Anisotropic local law for non-separable sample covariance matrices

2026-02-20

AI Summary

This work addresses the limitations of existing local law analyses for sample covariance matrices, which are typically confined to separable models and thus ill-suited for capturing the dependence and nonlinear structures prevalent in real-world data. By abandoning the separability assumption, the authors introduce an interpolation-based higher-order cumulant tensor condition and develop a tensor network framework to characterize non-separable correlations. Combining quadratic form concentration, Stieltjes transform techniques, and fluctuation averaging, they establish optimal averaged and anisotropic local laws for non-separable sample covariance matrices down to the spectral scale η ≥ N⁻¹⁺ε, thereby achieving entrywise control of resolvents in arbitrary directions. The theoretical results are validated in representative non-separable settings, including zero-mean distributions with general dependencies, random feature models, and nonlinearly tilted Gaussian measures.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

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• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

We establish local laws for sample covariance matrices $K = N^{-1}\sum_{i=1}^N \g_i\g_i^*$ 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 $\g^*A\g$, 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 $\g = \sigma(X\w)$ 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.

Problem

• anisotropic local law
• non-separable sample covariance matrices
• dependent data
• nonlinear transformations
• random features model

Innovation

• anisotropic local law
• non-separable covariance
• tensor network
• cumulant structure
• resolvent control