FastICA for Non-Whitened Data
Matthew Stephens
2026-07-16
Last updated: 2026-07-16
Checks: 7 0
Knit directory: misc/analysis/
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| File | Version | Author | Date | Message |
|---|---|---|---|---|
| Rmd | c9c23dd | Matthew Stephens | 2026-07-16 | Correct Newton update, rewrite introduction for exact equivariance |
| Rmd | 3341f07 | Matthew Stephens | 2026-07-16 | Add fastica_nonwhitened.Rmd: FastICA for non-whitened data |
| html | 3341f07 | Matthew Stephens | 2026-07-16 | Add fastica_nonwhitened.Rmd: FastICA for non-whitened data |
Introduction
Standard whitened FastICA
Given \(X\) (\(p \times n\)) with thin SVD \(X = UDV'\), standard FastICA pre-whitens by setting \(Z_{\text{wh}} = \sqrt{n}\,V'\) (\(k \times n\)), so that the sample covariance \(E[zz'] = I_k\). It maximises \(E[G(w'z)]\) subject to \(\|w\|^2 = 1\) (equivalently \(E[(w'z)^2] = 1\)). Because the covariance is the identity, the Newton fixed-point update requires no matrix inversion: \[w \leftarrow E[z\,g(w'z)] - E[g'(w'z)]\,w, \qquad w \leftarrow w/\|w\|.\]
Non-whitened FastICA
We instead work with \[Z = DV' \quad (k \times n), \qquad
C = E[zz'] = \tfrac{1}{n}D^2.\] The scale-free constraint is \(E[(w'z)^2] = 1\), i.e. \(w'Cw = 1\), i.e.
\(w'D^2w = n\). The Newton update for a constrained problem with non-identity covariance requires left-multiplying the gradient by \(C^{-1} = nD^{-2}\): \[w \leftarrow C^{-1}E[z\,g(w'z)] - E[g'(w'z)]\,w
= nD^{-2}E[z\,g(w'z)] - E[g'(w'z)]\,w, \qquad
w \leftarrow w\sqrt{n}/\sqrt{w'D^2 w}.\] In code, since \(Z\mathbf{g}/n = E[zg(w'z)]\), the term \(nD^{-2}E[zg]\) is computed as (Z %*% G) / diag(D2) (elementwise division by \(d_i^2\)).
Exact equivariance
The two algorithms are exactly equivariant under the linear map \[w_{\text{wh}} = C^{1/2}w = \tfrac{1}{\sqrt{n}}Dw, \qquad w_{\text{nw}} = C^{-1/2}w_{\text{wh}} = \sqrt{n}D^{-1}w_{\text{wh}},\] which preserves projections: \(w_{\text{wh}}'z_{\text{wh}} = w_{\text{nw}}'z_{\text{nw}}\).
Proof. Substitute \(z_{\text{wh}} = C^{-1/2}z = \sqrt{n}D^{-1}z\) and \(w_{\text{wh}} = C^{1/2}w_{\text{nw}} = \tfrac{D}{\sqrt{n}}w_{\text{nw}}\) into the whitened update: \[\tfrac{D}{\sqrt{n}}\,w^+ = E\!\left[\sqrt{n}D^{-1}z\cdot g(w'z)\right] - E[g']\,\tfrac{D}{\sqrt{n}}w = \sqrt{n}D^{-1}E[zg(w'z)] - E[g']\tfrac{D}{\sqrt{n}}w.\] Left-multiplying by \(\tfrac{\sqrt{n}}{D} = \sqrt{n}D^{-1}\) recovers exactly the non-whitened update: \[w^+ = nD^{-2}E[zg(w'z)] - E[g']\,w. \qquad \checkmark\] The normalisation steps also correspond: \(\|w_{\text{wh}}\|=1 \Leftrightarrow w_{\text{nw}}'D^2w_{\text{nw}} = n\). Therefore the two iterations trace identical trajectories at every step, and have the same fixed points with the same stability.
