Anisotropic local law for non-separable sample covariance matrices

We establish local laws for sample covariance matrices K = N-1sumi=1N gigi^* where the random vectors g1, ldots, gN ∈ Rn are independent with common covariance Sigma. Previous work has largely focused on the separable model g = Sigma1/2w 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 η ≥ 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(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.