Entropic CLT and phase transition in high-dimensional Wishart matrices

TL;DR

This paper establishes an entropy-based phase transition in high-dimensional Wishart matrices with log-concave entries, showing they approximate Gaussian matrices when the dimension is significantly larger than the cube of the sample size.

Contribution

It introduces a novel entropy-based method to analyze phase transitions in Wishart matrices with log-concave entries, revealing precise conditions for Gaussian approximation.

Findings

Matrices approximate Gaussian ensemble when d >> n^3

Entropy methods effectively analyze high-dimensional random matrices

Spectral norm concentration and small ball probabilities are key tools

Abstract

We consider high dimensional Wishart matrices $\mathbb{X} \mathbb{X}^{\top}$ where the entries of $\mathbb{X} \in {\mathbb{R}^{n \times d}}$ are i.i.d. from a log-concave distribution. We prove an information theoretic phase transition: such matrices are close in total variation distance to the corresponding Gaussian ensemble if and only if $d$ is much larger than $n^3$. Our proof is entropy-based, making use of the chain rule for relative entropy along with the recursive structure in the definition of the Wishart ensemble. The proof crucially relies on the well known relation between Fisher information and entropy, a variational representation for Fisher information, concentration bounds for the spectral norm of a random matrix, and certain small ball probability estimates for log-concave measures.