Why non-whitened outperforms whitened in practice
The dynamics are identical given corresponding initialisations, but a random \(w_{\text{nw}}\) does not correspond to a random \(w_{\text{wh}}\). From a random \(w_{\text{nw}}\), the equivalent whitened start is \[w_{\text{wh}} \propto \tfrac{D}{\sqrt{n}}w_{\text{nw}},\] which is biased toward large-\(d_i\) (signal) directions. Whitened FastICA starts from a uniformly random direction on the \(k\)-sphere, giving only a \(\sim k_{\text{true}}/k\) chance of landing near the signal subspace.
In other words: the non-whitened parameterisation gives every random start an implicit warm start toward the signal, without changing the algorithm’s fixed-point structure at all.
Motivation: robustness to the number of PCs
In practice we do not know how many independent components are present. Standard whitening requires a hard choice of \(k\): too small and we miss components, too large and we include pure-noise PCs. Whitening places all \(k\) PCs on equal footing, so inflating \(k\) adds pure-noise directions that are equally likely starting points for the iteration.
The non-whitened parameterisation’s random initialisation is naturally biased toward high-variance (signal) PCs, so extra noise PCs cause much less harm. We test this by using \(k = 20\) PCs throughout, even when the true number of sources is 1, 4, or 9.
Implementation
# ---- whitened helpers (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_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 = function(X, w) mean(log(cosh(t(X) %*% w)))
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)$v[, 1:n.comp])
}
# ---- non-whitened helpers ----
# Returns list(Z, D2): Z = DV' (k x n), D2 = diag(d^2) (k x k)
preprocess_nonwhitened = function(X, n.comp) {
X = X - rowMeans(X)
sv = svd(X, nu = 0, nv = n.comp)
d = sv$d[1:n.comp]
Z = diag(d, nrow = length(d)) %*% t(sv$v) # k x n
D2 = diag(d^2, nrow = length(d)) # k x k
list(Z = Z, D2 = D2)
}
# Normalize w so that w'D2 w = n
normalize_nw = function(w, D2, n) {
w * sqrt(n) / sqrt(as.numeric(t(w) %*% D2 %*% w))
}
fastica_r1update_nw = function(Z, D2, w) {
n = ncol(Z)
w = normalize_nw(w, D2, n)
P = t(Z) %*% w
G = tanh(P)
G2 = 1 - tanh(P)^2
w = (Z %*% G) / diag(D2) - mean(G2) * w # n * D2^{-1} %*% E[zg] - E[g'] * w
normalize_nw(w, D2, n)
}
fastica_r1update_nw_tlc = function(Z, D2, w, lambda = 1) {
n = ncol(Z)
w = normalize_nw(w, D2, n)
P = t(Z) %*% w
G = tanh(P) + 2 * lambda * abs(P)
G2 = 1 - tanh(P)^2 + 2 * lambda * sign(P)
w = (Z %*% G) / diag(D2) - mean(G2) * w
normalize_nw(w, D2, n)
}
compute_objective_nw = function(Z, D2, w) {
n = ncol(Z)
w = normalize_nw(w, D2, n)
mean(log(cosh(t(Z) %*% w)))
}
compute_objective_nw_tlc = function(Z, D2, w, lambda = 1) {
n = ncol(Z)
w = normalize_nw(w, D2, n)
P = t(Z) %*% w
mean(log(cosh(P)) + lambda * abs(P) * P)
}
# ---- generic runner: works for both whitened and non-whitened ----
# update_fn(data, w, ...) and obj_fn(data, w, ...) where data is a list
# (non-whitened) or a matrix (whitened).
run_seeds = function(data, S_true, update_fn, obj_fn, n_seeds = 50,
n_iter = 200, ...) {
obj = numeric(n_seeds)
maxcor = numeric(n_seeds)
for (seed in seq_len(n_seeds)) {
set.seed(seed)
w = rnorm(nrow(if (is.list(data)) data$Z else data))
for (i in seq_len(n_iter))
w = update_fn(data, w, ...)
