Hadamard product of independent random sample covariance matrices with correlation structure
Lucas Benigni, Ziyad Zaklani
TL;DR
This work analyzes the spectral behavior of the Hadamard product of independent sample covariance matrices with internal correlation structures, under a high-dimensional scaling $\frac{n}{d_1\cdots d_k}\to \gamma$. The authors deploy a Bai–Zhou-type framework, after expressing the matrix as $M=\frac{1}{|\mathbf{d}|}\mathcal{A}^T\mathcal{A}$ and computing a covariance tensor $\mathcal{T}$ via an interlacing braid operator, to show the limit spectrum exists and equals $\mu_{\mathrm{MP}}^{\gamma} \boxtimes (\mu_1\circledast\dots\circledast\mu_k)$, where $\mu_i$ are the limiting spectra of the row-covariance matrices $\Sigma^{(i)}$ and $\mu_{\mathrm{MP}}^{\gamma}$ is the Marchenko–Pastur law. The key technical contributions are the explicit computation of $\mathcal{T}$ and a detailed concentration of quadratic forms using cumulant expansions, which together ensure the spectral convergence under Hadamard product with correlation. The results generalize classical MP behavior to a broad Hadamard-product setting with structured correlations and provide a closed-form description of the asymptotic spectrum in terms of free multiplicative and classical multiplicative convolutions, with potential implications for high-dimensional statistics and neural network kernel/matrix models.
Abstract
We compute the asymptotic empirical eigenvalue distribution of the matrix $M = \bigodot_{i=1}^k \frac{1}{d_i}X^{(i)}{X^{(i)}}^\top$ where $X^{(i)}\in\mathbb{R}^{n\times d_i}$ are independent matrices with independent rows but general correlation within each row under the dimension scaling $\frac{n}{d_1\dots d_k}\to γ$.