Statistical Guarantees for Covariance Estimation with Adversarial Corruption under Constraints

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 $\varepsilon<1/32$ on the fraction of the $N$ observations of data $X$ are arbitrarily corrupted by an omniscient adversary, we obtain the near-minimax rate (up to constants) of \begin{align*} \left(\varepsilon^2 \vee \delta^{\star2} \right)\wedge d^2 \lesssim \inf_{\hat\Sigma} \sup_{\Sigma \in K} \sup_{\mathrm{C}} \mathbb{E} \| \hat\Sigma(\mathrm{C}(X)) - \Sigma \|_F^2 \lesssim \left(\varepsilon^2\log^2(1/\varepsilon) \vee \delta^{\star2} \right)\wedge d^2 \end{align*} where $d$ is the diameter of the set $K$, $N \gtrsim \sup_{\delta>0} \log \mathrm{M}^{\operatorname{loc}}(\delta, 2c)$, 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.