Optimal transport and information geometry
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Optimal transport and information geometry are both mathematical frameworks for studying geometries on spaces of probability distribution, and their connections have attracted more and more attention. In this paper we show that the pseudo-Riemannian framework of Kim and McCann, a geometric approach to the celebrated Ma-Trudinger-Wang condition in the regularity theory of optimal transport maps, encodes the dualistic structure in information geometry. This general relation is described using the natural framework of c-divergence, a divergence defined by an optimal transport map. This connection sheds light on old and new aspects of information geometry. For example, the dually flat geometry of Bregman divergence corresponds to the quadratic cost and the pseudo-Euclidean space, and the L^(α)-divergence introduced by Pal and the first author has constant sectional curvature in a sense to be made precise. We also show that the L^(α)-divergence is equivalent to a conformal divergence and Kurose's geometric divergence. Finally, we study canonical divergences in information geometry and interpret them using the pseudo-Riemannian framework.
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