High-dimensional covariance estimation is notoriously sensitive to outliers. While statistically optimal estimators exist for general heavy-tailed distributions, they often rely on computationally expensive techniques like semidefinite programming or iterative M-estimation ($O(d^3)$). In this work, we target the specific regime of \textbf{Sub-Weibull distributions} (characterized by stretched exponential tails $\exp(-t^α)$). We investigate a computationally efficient alternative: the \textbf{Cross-Fitted Norm-Truncated Estimator}. Unlike element-wise truncation, our approach preserves the spectral geometry while requiring $O(Nd^2)$ operations, which represents the theoretical lower bound for constructing a full covariance matrix. Although spherical truncation is geometrically suboptimal for anisotropic data, we prove that within the Sub-Weibull class, the exponential tail decay compensates for this mismatch. Leveraging weighted Hanson-Wright inequalities, we derive non-asymptotic error bounds showing that our estimator recovers the optimal sub-Gaussian rate $\tilde{O}(\sqrt{r(Σ)/N})$ with high probability. This provides a scalable solution for high-dimensional data that exhibits tails heavier than Gaussian but lighter than polynomial decay.

翻译：高维协方差估计对异常值极为敏感。尽管针对一般重尾分布存在统计最优估计器，但它们通常依赖于计算成本高昂的技术，如半定规划或迭代M估计（$O(d^3)$）。本研究针对**次韦布尔分布**（以拉伸指数尾 $\exp(-t^α)$ 为特征）这一特定体系，探索一种计算高效的替代方案：**交叉拟合范数截断估计器**。与逐元素截断不同，该方法在保持谱几何结构的同时仅需 $O(Nd^2)$ 次运算，这构成了构建完整协方差矩阵的理论下界。虽然球面截断对于各向异性数据在几何上并非最优，但我们证明在次韦布尔分布类别中，指数尾衰减补偿了这种不匹配性。通过加权Hanson-Wright不等式，我们推导出非渐近误差界，表明该估计器能以高概率恢复最优次高斯速率 $\tilde{O}(\sqrt{r(Σ)/N})$。这为尾部比高斯分布更重但比多项式衰减更轻的高维数据提供了一种可扩展的解决方案。