Eigenvalue distribution of the Hadamard product of sample covariance matrices in a quadratic regime

TL;DR

This paper proves that the eigenvalue distribution of the Hadamard product of two independent sample covariance matrices converges to the Marchenko--Pastur distribution in a specific high-dimensional regime.

Contribution

It establishes the spectral distribution convergence for the Hadamard product of independent sample covariance matrices in a quadratic asymptotic regime.

Findings

Eigenvalue distribution converges to Marchenko--Pastur law

Results hold in the quadratic regime where dimensions grow proportionally

Provides theoretical insight into spectral behavior of Hadamard products

Abstract

In this note, we prove that if $X\in\mathbb{R}^{n\times d}$ and $Y\in\mathbb{R}^{n\times p}$ are two independent matrices with i.i.d entries then the empirical spectral distribution of $\frac{1}{d}XX^\top \odot \frac{1}{p}YY^\top$, where $\odot$ denotes the Hadamard product, converges to the Marchenko--Pastur distribution of shape $\gamma$ in the quadratic regime of dimension $\frac{n}{dp}\to \gamma$ and $\frac{p}{d}\to a$.