Eigenvalue distribution of the Hadamard product of sample covariance matrices in a quadratic regime
Sebastien Abou Assaly, Lucas Benigni
TL;DR
This work analyzes the eigenvalue distribution of the Hadamard product of two independent, centered sample covariance matrices in a quadratic regime where $\frac{n}{pd}\to \gamma>0$ and $\frac{p}{d}\to a>0$. Using the method of moments, the authors compute limiting moments of the rescaled Hadamard product and show they match the moments of the Marchenko–Pastur distribution $\mu_{\mathrm{MP}_\gamma}$, with a combinatorial argument showing only double-tree index patterns contribute. They establish moment convergence and concentration, and then remove moment assumptions via truncation and Lévy-distance arguments, proving weak (and almost sure, under finite moments) convergence of the empirical spectral distribution to $\mu_{\mathrm{MP}_\gamma}$. This extends MP universality to a Hadamard product of independent null covariance matrices, with implications for high-dimensional statistics and kernel-type random matrices. The result provides a precise description of the spectrum in regimes relevant to large-scale data analyses and neural-network kernel theories.
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 $γ$ in the quadratic regime of dimension $\frac{n}{dp}\to γ$ and $\frac{p}{d}\to a$.