We address structured covariance estimation in elliptical distributions by
assuming that the covariance is a priori known to belong to a given convex set,
e.g., the set of Toeplitz or banded matrices. We consider the General Method of
Moments (GMM) optimization applied to robust Tyler's scatter M-estimator
subject to these convex constraints. Unfortunately, GMM turns out to be
non-convex due to the objective. Instead, we propose a new COCA estimator - a
convex relaxation 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 advantages of COCA in
synthetic simulations with structured compound Gaussian distributions. In these
examples, COCA outperforms competing methods such as Tyler's estimator and its
projection onto the structure set.Comment: arXiv admin note: text overlap with arXiv:1311.059