Anisotropic local law for non-separable sample covariance matrices

arXiv:2602.17960v2 Announce Type: replace-cross
Abstract: We establish local laws for sample covariance matrices $K = N^{-1}sum_{i=1}^N g_ig_i^*$ where the random vectors $g_1, ldots, g_N in R^n$ are independent with common covariance $Sigma$. Previous work has largely focused on the separable model $g = Sigma^{1/2}w$ 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 $g^*Ag$, 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 $eta geq N^{-1+eps}$. 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 $g = sigma(Xw)$ 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.