Anisotropic local law for non-separable sample covariance matrices
Abstract
We establish local laws for sample covariance matrices where the random vectors are independent with common covariance . Previous work has largely focused on the separable model with 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 , we prove an optimal averaged local law showing that the Stieltjes transform of converges to its deterministic limit uniformly down to the optimal scale . Under an additional structural assumption on the cumulant tensors of -- 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 in arbitrary directions. We discuss several classes of non-separable examples satisfying our assumptions, including conditionally mean-zero distributions, the random features model 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.
Cite
@article{arxiv.2602.17960,
title = {Anisotropic local law for non-separable sample covariance matrices},
author = {Zhou Fan and Renyuan Ma and Elliot Paquette and Zhichao Wang},
journal= {arXiv preprint arXiv:2602.17960},
year = {2026}
}