Structured Robust Covariance Estimation

Abstract

We consider robust covariance estimation with an emphasis on Tyler’s M-estimator. This method provides accurate inference of an unknown covariance in non-standard settings, including heavy-tailed distributions and outlier contaminated scenarios. We begin with a survey of the estimator and its various derivations in the classical unconstrained settings. The latter rely on the theory of g-convex analysis which we briefly review. Building on this background, we enhance robust covariance estimation via g-convex regularization, and allow accurate inference using a smaller number of samples. We consider shrinkage, diagonal loading, and prior knowledge in the form of symmetry and Kronecker structures. We introduce these concepts to the world of robust covariance estimation, and demonstrate how to exploit them in a computationally and statistically efficient manner. A. Wiesel and T. Zhang. Structured Robust Covariance Estimation. Foundations and Trends © in Signal Processing, vol. 8, no. 3, pp. 127–216, 2014. DOI: 10.1561/2000000053. Full text available at: http://dx.doi.org/10.1561/2000000053