Convex programming approach to robust estimation of a multivariate Gaussian model

Multivariate Gaussian distribution is often used as a first approximation to the distribution of high-dimensional data. Determining the parameters of this distribution under various constraints is a widely studied problem in statistics, and is often considered as a prototype for testing new algorithms or theoretical frameworks. In this paper, we develop a nonasymptotic approach to the problem of estimating the parameters of a multivariate Gaussian distribution when data are corrupted by outliers. We propose an estimator-efficiently computable by solving a convex program-that robustly estimates the population mean and the population covariance matrix even when the sample contains a significant proportion of outliers. In the case where the dimension $p$ of the data points is of smaller order than the sample size, our estimator of the corruption matrix is provably rate optimal simultaneously for the entry-wise $l_1$-norm, the Frobenius norm and the mixed $l_2/l_1$ norm. Furthermore, this optimality is achieved by a penalized square-root-of-least-squares method with a universal tuning parameter (calibrating the strength of the penalization). These results are partly extended to the case where $p$ is potentially larger than $n$, under the additional condition that the inverse covariance matrix is sparse.