FastICA for Non-Whitened Data
Matthew Stephens
2026-07-16
Last updated: 2026-07-16
Checks: 6 1
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Introduction
Standard FastICA pre-whitens data: given \(X\) (\(p \times n\)) with thin SVD \(X = UDV'\), it sets \(Z = \sqrt{n}\,V'\) (\(k \times n\)) and maximizes \(E[G(w'z)]\) subject to \(\|w\|^2 = 1\). The constraint \(\|w\|^2 = 1\) is equivalent to \(E[(w'z)^2] = n\) because \(ZZ' = nI_k\) for whitened data.
Here we skip the whitening step and instead work with \[Z = DV' \qquad (k \times n),\] so that \(ZZ' = D^2\). The natural constraint to keep evaluation scale-free is \[w'D^2 w = n,\] which again enforces \(E[(w'z)^2] = n\). The FastICA fixed-point update itself is unchanged; only the normalization step changes from \(w \leftarrow w/\|w\|\) to \(w \leftarrow w\sqrt{n}/\sqrt{w'D^2 w}\).
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. The problem with whitening is that all \(k\) retained PCs are placed on equal footing — the noise PCs get the same unit variance as the signal PCs. This can overwhelm the signal when \(k\) is much larger than the true number of components.
With the non-whitened approach the \(k\) PCs enter with their original singular values \(d_1 \ge d_2 \ge \cdots \ge d_k\). Signal PCs have large \(d_i\); excess noise PCs have small \(d_i\) and therefore small weight in the objective. The constraint \(w'D^2 w = n\) automatically concentrates \(w\) on the high-variance directions, so inflating \(k\) adds far less noise than in the whitened case.
To test this we use \(k = 20\) PCs for all experiments below, even though the true number of sources is smaller (1, 4, or 9). We expect whitened FastICA to degrade noticeably with the extra noise PCs, while non-whitened FastICA should remain robust.
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$v) # k x n
D2 = diag(d^2, nrow = length(d)) # k x k
list(Z = Z, D2 = D2)
}
# Normalize w so that w'D2 w = n
normalize_nw = function(w, D2, n) {
w * sqrt(n) / sqrt(as.numeric(t(w) %*% D2 %*% w))
}
fastica_r1update_nw = function(Z, D2, w) {
n = ncol(Z)
w = normalize_nw(w, D2, n)
P = t(Z) %*% w
G = tanh(P)
G2 = 1 - tanh(P)^2
w = Z %*% G - mean(G2) * n * w
normalize_nw(w, D2, n)
}
fastica_r1update_nw_tlc = function(Z, D2, w, lambda = 1) {
n = ncol(Z)
w = normalize_nw(w, D2, n)
P = t(Z) %*% w
G = tanh(P) + 2 * lambda * abs(P)
G2 = 1 - tanh(P)^2 + 2 * lambda * sign(P)
w = Z %*% G - mean(G2) * n * w
normalize_nw(w, D2, n)
}
compute_objective_nw = function(Z, D2, w) {
n = ncol(Z)
w = normalize_nw(w, D2, n)
mean(log(cosh(t(Z) %*% w)))
}
compute_objective_nw_tlc = function(Z, D2, w, lambda = 1) {
n = ncol(Z)
w = normalize_nw(w, D2, n)
P = t(Z) %*% w
mean(log(cosh(P)) + lambda * abs(P) * P)
}
# ---- generic runner: works for both whitened and non-whitened ----
# update_fn(data, w, ...) and obj_fn(data, w, ...) where data is a list
# (non-whitened) or a matrix (whitened).
run_seeds = function(data, S_true, update_fn, obj_fn, n_seeds = 50,
n_iter = 200, ...) {
obj = numeric(n_seeds)
maxcor = numeric(n_seeds)
for (seed in seq_len(n_seeds)) {
set.seed(seed)
w = rnorm(nrow(if (is.list(data)) data$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")
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 = 0.3834 max|cor| = 1 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")
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
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| = 1 frac > 0.9: 1 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| = 1 frac > 0.9: 1 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|")
par(mfrow = c(1, 1))Test 3: Sparse binary sources (4 true sources, p=0.2, 20 PCs)
set.seed(2)
n = 500; 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| = 0.347 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| = 1 frac > 0.9: 1 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| = 0.738 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| = 1 frac > 0.9: 1 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)
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
}
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))
Z9 = prewhiten(X9, k_pca)
nw9 = preprocess_nonwhitened(X9, k_pca)
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] 0.714 0.603 0.551 0.664 0.690 0.567 0.612 0.607 0.525cat("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] 0.996 0.990 0.990 0.964 0.974 0.969 0.945 0.937 0.846Extracted 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 stringr_1.6.0
[25] compiler_4.4.2 fs_1.6.6 Rcpp_1.1.1 pkgconfig_2.0.3
[29] later_1.4.6 digest_0.6.39 R6_2.6.1 pillar_1.11.1
[33] magrittr_2.0.4 bslib_0.10.0 tools_4.4.2 cachem_1.1.0