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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2512.17632 |
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| _version_ | 1866914233694814208 |
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| author | He, Even |
| author_facet | He, Even |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_17632 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fast and Robust: Computationally Efficient Covariance Estimation for Sub-Weibull Vectors He, Even Machine Learning Computation 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. |
| title | Fast and Robust: Computationally Efficient Covariance Estimation for Sub-Weibull Vectors |
| topic | Machine Learning Computation |
| url | https://arxiv.org/abs/2512.17632 |