Statistically Optimal Robust Mean and Covariance Estimation for Anisotropic Gaussians

Abstract

Assume that $X_{1}, \ldots, X_{N}$ is an $\varepsilon$-contaminated sample of $N$ independent Gaussian vectors in $\mathbb{R}^d$ with mean $\mu$ and covariance $\Sigma$. In the strong $\varepsilon$-contamination model we assume that the adversary replaced an $\varepsilon$ fraction of vectors in the original Gaussian sample by any other vectors. We show that there is an estimator $\widehat \mu$ of the mean satisfying, with probability at least $1 - \delta$, a bound of the form \[ \|\widehat{\mu} - \mu\|_2 \le c\left(\sqrt{\frac{\operatorname{Tr}(\Sigma)}{N}} + \sqrt{\frac{\|\Sigma\|\log(1/\delta)}{N}} + \varepsilon\sqrt{\|\Sigma\|}\right), \] where $c > 0$ is an absolute constant and $\|\Sigma\|$ denotes the operator norm of $\Sigma$. In the same contaminated Gaussian setup, we construct an estimator $\widehat \Sigma$ of the covariance matrix $\Sigma$ that satisfies, with probability at least $1 - \delta$, \[ \left\|\widehat{\Sigma} - \Sigma\right\| \le c\left(\sqrt{\frac{\|\Sigma\|\operatorname{Tr}(\Sigma)}{N}} + \|\Sigma\|\sqrt{\frac{\log(1/\delta)}{N}} + \varepsilon\|\Sigma\|\right). \] Both results are optimal up to multiplicative constant factors. Despite the recent significant interest in robust statistics, achieving both dimension-free bounds in the canonical Gaussian case remained open. In fact, several previously known results were either dimension-dependent and required $\Sigma$ to be close to identity, or had a sub-optimal dependence on the contamination level $\varepsilon$. As a part of the analysis, we derive sharp concentration inequalities for central order statistics of Gaussian, folded normal, and chi-squared distributions.