obj[seed] = obj_fn(data, w, ...)
proj = if (is.list(data)) t(data$Z) %*% w else t(data) %*% w
maxcor[seed] = max(abs(cor(t(S_true), proj)))
}
list(obj = obj, maxcor = maxcor)
}
# Wrappers so run_seeds gets a single 'data' argument
update_nw = function(data, w, lambda = NULL) {
if (is.null(lambda)) fastica_r1update_nw(data$Z, data$D2, w)
else fastica_r1update_nw_tlc(data$Z, data$D2, w, lambda = lambda)
}
obj_nw = function(data, w, lambda = NULL) {
if (is.null(lambda)) compute_objective_nw(data$Z, data$D2, w)
else compute_objective_nw_tlc(data$Z, data$D2, w, lambda = lambda)
}
k_pca = 20 # number of PCs used for all tests (larger than true k)Test 1: Symmetric binary source (1 true source, 20 PCs)
The data have a single Rademacher source, but we extract k_pca = 20 PCs. Whitening inflates all 20 to unit variance; non-whitening retains the true signal-to-noise contrast in the singular values.
Singular values
set.seed(10)
n = 200; p = 1000; k_true = 1
A = matrix(rnorm(p * k_true), nrow = p)
S_rad = matrix(sample(c(-1, 1), n, replace = TRUE), nrow = 1)
sigma = 0.1
X_rad = A %*% S_rad + matrix(rnorm(p * n, 0, sigma), nrow = p)
sv_rad = svd(X_rad - rowMeans(X_rad), nu = 0, nv = k_pca)
plot(sv_rad$d[1:k_pca], type = "b", pch = 19, cex = 0.7,
xlab = "PC index", ylab = "singular value",
main = "Top 20 singular values (1 true source)")
abline(v = k_true + 0.5, lty = 2, col = "red")
legend("topright", "true k", lty = 2, col = "red")
| Version | Author | Date |
|---|---|---|
| 3341f07 | Matthew Stephens | 2026-07-16 |
The first singular value is much larger than the rest, which are pure noise. Whitening would equalise all 20; non-whitening keeps this gap intact.
Single-seed detailed comparison
Z_rad = prewhiten(X_rad, k_pca)
nw_rad = preprocess_nonwhitened(X_rad, k_pca)
run_one = function(data, update_fn, obj_fn, seed = 1, n_iter = 200, ...) {
set.seed(seed)
w = rnorm(nrow(if (is.list(data)) data$Z else data))
for (i in seq_len(n_iter))
w = update_fn(data, w, ...)
proj = if (is.list(data)) t(data$Z) %*% w else t(data) %*% w
list(w = w, proj = as.vector(proj), obj = obj_fn(data, w, ...))
}
r_wh_lc = run_one(Z_rad, fastica_r1update, compute_objective)
r_wh_tlc = run_one(Z_rad, fastica_r1update_tlc, compute_objective_tlc, lambda = 1)
r_nw_lc = run_one(nw_rad, update_nw, obj_nw)
r_nw_tlc = run_one(nw_rad, update_nw, obj_nw, lambda = 1)
cat("whitened log-cosh: obj =", round(r_wh_lc$obj, 4),
" max|cor| =", round(max(abs(cor(as.vector(S_rad), r_wh_lc$proj))), 3), "\n")whitened log-cosh: obj = 0.329 max|cor| = 0.069 cat("whitened TLC: obj =", round(r_wh_tlc$obj, 4),
