Fast and Robust: Computationally Efficient Covariance Estimation for Sub-Weibull Vectors

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 $\mathbf{\text{Sub-Weibull}}\mathbf{\text{distributions}}$ (characterized by stretched exponential tails $\exp (-t^{\alpha })$). We investigate a computationally efficient alternative: the $\mathbf{\text{Cross-Fitted}}\mathbf{\text{Norm-Truncated}}\mathbf{\text{Estimator}}$. Unlike element-wise truncation, our approach preserves the spectral geometry while requiring $O(N{d}^2)$ 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 $\tilde{O}(\sqrt{r(\mathrm{\Sigma })\mathrm{/}N})$ with high probability. This provides a scalable solution for high-dimensional data that exhibits tails heavier than Gaussian but lighter than polynomial decay.