Riemannian Gaussian Distributions on the Space of Symmetric Positive Definite Matrices

TL;DR

This paper introduces Riemannian Gaussian distributions on the space of symmetric positive definite matrices, providing exact density functions, statistical inference methods, and an EM algorithm for mixture modeling, with applications in texture classification.

Contribution

It provides the first exact expression of Riemannian Gaussian densities, links maximum likelihood to Riemannian center of mass, and develops an EM algorithm for mixture models on $ ext{P}_m$, improving classification performance.

Findings

Exact density functions for Riemannian Gaussian distributions derived

Maximum likelihood estimation linked to Riemannian center of mass

EM algorithm for mixture modeling improves classification accuracy

Abstract

Data which lie in the space $\mathcal{P}_{m\,}$, of $m \times m$ symmetric positive definite matrices, (sometimes called tensor data), play a fundamental role in applications including medical imaging, computer vision, and radar signal processing. An open challenge, for these applications, is to find a class of probability distributions, which is able to capture the statistical properties of data in $\mathcal{P}_{m\,}$, as they arise in real-world situations. The present paper meets this challenge by introducing Riemannian Gaussian distributions on $\mathcal{P}_{m\,}$. Distributions of this kind were first considered by Pennec in $2006$. However, the present paper gives an exact expression of their probability density function for the first time in existing literature. This leads to two original contributions. First, a detailed study of statistical inference for Riemannian Gaussian distributions, uncovering the connection between maximum likelihood estimation and the concept of Riemannian centre of mass, widely used in applications. Second, the derivation and implementation of an expectation-maximisation algorithm, for the estimation of mixtures of Riemannian Gaussian distributions. The paper applies this new algorithm, to the classification of data in $\mathcal{P}_{m\,}$, (concretely, to the problem of texture classification, in computer vision), showing that it yields significantly better performance, in comparison to recent approaches.