = 0.077
par(mfrow = c(1, 2))
plot_obj_lc(mc_k20_02_lc,  "log-cosh, 1-source k=20 p=0.2",  lc_gauss)
plot_obj_asym(mc_k20_02_gr, "asymmetric, 1-source k=20 p=0.2")

par(mfrow = c(1, 1))

All methods struggle at \(p = 0.2\) in the \(k = 20\) setting. Log-cosh achieves 0% even though it succeeds at \(p = 0.1\). The asymmetric fastICA Hessian reaches 31% from random starts, which is better than the 6% at \(p = 0.5\) — the larger \(y_1\) value (\(\approx 2\)) creates a somewhat more tractable landscape.

\(p = 0.1\), \(k = 20\)

mc_k20_01_lc     = run_seeds_lc(Z_k20_01,       S_k20_01, hess = "fastICA", n_seeds = 100)
mc_k20_01_lc_tr  = run_seeds_lc(Z_k20_01,       S_k20_01, hess = "trace",   n_seeds = 100)
mc_k20_01_gr     = run_seeds_asym(Z_k20_01,      S_k20_01, "golden", hess = "fastICA", n_seeds = 100)
mc_k20_01_gr_tr  = run_seeds_asym(Z_k20_01,      S_k20_01, "golden", hess = "trace",   n_seeds = 100)
mc_k20_01_grw    = run_seeds_asym_warm(Z_k20_01, S_k20_01, "golden", hess = "fastICA", n_seeds = 100)
mc_k20_01_grw_tr = run_seeds_asym_warm(Z_k20_01, S_k20_01, "golden", hess = "trace",   n_seeds = 100)

cat("Single source, k=20 whitening, p=0.1 (n_seeds=100):\n")
Single source, k=20 whitening, p=0.1 (n_seeds=100):
cat(sprintf("  log-cosh (fastICA)            mean = %.3f   frac > 0.9 = %.2f\n",
    mean(mc_k20_01_lc), mean(mc_k20_01_lc > 0.9)))
  log-cosh (fastICA)            mean = 0.774   frac > 0.9 = 0.76
cat(sprintf("  log-cosh (trace)              mean = %.3f   frac > 0.9 = %.2f\n",
    mean(mc_k20_01_lc_tr), mean(mc_k20_01_lc_tr > 0.9)))
  log-cosh (trace)              mean = 0.360   frac > 0.9 = 0.29
cat(sprintf("  asym golden (random, fastICA) mean = %.3f   frac > 0.9 = %.2f   mean_p = %.3f\n",
    mean(mc_k20_01_gr$maxcor),     mean(mc_k20_01_gr$maxcor     > 0.9), mean(mc_k20_01_gr$p)))
  asym golden (random, fastICA) mean = 0.796   frac > 0.9 = 0.78   mean_p = 0.037
cat(sprintf("  asym golden (random, trace)   mean = %.3f   frac > 0.9 = %.2f   mean_p = %.3f\n",
    mean(mc_k20_01_gr_tr$maxcor),  mean(mc_k20_01_gr_tr$maxcor  > 0.9), mean(mc_k20_01_gr_tr$p)))
  asym golden (random, trace)   mean = 0.712   frac > 0.9 = 0.68   mean_p = 0.046
cat(sprintf("  asym golden (warm, fastICA)   mean = %.3f   frac > 0.9 = %.2f   mean_p = %.3f\n",
    mean(mc_k20_01_grw$maxcor),    mean(mc_k20_01_grw$maxcor    > 0.9), mean(mc_k20_01_grw$p)))
  asym golden (warm, fastICA)   mean = 0.799   frac > 0.9 = 0.78   mean_p = 0.034
cat(sprintf("  asym golden (warm, trace)     mean = %.3f   frac > 0.9 = %.2f   mean_p = %.3f\n",
    mean(mc_k20_01_grw_tr$maxcor), mean(mc_k20_01_grw_tr$maxcor > 0.9), mean(mc_k20_01_grw_tr$p)))
  asym golden (warm, trace)     mean = 0.799   frac > 0.9 = 0.78   mean_p = 0.034
par(mfrow = c(1, 2))
plot_obj_lc(mc_k20_01_lc,  "log-cosh, 1-source k=20 p=0.1",  lc_gauss)
plot_obj_asym(mc_k20_01_gr, "asymmetric, 1-source k=20 p=0.1")

par(mfrow = c(1, 1))