" max|cor| =", round(max(abs(cor(as.vector(S_rad), r_wh_tlc$proj))), 3), "\n")whitened TLC: obj = 0.6942 max|cor| = 0.025 cat("nonwhiten log-cosh: obj =", round(r_nw_lc$obj, 4),
" max|cor| =", round(max(abs(cor(as.vector(S_rad), r_nw_lc$proj))), 3), "\n")nonwhiten log-cosh: obj = 0.4334 max|cor| = 1 cat("nonwhiten TLC: obj =", round(r_nw_tlc$obj, 4),
" max|cor| =", round(max(abs(cor(as.vector(S_rad), r_nw_tlc$proj))), 3), "\n")nonwhiten TLC: obj = 0.6743 max|cor| = 0.201 Projections w’z
par(mfrow = c(2, 2))
ord = order(as.vector(S_rad))
col = ifelse(as.vector(S_rad)[ord] == 1, "tomato", "steelblue")
plot_proj = function(proj, title) {
plot(proj[ord], col = col, pch = 19, cex = 0.5,
main = title, xlab = "sample (sorted by true source)", ylab = "w'z",
cex.main = 0.85)
abline(h = 0, lty = 2)
}
plot_proj(r_wh_lc$proj, "whitened log-cosh")
plot_proj(r_wh_tlc$proj, "whitened TLC")
plot_proj(r_nw_lc$proj, "nonwhitened log-cosh")
plot_proj(r_nw_tlc$proj, "nonwhitened TLC")
| Version | Author | Date |
|---|---|---|
| 3341f07 | Matthew Stephens | 2026-07-16 |
par(mfrow = c(1, 1))Samples are sorted by true source value (+1 = red, -1 = blue). A good recovery shows two clearly separated bands.
Many seeds (200 iterations)
res_wh_lc = run_seeds(Z_rad, S_rad, fastica_r1update, compute_objective)
res_wh_tlc = run_seeds(Z_rad, S_rad, fastica_r1update_tlc, compute_objective_tlc, lambda = 1)
res_nw_lc = run_seeds(nw_rad, S_rad, update_nw, obj_nw)
res_nw_tlc = run_seeds(nw_rad, S_rad, update_nw, obj_nw, lambda = 1)
cat("whitened log-cosh: mean max|cor| =", round(mean(res_wh_lc$maxcor), 3),
" frac > 0.9:", mean(res_wh_lc$maxcor > 0.9), "\n")whitened log-cosh: mean max|cor| = 0.176 frac > 0.9: 0.06 cat("whitened TLC: mean max|cor| =", round(mean(res_wh_tlc$maxcor), 3),
" frac > 0.9:", mean(res_wh_tlc$maxcor > 0.9), "\n")whitened TLC: mean max|cor| = 0.11 frac > 0.9: 0 cat("nonwhiten log-cosh: mean max|cor| =", round(mean(res_nw_lc$maxcor), 3),
" frac > 0.9:", mean(res_nw_lc$maxcor > 0.9), "\n")nonwhiten log-cosh: mean max|cor| = 0.967 frac > 0.9: 0.96 cat("nonwhiten TLC: mean max|cor| =", round(mean(res_nw_tlc$maxcor), 3),
" frac > 0.9:", mean(res_nw_tlc$maxcor > 0.9), "\n")nonwhiten TLC: mean max|cor| = 0.189 frac > 0.9: 0 Test 2: Non-overlapping groups (4 true sources, 20 PCs)
set.seed(1)
n = 100; p = 1000; k_true = 4
A = matrix(rnorm(p * k_true), nrow = p)
S = matrix(0, nrow = k_true, 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_pca)
nw = preprocess_nonwhitened(X, k_pca)
res_wh_lc = run_seeds(Z, S, fastica_r1update, compute_objective)
res_wh_tlc = run_seeds(Z, S, fastica_r1update_tlc, compute_objective_tlc, lambda = 1)
res_nw_lc = run_seeds(nw, S, update_nw, obj_nw)
res_nw_tlc = run_seeds(nw, S, update_nw, obj_nw, lambda = 1)
