# Anisotropic local law for non-separable sample covariance matrices

arXiv: https://arxiv.org/abs/2602.17960 · HTML: https://mlsanity.com/paper/2602.17960/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Feb 2026.

## Built on by (most cited first, 3 of 3)

- [Generalization, memorization, and overfitting for diffusion models trained in the lazy high-dimensional regime](https://mlsanity.com/paper/2608.23938/) (2026)
- [Eigen-Spike Emergence and Quadratic Equivalents for Conjugate Kernels on Nonlinearly Separable Data](https://mlsanity.com/paper/2605.29669/) (2026)
- [Multi-kernel spectral clustering: Entrywise eigenvector perturbation bounds and exact recovery](https://mlsanity.com/paper/2608.08704/) (2026)

## Abstract (arXiv)

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.