Gaussian distributions on Riemannian symmetric spaces: statistical learning with structured covariance matrices

The Riemannian geometry of covariance matrices has been essential to several successful applications, in computer vision, biomedical signal and image processing, and radar data processing. For these applications, an important ongoing challenge is to develop Riemannian-geometric tools which are adapted to structured covariance matrices. The present paper proposes to meet this challenge by introducing a new class of probability distributions, Gaussian distributions of structured covariance matrices. These are Riemannian analogs of Gaussian distributions, which only sample from covariance matrices having a preassigned structure, such as complex, Toeplitz, or block-Toeplitz. The usefulness of these distributions stems from three features: (1) they are completely tractable, analytically or numerically, when dealing with large covariance matrices, (2) they provide a statistical foundation to the concept of structured Riemannian barycentre (i.e. Fr\'echet or geometric mean), (3) they lead to efficient statistical learning algorithms, which realise, among others, density estimation and classification of structured covariance matrices. The paper starts from the observation that several spaces of structured covariance matrices, considered from a geometric point of view, are Riemannian symmetric spaces. Accordingly, it develops an original theory of Gaussian distributions on Riemannian symmetric spaces, of their statistical inference, and of their relationship to the concept of Riemannian barycentre. Then, it uses this original theory to give a detailed description of Gaussian distributions of three kinds of structured covariance matrices, complex, Toeplitz, and block-Toeplitz. Finally, it describes algorithms for density estimation and classification of structured covariance matrices, based on Gaussian distribution mixture models.