Abstract

The concept of Gaussian distribution can be based on several different definitions: maximum entropy, minimum uncertainty, the central limit theorem, or the kinetic theory of gases. When considered in Euclidean space, all of these definitions lead to the same expression of the Gaussian distribution, but in more general spaces, different definitions lead to different expressions. This talk will propose an original definition of the concept of Gaussian distribution, which is valid in Riemannian symmetric spaces of negative curvature. Namely, the definition is given by the property that maximum likelihood is equivalent to Riemannian barycentre. There are no good or bad definitions, only more or less useful ones. The proposed definition offers two advantages (1) many spaces of covariance matrices (real, complex, quaternion, Toeplitz, block-Toeplitz) are Riemannian symmetric spaces of negative curvature (2) it provides a statistical foundation to the use of Riemannian barycentres, which is a popular technique in many applications. The talk will compare the proposed definition to other possible definitions, develop its theoretical consequences, and finally explain how it gives rise to new statistical learning algorithms, specifically adapted to big data and high-dimensional data, all of this being illustrated by examples. Details may be found in

https://arxiv.org/abs/1507.01760

https://arxiv.org/abs/1607.06929

https://arxiv.org/abs/1707.07163