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Chapitre D'ouvrage Année : 2022

Gaussian distributions on Riemannian symmetric spaces of non-positive curvature

Cyrus Mostajeran
  • Fonction : Auteur
Simon Heuveline
  • Fonction : Auteur

Résumé

This article aims to give a coherent presentation of the theory of Gaussian distributions on Riemannian symmetric spaces and also to report on recent original developments of this theory. The initial goal is to define a family of probability distributions, on any suitable Riemannian manifold, for which maximum-likelihood estimation, based on a finite sequence of observations, is equivalent to computation of the Riemannian barycentre of these observations. As it turns out, this goal is achievable whenever the underlying Riemannian manifold is a Riemannian symmetric space of non-positive curvature. In this case, the required Gaussian distributions are exactly the maximum-entropy distributions, for fixed barycentre and dispersion. The second step is the search for efficient means of computing the normalising factors associated with these distributions. This leads to a fascinating connection with random matrix theory, and even with theoretical physics (Chern-Simons theory), which yields a series of original results that provide exact expressions, as well as high-dimensional asymptotic expansions of the normalising factors. Another outcome of this connection with random matrix theory is the new idea of duality, between Gaussian distributions on Riemannian symetric spaces of opposite curvatures. The present article also investigates Bayesian inference for Gaussian distributions on symmetric spaces. This investigation motivates original results regarding Markov-Chain Monte Carlo and convex optimisation on Riemannian manifolds. It also reveals a new open problem (roughly, this concerns the equality of a posteriori mode with a posteriori barycentre), which should be the focus of future developments.
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Dates et versions

hal-03865789 , version 1 (22-11-2022)

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  • HAL Id : hal-03865789 , version 1

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Salem Said, Cyrus Mostajeran, Simon Heuveline. Gaussian distributions on Riemannian symmetric spaces of non-positive curvature. Handbook of Statistics Vol 46, pp.357-400, 2022. ⟨hal-03865789⟩
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