Group Symmetric Covariance Estimation

We focus on the problems of Gaussian and robust covariance estimation in high dimensions under group symmetry constraints. We assume the true covariance matrix to commute with a finite unitary matrix group, which is referred to as the group symmetry property. Examples of group symmetric structures include circulant, persymmetric, proper quaternion and many others. We develop the group symmetric versions of the Sample Covariance and Tyler's M-estimator and determine their performance benefits. In particular, the classical results claim that at least n = p and n = p+1 sample points in general position are necessary to ensure the existence and uniqueness of the Sample Covariance and Tyler's estimator respectively, where p is the ambient dimension. We significantly improve these requirements for both estimates and show that in many cases even 1 or 2 samples are enough to guarantee the existence and uniqueness regardless of p. We build a unified framework for group symmetric estimation and explain the nature and consequences of the underlying block-diagonalization phenomenon. In addition, we develop a group symmetric detection framework and investigate the gains it yields.