Statistically Optimal Robust Mean and Covariance Estimation for Anisotropic Gaussians

• URL: http://arxiv.org/abs/2301.09024v1
• Date: Sat, 21 Jan 2023 23:28:55 GMT
• Title: Statistically Optimal Robust Mean and Covariance Estimation for Anisotropic Gaussians
• Authors: Arshak Minasyan and Nikita Zhivotovskiy
• Abstract summary: 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 construct an estimator $widehat Sigma of the cofrac matrix $Sigma that satisfies, with probability at least $1 - delta.
• Score: 3.5788754401889014
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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