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.

set.seed(99)
n_rec = 500; p_dim_rec = 1000
A_rec = matrix(rnorm(p_dim_rec), nrow = p_dim_rec)

p_trues  = c(0.05, 0.10, 0.15, 0.20, 0.30, 0.40, 0.50)
p_hat_M  = numeric(length(p_trues))
p_hat_gr = numeric(length(p_trues))

for (j in seq_along(p_trues)) {
  pt = p_trues[j]
  S  = matrix(as.numeric(runif(n_rec) < pt), nrow = 1)
  S  = (S - pt) / sqrt(pt * (1-pt))
  X  = A_rec %*% S + matrix(rnorm(p_dim_rec * n_rec, 0, 0.1), nrow = p_dim_rec)
  Z  = prewhiten(X, 1)
  set.seed(1)
  r_M  = fastica_asym_r1(Z, anchor = "M",      w_init = rnorm(1))
  r_gr = fastica_asym_r1(Z, anchor = "golden",  w_init = rnorm(1))
  p_hat_M[j]  = min(r_M$p,  1 - r_M$p)
  p_hat_gr[j] = min(r_gr$p, 1 - r_gr$p)
}

plot(p_trues, p_hat_M, pch = 19, col = "steelblue",
     xlim = c(0, 0.52), ylim = c(0, 0.52),
     xlab = "true p  (sparse fraction)",
     ylab = "estimated p  (rare-state probability)",
     main = "Asymmetry parameter recovery  (k = 1 whitening)")
points(p_trues, p_hat_gr, pch = 17, col = "tomato")
abline(0, 1, lty = 2, col = "grey50")
legend("topleft", c("M-estimator", "golden-ratio"),
       col = c("steelblue","tomato"), pch = c(19,17), bty = "n")

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
9 groups (\(p\approx 0.1\), \(k=9\)) 0.82 0.16 0.98 0.94 1.00 1.00
1 source (\(p=0.5\), \(k=20\), \(n=200\)) 0.06 0.19 0.03 0.04 0.06 0.06
1 source (\(p=0.2\), \(k=20\), \(n=200\)) 0.00 0.00 0.31 0.24 0.17 0.17
1 source (\(p=0.1\), \(k=20\), \(n=200\)) 0.76 0.29 0.78 0.68 0.78 0.78
  • 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.
  • Log-cosh finds sparse sources as local minima. \(E[\log\cosh]\) falls below the Gaussian baseline for \(p \lesssim 0.25\), so the true source is a local minimum (not maximum) of the log-cosh contrast on the sphere. The fixed-point iteration converges to stationary points of either sign; for \(p = 0.1\) (\(y_1 \approx 3\), very large active-sample values) the minimum is deep enough that random starts land in its basin 82% (9 groups) or 76% (single source) of the time. For \(p = 0.2\) (\(y_1 = 2\)) the basin is too shallow and success drops to 0%. The asymmetric method avoids this threshold entirely by adapting \(p\) to match the source.
  • 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 it collapses log-cosh from 82% to 16% (9 groups, \(p=0.1\)) and from 76% to 29% (single source, \(k=20\), \(p=0.1\)). The mechanism is \(n/k\) amplification: \(c_{\text{trace}} = \overline{M'} + (n/k)\,\text{Cov}(M', S)\). For symmetric sources \(\text{Cov}(M', S) \approx 0\); for sparse sources the strong negative covariance (active samples: small \(M'\), large \(S_{ii}\)) is amplified by \(n/k\), 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] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

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    jquerylib_0.1.4   evaluate_1.0.5    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   rstudioapi_0.18.0 later_1.4.6       digest_0.6.39    
[33] R6_2.6.1          pillar_1.11.1     magrittr_2.0.4    bslib_0.10.0     
[37] tools_4.4.2       cachem_1.1.0