Anisotropic local law for non-separable sample covariance matrices

• URL: http://arxiv.org/abs/2602.17960v2
• Date: Mon, 23 Feb 2026 04:32:25 GMT
• Title: Anisotropic local law for non-separable sample covariance matrices
• Authors: Zhou Fan, Renyuan Ma, Elliot Paquette, Zhichao Wang,
• Abstract summary: We establish local laws for sample covariance matrices $K = N-1sum_i=1N g_ig_ig_i*$ where the random vectors $g_1, ldots, g_N in Rn$ are independent with common covariance $$.<n>We discuss several classes of non-separable examples satisfying our assumptions, including conditionally mean-zero distributions, the random features model $g = (Xw)$ arising in machine learning, and Gaussian measures with
• Score: 10.181748307494608
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We establish local laws for sample covariance matrices $K = N^{-1}\sum_{i=1}^N \g_i\g_i^*$ where the random vectors $\g_1, \ldots, \g_N \in \R^n$ are independent with common covariance $Σ$. Previous work has largely focused on the separable model $\g = Σ^{1/2}\w$ with $\w$ having independent entries, but this structure is rarely present in statistical applications involving dependent or nonlinearly transformed data. Under a concentration assumption for quadratic forms $\g^*A\g$, we prove an optimal averaged local law showing that the Stieltjes transform of $K$ converges to its deterministic limit uniformly down to the optimal scale $η\geq N^{-1+\eps}$. Under an additional structural assumption on the cumulant tensors of $\g$ -- which interpolates between the highly structured case of independent entries and generic dependence -- we establish the full anisotropic local law, providing entrywise control of the resolvent $(K-zI)^{-1}$ in arbitrary directions. We discuss several classes of non-separable examples satisfying our assumptions, including conditionally mean-zero distributions, the random features model $\g = σ(X\w)$ arising in machine learning, and Gaussian measures with nonlinear tilting. The proofs introduce a tensor network framework for analyzing fluctuation averaging in the presence of higher-order cumulant structure.

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