par(mfrow = c(2, 2))
hist(res_wh_lc$maxcor, breaks = seq(0, 1, by = 0.05), main = "whitened log-cosh", xlab = "max |cor|")
hist(res_wh_tlc$maxcor, breaks = seq(0, 1, by = 0.05), main = "whitened TLC", xlab = "max |cor|")
hist(res_nw_lc$maxcor, breaks = seq(0, 1, by = 0.05), main = "nonwhitened log-cosh", xlab = "max |cor|")
hist(res_nw_tlc$maxcor, breaks = seq(0, 1, by = 0.05), main = "nonwhitened TLC", xlab = "max |cor|")
| Version | Author | Date |
|---|---|---|
| 3341f07 | Matthew Stephens | 2026-07-16 |
par(mfrow = c(1, 1))Test 3: Sparse binary sources (4 true sources, p=0.2, 20 PCs)
set.seed(2)
n = 100; p = 1000; k_true = 4
prob_active = 0.2
A = matrix(rnorm(p * k_true), nrow = p)
S_sparse = matrix(0, nrow = k_true, ncol = n)
for (j in 1:k_true)
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_pca)
nw_sp = preprocess_nonwhitened(X_sp, k_pca)
res_wh_lc_sp = run_seeds(Z_sp, S_sparse, fastica_r1update, compute_objective, n_seeds = 100)
res_wh_tlc_sp = run_seeds(Z_sp, S_sparse, fastica_r1update_tlc, compute_objective_tlc, n_seeds = 100, lambda = 1)
res_nw_lc_sp = run_seeds(nw_sp, S_sparse, update_nw, obj_nw, n_seeds = 100)
res_nw_tlc_sp = run_seeds(nw_sp, S_sparse, update_nw, obj_nw, n_seeds = 100, lambda = 1)
cat("whitened log-cosh: mean max|cor| =", round(mean(res_wh_lc_sp$maxcor), 3),
" frac > 0.9:", mean(res_wh_lc_sp$maxcor > 0.9), "\n")whitened log-cosh: mean max|cor| = 0.337 frac > 0.9: 0 cat("whitened TLC: mean max|cor| =", round(mean(res_wh_tlc_sp$maxcor), 3),
" frac > 0.9:", mean(res_wh_tlc_sp$maxcor > 0.9), "\n")whitened TLC: mean max|cor| = 0.856 frac > 0.9: 0.8 cat("nonwhiten log-cosh: mean max|cor| =", round(mean(res_nw_lc_sp$maxcor), 3),
" frac > 0.9:", mean(res_nw_lc_sp$maxcor > 0.9), "\n")nonwhiten log-cosh: mean max|cor| = 0.378 frac > 0.9: 0 cat("nonwhiten TLC: mean max|cor| =", round(mean(res_nw_tlc_sp$maxcor), 3),
" frac > 0.9:", mean(res_nw_tlc_sp$maxcor > 0.9), "\n")nonwhiten TLC: mean max|cor| = 0.992 frac > 0.9: 0.99 par(mfrow = c(2, 2))
hist(res_wh_lc_sp$maxcor, breaks = seq(0, 1, by = 0.05), main = "whitened log-cosh", xlab = "max |cor|")
hist(res_wh_tlc_sp$maxcor, breaks = seq(0, 1, by = 0.05), main = "whitened TLC", xlab = "max |cor|")
hist(res_nw_lc_sp$maxcor, breaks = seq(0, 1, by = 0.05), main = "nonwhitened log-cosh", xlab = "max |cor|")
hist(res_nw_tlc_sp$maxcor, breaks = seq(0, 1, by = 0.05), main = "nonwhitened TLC", xlab = "max |cor|")
par(mfrow = c(1, 1))Test 4: 9 overlapping groups, 20 members each, n=100 (20 PCs, no deflation)
Single-component extraction over many seeds, same as Tests 1–3. Each seed finds one direction; we report the max |cor| with any of the 9 true sources.
