Even He | Searcharxiv

arXiv2026

Computable Bernstein Certificates for Cross-Fitted Clipped Covariance Estimation

We study operator-norm covariance estimation from heavy-tailed samples that may include a small fraction of arbitrary outliers. A simple and widely used safeguard is \emph{Euclidean norm clipping}, but its accuracy depends critically on an unknown clipping level. We propose a cross-fitted clipped covariance estimator equipped with \emph{fully computable} Bernstein-type deviation certificates, enabling principled data-driven tuning via a selector (\emph{MinUpper}) that balances certified stochastic error and a robust hold-out proxy for clipping bias. The resulting procedure adapts to intrinsic complexity measures such as effective rank under mild tail regularity and retains meaningful guarantees under only finite fourth moments. Experiments on contaminated spiked-covariance benchmarks illustrate stable performance and competitive accuracy across regimes.

stat.ML↗

arXiv2025

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 \textbf{Sub-Weibull distributions} (characterized by stretched exponential tails $\exp(-t^\alpha)$). 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 $\tilde{O}(\sqrt{r(\Sigma)/N})$ with high probability. This provides a scalable solution for high-dimensional data that exhibits tails heavier than Gaussian but lighter than polynomial decay.

stat.ML↗