English

Anisotropic local law for non-separable sample covariance matrices

Probability 2026-02-24 v2 Applications Machine Learning

Abstract

We establish local laws for sample covariance matrices K=N1i=1N\gi\giK = N^{-1}\sum_{i=1}^N \g_i\g_i^* where the random vectors \g1,,\gNRn\g_1, \ldots, \g_N \in \R^n are independent with common covariance Σ\Sigma. Previous work has largely focused on the separable model \g=Σ1/2\w\g = \Sigma^{1/2}\w with \w\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 \gA\g\g^*A\g, we prove an optimal averaged local law showing that the Stieltjes transform of KK converges to its deterministic limit uniformly down to the optimal scale ηN1+\eps\eta \geq N^{-1+\eps}. Under an additional structural assumption on the cumulant tensors of \g\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 (KzI)1(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)\g = \sigma(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.

Keywords

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}
}
R2 v1 2026-07-01T10:43:48.501Z