A Riemannian Framework for Low-Rank Structured Elliptical Models

Abstract

This paper proposes an original Riemmanian geometry for low-rank structured elliptical models, i.e., when samples are elliptically distributed with a covariance matrix that has a low-rank plus identity structure. The considered geometry is the one induced by the product of the Stiefel manifold and the manifold of Hermitian positive definite matrices, quotiented by the unitary group. One of the main contribution is to consider an original Riemannian metric, leading to new representations of tangent spaces and geodesics. From this geometry, we derive a new Riemannian optimization framework for robust covariance estimation, which is leveraged to minimize the popular Tyler's cost function on the considered quotient manifold. We also obtain a new divergence function, which is exploited to define a geometrical error measure on the quotient, and the corresponding intrinsic Cramér-Rao lower bound is derived. Thanks to the structure of the chosen parametrization, we further consider the subspace estimation error on the Grassmann manifold and provide its intrinsic Cramér-Rao lower bound. Our theoretical results are illustrated on some numerical experiments, showing the interest of the proposed optimization framework and that performance bounds can be reached.