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))
sv9 = svd(X9 - rowMeans(X9), nu = 0, nv = k_pca)
Z9 = prewhiten(X9, k_pca)
nw9 = preprocess_nonwhitened(X9, k_pca)Singular values
plot(sv9$d[1:k_pca], type = "b", pch = 19, cex = 0.7,
xlab = "PC index", ylab = "singular value",
main = "Top 20 singular values (9 true sources)")
abline(v = K + 0.5, lty = 2, col = "red")
legend("topright", "true K", lty = 2, col = "red")
Single-seed detailed comparison
r9_wh_lc = run_one(Z9, fastica_r1update, compute_objective)
r9_wh_tlc = run_one(Z9, fastica_r1update_tlc, compute_objective_tlc, lambda = 1)
r9_nw_lc = run_one(nw9, update_nw, obj_nw)
r9_nw_tlc = run_one(nw9, update_nw, obj_nw, lambda = 1)
cor_fmt = function(proj) round(max(abs(cor(L, proj))), 3)
cat("whitened log-cosh: obj =", round(r9_wh_lc$obj, 4), " max|cor| =", cor_fmt(r9_wh_lc$proj), "\n")whitened log-cosh: obj = 0.3031 max|cor| = 0.494 cat("whitened TLC: obj =", round(r9_wh_tlc$obj, 4), " max|cor| =", cor_fmt(r9_wh_tlc$proj), "\n")whitened TLC: obj = 0.961 max|cor| = 1 cat("nonwhiten log-cosh: obj =", round(r9_nw_lc$obj, 4), " max|cor| =", cor_fmt(r9_nw_lc$proj), "\n")nonwhiten log-cosh: obj = 0.4131 max|cor| = 0.294 cat("nonwhiten TLC: obj =", round(r9_nw_tlc$obj, 4), " max|cor| =", cor_fmt(r9_nw_tlc$proj), "\n")nonwhiten TLC: obj = 0.961 max|cor| = 1 lbls = c("whitened log-cosh", "whitened TLC", "nonwhitened log-cosh", "nonwhitened TLC")
par(mfrow = c(2, 2))
for (i in seq_along(lbls)) {
res = list(r9_wh_lc, r9_wh_tlc, r9_nw_lc, r9_nw_tlc)[[i]]
best_src = which.max(abs(cor(L, res$proj)))
ord = order(L[, best_src])
col = ifelse(L[ord, best_src] == 1, "tomato", "steelblue")
plot(res$proj[ord], col = col, pch = 19, cex = 0.5,
main = paste0(lbls[i], "\nmax|cor|=", cor_fmt(res$proj), " (src ", best_src, ")"),
xlab = "sample (sorted by best-matching source)", ylab = "w'z", cex.main = 0.8)
abline(h = 0, lty = 2)
}
par(mfrow = c(1, 1))Many seeds
res9_wh_lc = run_seeds(Z9, t(L), fastica_r1update, compute_objective, n_seeds = 100)
res9_wh_tlc = run_seeds(Z9, t(L), fastica_r1update_tlc, compute_objective_tlc, n_seeds = 100, lambda = 1)
res9_nw_lc = run_seeds(nw9, t(L), update_nw, obj_nw, n_seeds = 100)
res9_nw_tlc = run_seeds(nw9, t(L), update_nw, obj_nw, n_seeds = 100, lambda = 1)
cat("whitened log-cosh: mean max|cor| =", round(mean(res9_wh_lc$maxcor), 3),
" frac > 0.9:", mean(res9_wh_lc$maxcor > 0.9), "\n")whitened log-cosh: mean max|cor| = 0.408 frac > 0.9: 0 cat("whitened TLC: mean max|cor| =", round(mean(res9_wh_tlc$maxcor), 3),
" frac > 0.9:", mean(res9_wh_tlc$maxcor > 0.9), "\n")whitened TLC: mean max|cor| = 0.987 frac > 0.9: 0.98 cat("nonwhiten log-cosh: mean max|cor| =", round(mean(res9_nw_lc$maxcor), 3),
" frac > 0.9:", mean(res9_nw_lc$maxcor > 0.9), "\n")nonwhiten log-cosh: mean max|cor| = 0.412 frac > 0.9: 0 cat("nonwhiten TLC: mean max|cor| =", round(mean(res9_nw_tlc$maxcor), 3),
