Riemannian Lp Center of Mass for Scatter Matrix Estimation in Complex Elliptically Symmetric Distributions

Abstract

A popular method for fusing a set of covariance matrix estimates (with unavailable correlation) is to solve their geometrical mean or median, which is defined by a Riemannian geometry of Hermitian positive-definite (HPD) matrices. The most well-known such geometry is identical to the Fisher information geometry of multivariate Gaussian distributions with a fixed mean. This paper identifies the space of HPD matrices with the manifold of centered (i.e., zero-mean) complex elliptically symmetric (CES) distributions. First, the Fisher information matrix for the CES distributions defines a different Riemannian metric on HPD matrices, and the induced Riemannian geometry is studied. Then, the Riemannian L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> mean of some HPD matrices is calculated to produce a final estimation for the scatter matrix (proportional to the covariance matrix) of a CES distribution. While the corresponding objective function is proven to be gconvex, a Riemannian gradient descent algorithm is given to compute the solution. Finally, numerical examples are provided to illustrate the derived geometrical structure and its application to target detection.