Stochastic EM algorithm for mixture estimation on manifolds

Abstract

This paper presents a novel algorithm for estimating parameters of a mixture of Gaussian laws when data lie in a Riemannian manifold. We consider the stochastic variant of the well-known Expectation-Maximization (EM) algorithm in the case of Riemannian geometry. The Riemannian mixture is devoted, here, to the case of Riemannian manifold of Symmetric Positive Definite (SPD) matrices. With a slight modification, the stochastic EM algorithm developed originally for Euclidean case can also be derived for SPD manifold. We provide some Monte- Carlo numerical simulations in order to analyse, in details, the proposed algorithm in comparison with the conventional EM one.