COVARIANCE ESTIMATION IN ELLIPTICAL MODELS WITH CONVEX STRUCTURE

We address structured covariance estimation in Elliptical distribu-tion. We assume it is a priori known that the covariance belongs to a given convex set, e.g., the set of Toeplitz or banded matrices. We consider the General Method of Moments (GMM) optimization subject to these convex constraints. Unfortunately, GMM is still non-convex due to objective. Instead, we propose COCA- a convex re-laxation which can be efficiently solved. We prove that the relaxation is tight in the unconstrained case for a finite number of samples, and in the constrained case asymptotically. We then illustrate the advan-tages of COCA in synthetic simulations with structured Compound Gaussian distributions. In these examples, COCA outperforms com-peting methods as Tyler’s estimate and its projection onto a convex set. Index Terms — Elliptical distribution, Tyler’s scatter estimator, Generalized Method of Moments, robust covariance estimation