Geometry of covariance matrices, covariance operators, and Gaussian processes – theory and applications

Covariance matrices, covariance operators, Gaussian measures, and Gaussian processes play important roles in many areas of machine learning, statistics, and numerous application domains. Of particular importance in the study of these objects and their applications are the similarity functions between them. In this talk, we present an overview of some recent generalizations of non-Euclidean distances between covariance matrices and Gaussian measures on Euclidean space to the infinite-dimensional setting of covariance operators and Gaussian processes. Our focus is on the generalization of the Fisher-Rao distance and related quantities (from Information Geometry) and the Entropic Regularization of the 2-Wasserstein distance (from Optimal Transport). Our formulations all admit closed form expressions that can be readily applied in practice, particularly in the reproducing kernel Hilbert space (RKHS) setting. The mathematical formulations will be accompanied by numerical experiments in computer vision.