We address structured covariance estimation in elliptical distributions byassuming 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 ofMoments (GMM) optimization applied to robust Tyler's scatter M-estimatorsubject to these convex constraints. Unfortunately, GMM turns out to benon-convex due to the objective. Instead, we propose a new COCA estimator - aconvex relaxation which can be efficiently solved. We prove that the relaxationis tight in the unconstrained case for a finite number of samples, and in theconstrained case asymptotically. We then illustrate the advantages of COCA insynthetic simulations with structured compound Gaussian distributions. In theseexamples, COCA outperforms competing methods such as Tyler's estimator and itsprojection onto the structure set.
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