  In this paper we consider Tyler's robust covariance M-estimator under group
symmetry constraints. We assume that the covariance matrix is invariant to the
conjugation action of a unitary matrix group, referred to as group symmetry.
Examples of group symmetric structures include circulant, perHermitian and
proper quaternion matrices. We introduce a group symmetric version of Tyler's
estimator (STyler) and provide an iterative fixed point algorithm to compute
it. The classical results claim that at least n=p+1 sample points in general
position are necessary to ensure the existence and uniqueness of Tyler's
estimator, where p is the ambient dimension. We show that the STyler requires
significantly less samples. In some groups even two samples are enough to
guarantee its existence and uniqueness. In addition, in the case of elliptical
populations, we provide high probability bounds on the error of the STyler.
These too, quantify the advantage of exploiting the symmetry structure.
Finally, these theoretical results are supported by numerical simulations.ted
by numerical simulations.
