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.

翻译：本文建立了样本协方差矩阵 $K = N^{-1}\sum_{i=1}^N \g_i\g_i^*$ 的局部定律，其中随机向量 $\g_1, \ldots, \g_N \in \R^n$ 相互独立且具有公共协方差 $Σ$。先前研究主要集中于可分离模型 $\g = Σ^{1/2}\w$（其中 $\w$ 具有独立分量），但该结构在涉及依赖数据或非线性变换数据的统计应用中很少出现。在二次型 $\g^*A\g$ 满足集中性假设的条件下，我们证明了最优平均局部定律，表明 $K$ 的 Stieltjes 变换在最优尺度 $η\geq N^{-1+\eps}$ 范围内一致收敛于其确定性极限。在 $\g$ 的累积量张量满足附加结构性假设的条件下——该假设在高度结构化的独立分量情形与一般依赖情形之间建立了插值——我们建立了完整的各向异性局部定律，为预解式 $(K-zI)^{-1}$ 在任意方向上的分量提供了控制。我们讨论了满足假设的几类非可分离示例，包括条件均值零分布、机器学习中出现的随机特征模型 $\g = σ(X\w)$，以及具有非线性倾斜的高斯测度。证明过程引入了张量网络框架，用于分析高阶累积量结构存在时的波动平均效应。