Mathematical Derivation: SVD Whitening Implementation

This document explains the SVD Whitening method in SVD.R. While mathematically equivalent to PCA whitening for Symmetric Positive Definite (SPD) matrices, this method is framed in SVD notation; in eegwhiten it shares the same eigensolver as PCA.R.

1. Core Concept

The Singular Value Decomposition (SVD) generalizes eigendecomposition. For a covariance matrix (which is symmetric), the Singular Values are identical to the Eigenvalues, and Left/Right Singular Vectors are identical to Eigenvectors.

2. Mathematical Formulation

Step 1: Decomposition

Mathematically, we want the SVD of the covariance matrix $\Sigma$:

Since $\Sigma$ is symmetric and positive definite:

• $U = V$ (Left and Right singular vectors are the same).
• $D$ contains positive singular values (identical to eigenvalues $\lambda$).

Because the two decompositions coincide, SVD.R does not call svd(). It obtains $U$ and $D$ from the package’s shared SPD eigensolver .eigen_spd(), the same one used by PCA.R:

• By default this is base R’s eigen(Sigma, symmetric = TRUE).
• When only the leading n_comp components are requested (fewer than ncol(Sigma)) and the RSpectra package is available, it switches to the iterative RSpectra::eigs_sym() (selectable via eig_method).
• The singular values are returned in decreasing order, and the decomposition is truncated to the first n_comp components if requested.

Finally, the sign of each singular vector is fixed so that results are reproducible across runs: its largest-magnitude entry is made positive, or it is aligned with sign_ref when one is supplied.

R Code Correspondence:

dec <- .eigen_spd(Sigma, k = k, method = eig_method, fast = fast, precomp = precomp)
U <- dec$vectors   # singular vectors (= eigenvectors), one column per component
D <- dec$values    # singular values (= eigenvalues), decreasing

if (any(D <= 0)) stop("Sigma must be positive-definite.")

U <- .fix_component_sign(U, sign_ref = sign_ref)

Step 2: Constructing the Whitening Matrix

With $U$ and $D$ in hand, the whitening matrix takes the same form as in PCA whitening:

so that $Z = X W^\top$ satisfies $\text{Cov}(Z) = D^{-1/2} U^\top \Sigma \, U D^{-1/2} = D^{-1/2} D \, D^{-1/2} = I$.

R Code Correspondence:

# W = D^-1/2 * U^T
W <- sweep(t(U), 1L, 1 / sqrt(D), "*")