PseudoRiemannian geometry embeds information geometry in optimal transport
Abstract
Optimal transport and information geometry both study geometric structures on spaces of probability distributions. Optimal transport characterizes the costminimizing movement from one distribution to another, while information geometry originates from coordinateinvariant properties of statistical inference. Their connections and applications in statistics and machine learning have started to gain more attention. In this paper we give a new differential geometric connection between the two fields. Namely, the pseudoRiemannian framework of Kim and McCann, a geometric perspective on the fundamental MaTrudingerWang (MTW) condition in the regularity theory of optimal transport maps, encodes the dualistic structure of statistical manifold. This general relation is described using the natural framework of $c$divergence, a divergence defined by an optimal transport map. As a byproduct, we obtain a new informationgeometric interpretation of the MTW tensor. This connection sheds light on old and new aspects of information geometry. The dually flat geometry of Bregman divergence corresponds to the quadratic cost and the pseudoEuclidean space, and the $L^{(\alpha)}$divergence introduced by Pal and the first author has constant sectional curvature in a sense to be made precise. In these cases we give a geometric interpretation of the informationgeometric curvature in terms of the divergence between a primaldual pair of geodesics.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 DOI:
 10.48550/arXiv.1906.00030
 arXiv:
 arXiv:1906.00030
 Bibcode:
 2019arXiv190600030W
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Probability;
 Mathematics  Statistics Theory
 EPrint:
 28 pages, 2 figures. Substantially revised. [Originally titled "Optimal transport and information geometry.]