Anisotropic local law for non-separable sample covariance matrices
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866908848316481536 |
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| author | Fan, Zhou Ma, Renyuan Paquette, Elliot Wang, Zhichao |
| author_facet | Fan, Zhou Ma, Renyuan Paquette, Elliot Wang, Zhichao |
| contents | 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 $Σ$. Previous work has largely focused on the separable model $\g = Σ^{1/2}\w$ 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 $η\geq N^{-1+\eps}$. 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 $\g = σ(X\w)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_17960 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Anisotropic local law for non-separable sample covariance matrices Fan, Zhou Ma, Renyuan Paquette, Elliot Wang, Zhichao Probability Applications Machine Learning 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 $Σ$. Previous work has largely focused on the separable model $\g = Σ^{1/2}\w$ 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 $η\geq N^{-1+\eps}$. 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 $\g = σ(X\w)$ 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. |
| title | Anisotropic local law for non-separable sample covariance matrices |
| topic | Probability Applications Machine Learning |
| url | https://arxiv.org/abs/2602.17960 |