Statistical Guarantees for Covariance Estimation with Adversarial Corruption under Constraints

arXiv:2610.10840v1 Announce Type: new Abstract: We obtain a near-minimax rate on covariance estimation of adversarially corrupted Gaussian data in Frobenius norm when the covariance estimation is assumed to lie in a star-shaped set $K \subseteq \mathbb{R}^{n \times n}$. Assuming a known upper bound of $\varepsilon 0} \log \mathrm{M}^{\operatorname{loc}}(\delta, 2c)$, and \begin{align*} \delta^*=\sup\left\{ \delta\geq 0: \frac{N}{\Upsilon^2} \delta^2\leq \log \mathrm{M}_K^{\operatorname{loc}}(\delta,c) \right\} \end{align*} for an arbitrary large constant $c$ and the local metric entropy of $K$, $\mathrm{M}^{\operatorname{loc}}_K(\delta,c)$. We believe this is the first such general result on structured robust covariance estimation in the Frobenius norm.