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

• URL: http://arxiv.org/abs/2512.17632v1
• Date: Fri, 19 Dec 2025 14:34:30 GMT
• Title: Fast and Robust: Computationally Efficient Covariance Estimation for Sub-Weibull Vectors
• Authors: Even He,
• Abstract summary: In this work, we target the specific regime of textbfSub-Weibull (characterized by stretched exponential tails $exp(-t)$)<n>Unlike element-wise truncation, our approach preserves the spectral geometry while requiring $O(Nd2)$ operations.<n>We derive non-asymptotic error bounds showing that our estimator recovers the optimal sub-Gaussian rate $tildeO(sqrtr()/N)$ with high probability.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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