Riemannian Gaussian Distributions on the Space of Symmetric Positive Definite Matrices
Data, which lie in the space <inline-formula> <tex-math notation="LaTeX">$\mathcal {P}_{m\,}$ </tex-math></inline-formula>, of <inline-formula> <tex-math notation="LaTeX">$m \times m$ </tex-math></inline-formula> symmetric positive definite matrices, (sometimes called <italic>tensor data</italic>), 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 <inline-formula> <tex-math notation="LaTeX">$\mathcal {P}_{m\,}$ </tex-math></inline-formula>, as they arise in real-world situations. The present paper meets this challenge by introducing <italic>Riemannian Gaussian distributions</italic> on <inline-formula> <tex-math notation="LaTeX">$\mathcal {P}_{m\,}$ </tex-math></inline-formula>. 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 the maximum likelihood estimation and the concept of Riemannian centre of mass, widely used in applications. Second, the derivation and the 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 <inline-formula> <tex-math notation="LaTeX">$\mathcal {P}_{m\,}$ </tex-math></inline-formula>, (concretely, to the problem of texture classification, in computer vision), showing that it yields significantly better performance, in comparison to recent approaches.