发表机构
Yale University（耶鲁大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
在比例n≈N下，仅假设列向量二次型最优集中及标准非退化条件，证明了样本协方差矩阵的最优各向异性局部律，去除了高阶累积量张量假设，适用于对数凹分布、高斯倾斜、深度随机特征及高温球面4-spin模型。
AI 中文摘要
我们研究比例 regime $n \asymp N$ 下的样本协方差矩阵 $K = \frac{1}{N} \sum_{i=1}^N \x_i \x_i^* \in \R^{n \times n}$。列向量 $\x_1, \ldots, \x_N \in \R^n$ 独立且中心化，具有共同协方差 $\E \x_i \x_i^* = \Sigma$，但坐标之间可能具有强非线性依赖。仅假设列向量的二次型以最优速率 $| \x_i^* A \x_i - \Tr \Sigma A | \prec \\| A \\|_F$ 均匀集中，加上多项式范数矩和 $\Sigma$ 的标准非退化条件，我们证明了最优各向异性局部律：在正则谱域上，一致地向下到谱尺度 $\eta:= \Im z \geq N^{-1 + \tau}$，对所有确定性单位向量 $\u,\bv \in \C^n$，有 $\big| \\< \u, \big( (K-z)^{-1} - (-zI_n-z\widetilde m_0(z)\Sigma \big)^{-1} \bv \\> \big| \prec \sqrt{\frac{\Im \widetilde m_0 (z)}{N\eta}} + \frac{1}{N\eta}$，其中 $\widetilde m_0(z)$ 是变形 Marchenko-Pastur 律的 Stieltjes 变换。这去除了 Fan, Ma, Paquette, and Wang (2026) 的高阶累积量张量假设，从而回答了其工作中提出的问题。该结果适用于，除其他例子外，每个具有有界非退化协方差的中心化对数凹列分布、高斯向量的非线性倾斜、深度随机特征，以及一个高温球面 4-spin 模型，对于该模型已知累积量假设不成立。
英文摘要
We study sample covariance matrices $K = \frac{1}{N} \sum_{i=1}^N \mathbf{x}_i \mathbf{x}_i^* \in \mathbb{R}^{n \times n}$ in the proportional regime $n \asymp N$. The columns $ \mathbf{x}_1, \ldots, \mathbf{x}_N \in \mathbb{R}^n$ are independent and centered, with common covariance $\mathbb{E} \mathbf{x}_i \mathbf{x}_i^* = Σ$, but may otherwise have strongly and nonlinearly dependent coordinates. Assuming only that quadratic forms of the columns concentrate uniformly at the optimal rate $| \mathbf{x}_i^* A \mathbf{x}_i - \mathrm{Tr} ΣA | \prec \| A \|_F$, together with polynomial norm moments and a standard nondegeneracy condition on $Σ$, we prove the optimal anisotropic local law: on regular spectral domains, uniformly down to spectral scales $η:= \mathrm{Im}\, z \geq N^{-1 + τ}$, \[ \big| \langle \mathbf{u} , \big( (K-z)^{-1} - (-zI_n-z\widetilde m_0(z)Σ\big)^{-1} \big) \mathbf{v} \> \big| \prec \sqrt{\frac{\mathrm{Im}\, \widetilde m_0 (z)}{Nη}} + \frac{1}{Nη} \] for all deterministic unit vectors $ \mathbf{u}, \mathbf{v} \in \mathbb{C}^n$, where $\widetilde m_0(z)$ is the Stieltjes transform of the deformed Marchenko-Pastur law. This removes the higher-cumulant tensor assumption of Fan, Ma, Paquette, and Wang (2026), thereby answering the question raised in their work. The result applies, among other examples, to every centered log-concave column distribution with bounded, nondegenerate covariance, nonlinear tilts of Gaussian vectors, deep random features, and a high-temperature spherical 4-spin model for which the cumulant assumption is known to fail.
