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(d3)). In this work, we target the specific regime of Sub-Weibull distributions (characterized by stretched exponential tails exp(-tα)). We investigate a computationally efficient alternative: the Cross-Fitted Norm-Truncated Estimator. Unlike element-wise truncation, our approach preserves the spectral geometry while requiring O(Nd2) 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 tildeO(√r(Sigma)/N) with high probability. This provides a scalable solution for high-dimensional data that exhibits tails heavier than Gaussian but lighter than polynomial decay.