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, the SVD implementation offers numerical advantages in certain contexts.

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

We perform SVD on 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 $\lambda$).

R Code Correspondence: ```r svdSigma <- svd(Sigma) U <- svdSigma$u D <- svdSigma$d