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Rmd 159d466 Matthew Stephens 2026-07-16 Add tilted log-cosh FastICA analysis

Introduction

This implements and tests the “Tilted Log-Cosh” (TLC) contrast function for FastICA, motivated by the paper skew_fastICA.pdf. The standard log-cosh contrast fails to detect highly skewed (sparse) binary sources because its expectation falls below the Gaussian baseline as \(p \to 0\) or \(p \to 1\). The tilted version adds a \(\lambda z|z|\) term:

\[G(z) = \log(\cosh(z)) + \lambda z|z|\]

with derivatives: \[g(z) = \tanh(z) + 2\lambda|z|, \quad g'(z) = 1 - \tanh^2(z) + 2\lambda \text{sign}(z)\]

The odd term \(z|z|\) has expectation \(2p-1\) for a standardized binary source, so the tilted objective monotonically increases as \(p \to 1\), unlike log-cosh which peaks at \(p=0.5\).

Implementation

# Standard log-cosh FastICA (single component, rank-1 update)
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))
}

compute_objective = function(X, w) {
  P = t(X) %*% w
  mean(log(cosh(P)))
}

# Tilted log-cosh FastICA (single component)
fastica_r1update_tlc = function(X, w, lambda = 1) {
  w = w / sqrt(sum(w^2))
  P = t(X) %*% w
  G  = tanh(P) + 2 * lambda * abs(P)
  G2 = 1 - tanh(P)^2 + 2 * lambda * sign(P)
  w  = X %*% G - mean(G2) * ncol(X) * w
  w / sqrt(sum(w^2))
}

compute_objective_tlc = function(X, w, lambda = 1) {
  P = t(X) %*% w
  mean(log(cosh(P)) + lambda * abs(P) * P)
}

prewhiten = function(X, n.comp) {
  X = X - rowMeans(X)
  sqrt(ncol(X)) * t(svd(X)$v[, 1:n.comp])
}

run_seeds = function(X, S_true, update_fn, obj_fn, n_seeds = 50, n_iter = 200, ...) {
  obj    = numeric(n_seeds)
  maxcor = numeric(n_seeds)
  for (seed in seq_len(n_seeds)) {
    set.seed(seed)
    w = rnorm(nrow(X))
    for (i in seq_len(n_iter))
      w = update_fn(X, w, ...)
    obj[seed]    = obj_fn(X, w, ...)
    maxcor[seed] = max(abs(cor(t(S_true), t(X) %*% w)))
  }
  list(obj = obj, maxcor = maxcor)
}

Show theoretical motivation: log-cosh fails for skewed binary sources

# E[log(cosh(z))] for a standardized binary source with P(x=-1)=p
expected_logcosh = function(p) {
  z_neg =  -sqrt((1 - p) / p)
  z_pos =   sqrt(p / (1 - p))
  p * log(cosh(z_neg)) + (1 - p) * log(cosh(z_pos))
}

expected_tlc = function(p, lambda = 1) {
  expected_logcosh(p) + lambda * (2 * p - 1)
}

log_cosh_stable = function(z) abs(z) + log1p(exp(-2 * abs(z))) - log(2)
gaussian_baseline = integrate(function(z) log_cosh_stable(z) * dnorm(z), -Inf, Inf)$value

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

plot(pvec, lc, type = "l", col = "blue", ylim = range(c(lc, tlc)),
     xlab = "p (prob of -1)", ylab = "E[G(z)]",
     main = "Log-cosh vs Tilted Log-Cosh for binary source")
lines(pvec, tlc, col = "red")
abline(h = gaussian_baseline, lty = 2, col = "gray")
legend("bottomleft", c("log-cosh", "tilted log-cosh (λ=1)", "Gaussian baseline"),
       col = c("blue","red","gray"), lty = c(1,1,2))

Version Author Date
7da05b8 Matthew Stephens 2026-07-16

Standard log-cosh falls below the Gaussian baseline for skewed sources. The tilted version increases monotonically toward the sparse extreme.

Test 1: Symmetric binary source (k=1, p=0.5)

The simplest case: a single Rademacher source (-1 or +1 with equal probability). Both methods should work here since log-cosh peaks at p=0.5.

set.seed(10)
n = 200
p = 1000
k = 1
A = matrix(rnorm(p * k), nrow = p)
S_rad = matrix(sample(c(-1, 1), n, replace = TRUE), nrow = 1)
sigma = 0.1
X_rad = A %*% S_rad + matrix(rnorm(p * n, 0, sigma), nrow = p)
Z_rad = sqrt(n) * t(svd(X_rad)$v[, 1:k, drop = FALSE])

res_lc_rad  = run_seeds(Z_rad, S_rad, fastica_r1update,     compute_objective,     n_seeds = 50)
res_tlc_rad = run_seeds(Z_rad, S_rad, fastica_r1update_tlc, compute_objective_tlc, n_seeds = 50)

cat("log-cosh  : mean max|cor| =", round(mean(res_lc_rad$maxcor),  3),
    " fraction > 0.9:", mean(res_lc_rad$maxcor  > 0.9), "\n")
log-cosh  : mean max|cor| = 1  fraction > 0.9: 1 
cat("tilted lc : mean max|cor| =", round(mean(res_tlc_rad$maxcor), 3),
    " fraction > 0.9:", mean(res_tlc_rad$maxcor > 0.9), "\n")
tilted lc : mean max|cor| = 1  fraction > 0.9: 1