" frac > 0.9:", mean(res9_nw_tlc$maxcor > 0.9), "\n")nonwhiten TLC: mean max|cor| = 1 frac > 0.9: 1 par(mfrow = c(2, 2))
hist(res9_wh_lc$maxcor, breaks = seq(0, 1, by = 0.05), main = "whitened log-cosh", xlab = "max |cor|")
hist(res9_wh_tlc$maxcor, breaks = seq(0, 1, by = 0.05), main = "whitened TLC", xlab = "max |cor|")
hist(res9_nw_lc$maxcor, breaks = seq(0, 1, by = 0.05), main = "nonwhitened log-cosh", xlab = "max |cor|")
hist(res9_nw_tlc$maxcor, breaks = seq(0, 1, by = 0.05), main = "nonwhitened TLC", xlab = "max |cor|")
par(mfrow = c(1, 1))Cross-initialization test
results_cross9 = data.frame(
seed = seq_len(n_seeds_cross),
wh_maxcor = NA_real_,
nw_maxcor = NA_real_,
wh_from_nw = NA_real_,
nw_from_wh = NA_real_
)
for (seed in seq_len(n_seeds_cross)) {
set.seed(seed)
w0 = rnorm(k_pca)
w_wh = w0
for (i in seq_len(2000)) w_wh = fastica_r1update_tlc(Z9, w_wh)
w_nw = w0
for (i in seq_len(2000)) w_nw = update_nw(nw9, w_nw, lambda = 1)
d9 = sv9$d[1:k_pca]
w_wh_from_nw = d9 * w_nw / sqrt(n)
for (i in seq_len(2000)) w_wh_from_nw = fastica_r1update_tlc(Z9, w_wh_from_nw)
w_nw_from_wh = sqrt(n) * w_wh / d9
for (i in seq_len(2000)) w_nw_from_wh = update_nw(nw9, w_nw_from_wh, lambda = 1)
mc_wh = function(w) max(abs(cor(L, as.vector(t(Z9) %*% w))))
mc_nw = function(w) max(abs(cor(L, as.vector(t(nw9$Z) %*% w))))
results_cross9[seed, "wh_maxcor"] = mc_wh(w_wh)
results_cross9[seed, "nw_maxcor"] = mc_nw(w_nw)
results_cross9[seed, "wh_from_nw"] = mc_wh(w_wh_from_nw)
results_cross9[seed, "nw_from_wh"] = mc_nw(w_nw_from_wh)
}
cat("Whitened from random start: mean max|cor| =",
round(mean(results_cross9$wh_maxcor), 3), "\n")Whitened from random start: mean max|cor| = 0.989 cat("Nonwhitened from random start: mean max|cor| =",
round(mean(results_cross9$nw_maxcor), 3), "\n")Nonwhitened from random start: mean max|cor| = 1 cat("Whitened init'd from NW solution: mean max|cor| =",
round(mean(results_cross9$wh_from_nw), 3), "\n")Whitened init'd from NW solution: mean max|cor| = 1 cat("Nonwhitened init'd from WH solution: mean max|cor| =",
round(mean(results_cross9$nw_from_wh), 3), "\n")Nonwhitened init'd from WH solution: mean max|cor| = 0.989 Test 5: 9 overlapping groups, deflation (20 PCs)
Multi-component extraction via deflation. After finding weight \(w_1\), the next search is restricted to vectors satisfying \(w \perp_{D^2} w_1\), i.e. \(w_1' D^2 w = 0\), so the extracted components are orthogonal in the \(D^2\)-inner-product sense (equivalently, orthogonal projections in the original data space). We extract 9 components from a 20-PC space.
fastica_deflation_nw = function(nw, k, update_fn, obj_fn, n_iter = 200,
n_starts = 5, ...) {
Z = nw$Z
D2 = nw$D2
n = ncol(Z)
W = matrix(0, nrow(Z), 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(Z))
if (comp > 1) {
for (j in 1:(comp - 1))
w = w - as.numeric(t(W[, j]) %*% D2 %*% w) /
as.numeric(t(W[, j]) %*% D2 %*% W[, j]) * W[, j]
}
w = normalize_nw(w, D2, n)
for (i in seq_len(n_iter)) {
w = update_fn(nw, w, ...)
