Gaussian distributions in symmetric spaces: statistical learning from covariance matrices


Séminaire Données et Aléatoire Théorie & Applications

20/01/2022 - 14:00 Salem Said (LJK - DAO) Salle 106

There exist several definitions of the Gaussian distribution : maximum entropy, minimum uncertainty, the central limit theorem, or definitions arising from statistical physics. In a Euclidean space, all of these definitions lead to the same famous expression of the Gaussian density. However, in more general spaces, each definition leads to a quite different formulation. The presentation will introduce an original definition of the Gaussian distribution, which is valid in any Riemannian symmetric space of negative curvature. Namely, a Gaussian distribution is defined 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 one introduced here has two advantages : (i) many spaces of covariance matrices (real, complex, quaternion, Toeplitz, block-Toeplitz) are Riemannian symmetric spaces of negative curvature. (ii) it provides a statistical foundation to the use of the Riemannian barycentre, which has become the workhorse of data analysis on Riemannian manifolds. This definition will be compared to other definitions, its theoretical consequences will be developed, and it will be shown to lead to new practical statistical learning algorithms, well-adapted to high-dimensional big-data..Details may be found in the following paper and thesis :