Nearly Optimal Robust Covariance and Scatter Matrix Estimation Beyond Gaussians

We study the problem of computationally efficient robust estimation of the covariance/scatter matrix of elliptical distributions -- that is, affine transformations of spherically symmetric distributions -- under the strong contamination model in the high-dimensional regime $d \gtrsim 1/\varepsilon^2$, where $d$ is the dimension and $\varepsilon$ is the fraction of adversarial corruptions. We propose an algorithm that, under a very mild assumption on the scatter matrix $\Sigma$, and given a nearly optimal number of samples $n = \tilde{O}(d^2/\varepsilon^2)$, computes in polynomial time an estimator $\hat{\Sigma}$ such that, with high probability, $$\left\| \Sigma^{-1/2} \hat{\Sigma} \Sigma^{-1/2} - Id \right\|_{\text F} \le O(\varepsilon \log(1/\varepsilon))\,.$$ As an application of our result, we obtain the first efficiently computable, nearly optimal robust covariance estimators that extend beyond the Gaussian case. Specifically, for elliptical distributions satisfying the Hanson--Wright inequality (such as Gaussians and uniform distributions over ellipsoids), our estimator $\hat{\Sigma}$ of the covariance $\Sigma$ achieves the same error guarantee as in the Gaussian case. Moreover, for elliptical distributions with sub-exponential tails (such as the multivariate Laplace distribution), we construct an estimator $\hat{\Sigma}$ satisfying the spectral norm bound $$\left\| \Sigma^{-1/2} \hat{\Sigma} \Sigma^{-1/2} - Id \right\| \le O(\varepsilon \log(1/\varepsilon))\,.$$ Our approach is based on estimating the covariance of the spatial sign of elliptical distributions. The estimation proceeds in several stages, one of which involves a novel spectral covariance filtering algorithm. This algorithm combines covariance filtering techniques with degree-4 sum-of-squares relaxations, and we believe it may be of independent interest for future applications.