High-dimensional covariance estimation under the presence of outliers

This paper considers the problem of robust covariance estimation in the so-called "large p small n" setting. Its first contribution is the proposal of a novel (non-robust) high dimensional covariance estimation method that is based on eigenvalue regularization. The method is called Cover, short for COVariance Eigenvalue-Regularized estimation. It is fast to execute and enjoys excellent theoretical properties for the case when p is fixed. As a second contribution, this paper modifies Cover by incorporating Huber's loss function into the estimation procedure. By design, the resulting method is robust to outliers and is called RCover. The empirical performances of Cover and RCover are tested and compared with existing methods via a sequence of numerical experiments. It is shown that, with the presence of outliers, RCover almost always outperforms other methods tested.