Mathematical Derivation: Cholesky Whitening Implementation

This document details the Cholesky Whitening method implemented in Cholesky.R. This method is computationally efficient and results in a lower-triangular transformation matrix (in the context of the whitening operation), which induces a specific ordering dependency among features.

1. Core Concept

Cholesky decomposition factorizes a Symmetric Positive Definite (SPD) matrix into the product of a triangular matrix and its transpose. It is generally faster than eigendecomposition but is not scale-invariant or rotation-invariant (permutation of channels affects the result).

2. Mathematical Formulation

Step 1: Decomposition

Given the covariance matrix $\Sigma$, we compute the Cholesky factorization. In R, the chol() function returns an Upper Triangular matrix $R$ such that:

Note: In standard math texts, Cholesky is often defined as $\Sigma = L L^\top$ where $L$ is lower triangular. R returns the upper triangular factor $R = L^\top$.

R Code Correspondence: ```r chol_Sigma <- chol(Sigma) # R, upper triangular