Test 2: Non-overlapping groups (k=4, p=0.25 per source)

Here we have 4 non-overlapping groups of 25 samples each in n=100, so each source is active for 25% of samples (p=0.25). Without noise the 4 groups would perfectly partition all samples, making the centered data rank k-1 and the k-th whitened component the constant direction. The added noise breaks this exact rank deficiency.

set.seed(1)
n = 100
p = 1000
k = 4
A = matrix(rnorm(p * k), nrow = p)
S = matrix(0, nrow = k, ncol = n)
S[1, 1:25]   = 1
S[2, 26:50]  = 1
S[3, 51:75]  = 1
S[4, 76:100] = 1
sigma = 0.1
X  = A %*% S + matrix(rnorm(p * n, 0, sigma), nrow = p)
Z  = prewhiten(X, k)

res_lc  = run_seeds(Z, S, fastica_r1update,     compute_objective)
res_tlc = run_seeds(Z, S, fastica_r1update_tlc, compute_objective_tlc)

par(mfrow = c(1, 2))
hist(res_lc$maxcor,  breaks = seq(0, 1, by = 0.05), main = "log-cosh: max |cor| with true S",  xlab = "")
hist(res_tlc$maxcor, breaks = seq(0, 1, by = 0.05), main = "tilted log-cosh: max |cor| with true S", xlab = "")

Version Author Date
7da05b8 Matthew Stephens 2026-07-16 
par(mfrow = c(1, 1))

Best and worst seeds for tilted log-cosh on Test 2:

get_projection = function(X, seed, update_fn, n_iter = 200, ...) {
  set.seed(seed)
  w = rnorm(nrow(X))
  for (i in seq_len(n_iter))
    w = update_fn(X, w, ...)
  as.vector(t(X) %*% w)
}

best_seed  = which.max(res_tlc$maxcor)
worst_seed = which.min(res_tlc$maxcor)

proj_best  = get_projection(Z, best_seed,  fastica_r1update_tlc)
proj_worst = get_projection(Z, worst_seed, fastica_r1update_tlc)

par(mfrow = c(1, 2))
plot(proj_best,  main = paste0("TLC best seed (", best_seed, "), max|cor|=",
     round(res_tlc$maxcor[best_seed], 3)), xlab = "sample", ylab = "projection")
plot(proj_worst, main = paste0("TLC worst seed (", worst_seed, "), max|cor|=",
     round(res_tlc$maxcor[worst_seed], 3)), xlab = "sample", ylab = "projection")

Version Author Date
7da05b8 Matthew Stephens 2026-07-16 
par(mfrow = c(1, 1))

Test 3: Sparse binary sources (p=0.2)

Here the true source takes value 1 with probability 0.2. From the theoretical plot, p=0.2 is near where E[log(cosh(z))] is closest to the Gaussian baseline, making it hard for standard log-cosh to distinguish the source from Gaussian noise.

set.seed(2)
n   = 500
p   = 1000
k   = 4
prob_active = 0.2  # 20% of samples are "on" per source

A = matrix(rnorm(p * k), nrow = p)
S_sparse = matrix(0, nrow = k, ncol = n)
for (j in 1:k)
  S_sparse[j, sample(n, round(prob_active * n))] = 1

sigma = 0.1
X_sp = A %*% S_sparse + matrix(rnorm(p * n, 0, sigma), nrow = p)
Z_sp = prewhiten(X_sp, k)

res_lc_sp  = run_seeds(Z_sp, S_sparse, fastica_r1update,     compute_objective,     n_seeds = 100)
res_tlc_sp = run_seeds(Z_sp, S_sparse, fastica_r1update_tlc, compute_objective_tlc, n_seeds = 100)

cat("log-cosh:    mean max|cor| =", round(mean(res_lc_sp$maxcor),  3),
    " fraction > 0.9:", mean(res_lc_sp$maxcor  > 0.9), "\n")
log-cosh:    mean max|cor| = 0.723  fraction > 0.9: 0.07 
cat("tilted lc:   mean max|cor| =", round(mean(res_tlc_sp$maxcor), 3),
    " fraction > 0.9:", mean(res_tlc_sp$maxcor > 0.9), "\n")
tilted lc:   mean max|cor| = 1  fraction > 0.9: 1 
par(mfrow = c(1, 2))
hist(res_lc_sp$maxcor,  breaks = seq(0, 1, by = 0.05),
     main = "log-cosh: sparse binary source",    xlab = "max |cor| with true S")
hist(res_tlc_sp$maxcor, breaks = seq(0, 1, by = 0.05),
     main = "tilted log-cosh: sparse binary source", xlab = "max |cor| with true S")