if (comp > 1) {
for (j in 1:(comp - 1))
w = w - as.numeric(t(W[, j]) %*% D2 %*% w) /
as.numeric(t(W[, j]) %*% D2 %*% W[, j]) * W[, j]
w = normalize_nw(w, D2, n)
}
}
o = obj_fn(nw, w, ...)
if (o > best_obj) { best_obj = o; best_w = w }
}
W[, comp] = best_w
}
W
}
fastica_deflation_wh = 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))
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, ...)
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
}
W9_wh_lc = fastica_deflation_wh(Z9, K, fastica_r1update, compute_objective, n_starts = 10)
W9_wh_tlc = fastica_deflation_wh(Z9, K, fastica_r1update_tlc, compute_objective_tlc, n_starts = 10, lambda = 1)
W9_nw_lc = fastica_deflation_nw(nw9, K, update_nw, obj_nw, n_starts = 10)
W9_nw_tlc = fastica_deflation_nw(nw9, K, update_nw, obj_nw, n_starts = 10, lambda = 1)
cor9_wh_lc = cor(L, t(Z9) %*% W9_wh_lc)
cor9_wh_tlc = cor(L, t(Z9) %*% W9_wh_tlc)
cor9_nw_lc = cor(L, t(nw9$Z) %*% W9_nw_lc)
cor9_nw_tlc = cor(L, t(nw9$Z) %*% W9_nw_tlc)
cat("whitened log-cosh — max |cor| per true source:\n")whitened log-cosh <U+2014> max |cor| per true source:print(round(apply(abs(cor9_wh_lc), 2, max), 3))[1] 0.514 0.619 0.532 0.317 0.482 0.565 0.330 0.334 0.467cat("whitened TLC — max |cor| per true source:\n")whitened TLC <U+2014> max |cor| per true source:print(round(apply(abs(cor9_wh_tlc), 2, max), 3))[1] 1.000 1.000 0.998 0.996 0.992 0.984 0.986 0.982 0.931cat("nonwhitened log-cosh — max |cor| per true source:\n")nonwhitened log-cosh <U+2014> max |cor| per true source:print(round(apply(abs(cor9_nw_lc), 2, max), 3))[1] 0.492 0.473 0.408 0.596 0.318 0.383 0.480 0.431 0.349cat("nonwhitened TLC — max |cor| per true source:\n")nonwhitened TLC <U+2014> max |cor| per true source:print(round(apply(abs(cor9_nw_tlc), 2, max), 3))[1] 1.000 1.000 0.998 0.998 0.996 0.991 0.968 0.961 0.953Extracted projections (whitened TLC vs nonwhitened TLC):
proj9_wh_tlc = t(Z9) %*% W9_wh_tlc
proj9_nw_tlc = t(nw9$Z) %*% W9_nw_tlc
par(mfrow = c(K, 2), mar = c(1, 2, 1.5, 0.5))
for (i in 1:K) {
plot(proj9_wh_tlc[, i], main = paste0("whitened TLC comp ", i),
ylab = "", xlab = "", cex.main = 0.8)
plot(proj9_nw_tlc[, i], main = paste0("nonwhitened TLC comp ", i),
ylab = "", xlab = "", cex.main = 0.8)
}
par(mfrow = c(1, 1))Summary
The non-whitened FastICA replaces the whitened data \(Z = \sqrt{n}V'\) with \(Z = DV'\) and changes only the normalization: \(\|w\|^2 = 1\) becomes \(w'D^2 w = n\). This preserves the fixed-point structure of the algorithm. The deflation step uses the \(D^2\)-inner-product for orthogonalization instead of the standard inner product. When \(k\) is chosen larger than the true number of components, the non-whitened approach is more robust because excess noise PCs — which have small singular values \(d_i\) — contribute little to the objective, whereas whitening equalises all \(k\) PCs and amplifies noise.
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