Version Author Date
7da05b8 Matthew Stephens 2026-07-16 
par(mfrow = c(1, 1))

Multi-component extraction via deflation

To extract \(k\) components we use deflation: find one component at a time, then project it out of the weight space before searching for the next. Specifically, after finding weight vector \(w_1\), the next search is restricted to vectors orthogonal to \(w_1\) by subtracting the projection onto \(w_1\) after each update. This ensures each extracted component is distinct. We use multiple random starts per component and keep the one with the highest objective value.

fastica_deflation = function(X, k, update_fn, obj_fn, n_iter = 200, n_starts = 5, ...) {
  W = matrix(0, nrow(X), k)
  for (comp in seq_len(k)) {
    best_obj = -Inf
    best_w   = NULL
    for (s in seq_len(n_starts)) {
      set.seed(s + comp * 1000)
      w = rnorm(nrow(X))
      # project out already-found components
      if (comp > 1)
        w = w - W[, 1:(comp-1), drop=FALSE] %*% (t(W[, 1:(comp-1), drop=FALSE]) %*% w)
      w = w / sqrt(sum(w^2))
      for (i in seq_len(n_iter)) {
        w = update_fn(X, w, ...)
        # deflate
        if (comp > 1)
          w = w - W[, 1:(comp-1), drop=FALSE] %*% (t(W[, 1:(comp-1), drop=FALSE]) %*% w)
        w = w / sqrt(sum(w^2))
      }
      o = obj_fn(X, w, ...)
      if (o > best_obj) { best_obj = o; best_w = w }
    }
    W[, comp] = best_w
  }
  W
}

# Extract 4 components on the sparse simulation
W_lc  = fastica_deflation(Z_sp, k, fastica_r1update,     compute_objective)
W_tlc = fastica_deflation(Z_sp, k, fastica_r1update_tlc, compute_objective_tlc)

cor_lc  = cor(t(S_sparse), t(Z_sp) %*% W_lc)
cor_tlc = cor(t(S_sparse), t(Z_sp) %*% W_tlc)

cat("log-cosh deflation — max |cor| per true source:\n")
log-cosh deflation <U+2014> max |cor| per true source:
print(round(apply(abs(cor_lc),  1, max), 3))
[1] 0.951 0.720 0.998 0.699
cat("tilted log-cosh deflation — max |cor| per true source:\n")
tilted log-cosh deflation <U+2014> max |cor| per true source:
print(round(apply(abs(cor_tlc), 1, max), 3))
[1] 1.000 1.000 1.000 0.995

Test 4: 9 overlapping groups, 20 members each, n=100

Here each source is active for 20 out of 100 samples (p=0.2), and groups can overlap. This is harder than the non-overlapping case and matches the simulation from the original file. We use deflation to extract all 9 components.

set.seed(1)
n = 100
p = 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 * K), nrow = p, ncol = K)
sigma = 0.1
X9 = t(L %*% t(FF) + matrix(rnorm(n * p, 0, sigma), nrow = n))  # p x n
Z9 = prewhiten(X9, K)

W9_lc  = fastica_deflation(Z9, K, fastica_r1update,     compute_objective,     n_starts = 10)
W9_tlc = fastica_deflation(Z9, K, fastica_r1update_tlc, compute_objective_tlc, n_starts = 10)

cor9_lc  = cor(L, t(Z9) %*% W9_lc)
cor9_tlc = cor(L, t(Z9) %*% W9_tlc)

cat("log-cosh — max |cor| per true source:\n")
log-cosh <U+2014> max |cor| per true source:
print(round(apply(abs(cor9_lc),  2, max), 3))
[1] 0.612 0.546 0.711 0.747 0.584 0.497 0.609 0.609 0.614
cat("tilted log-cosh — max |cor| per true source:\n")
tilted log-cosh <U+2014> max |cor| per true source:
print(round(apply(abs(cor9_tlc), 2, max), 3))
[1] 1.000 1.000 0.998 0.998 0.996 0.975 0.990 0.968 0.954

Extracted sources (rows = components, columns = samples):

proj9_lc  = t(Z9) %*% W9_lc
proj9_tlc = t(Z9) %*% W9_tlc

par(mfrow = c(K, 2), mar = c(1, 2, 1.5, 0.5))
for (i in 1:K) {
  plot(proj9_lc[, i],  main = paste0("log-cosh comp ", i),     ylab = "", xlab = "", cex.main = 0.8)
  plot(proj9_tlc[, i], main = paste0("tilted lc comp ", i), ylab = "", xlab = "", cex.main = 0.8)
}

par(mfrow = c(1, 1))

Summary

The tilted log-cosh contrast G(z) = log(cosh(z)) + λz|z| addresses the fundamental failure of standard log-cosh for sparse binary sources. The odd term z|z| has expectation 2p−1, so sparse sources (p→1) score highly rather than falling below the Gaussian baseline.


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/Detroit
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