Pseudo-Riemannian geometry encodes information geometry in optimal transport

Abstract

Optimal transport and information geometry both study geometric structures on spaces of probability distributions. Optimal transport characterizes the cost-minimizing movement from one distribution to another, while information geometry originates from coordinate invariant properties of statistical inference. Their relations and applications in statistics and machine learning have started to gain more attention. In this paper we give a new differential-geometric relation between the two fields. Namely, the pseudo-Riemannian framework of Kim and McCann, which provides a geometric perspective on the fundamental Ma–Trudinger–Wang (MTW) condition in the regularity theory of optimal transport maps, encodes the dualistic structure of statistical manifold. This general relation is described using the framework of c-divergence under which divergences are defined by optimal transport maps. As a by-product, we obtain a new information-geometric interpretation of the MTW tensor on the graph of the transport map. This relation sheds light on old and new aspects of information geometry. The dually flat geometry of Bregman divergence corresponds to the quadratic cost and the pseudo-Euclidean space, and the logarithmic $$L^{(\alpha )}$$ L ( α ) -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 information-geometric curvature in terms of the divergence between a primal-dual pair of geodesics.

Geometric thermodynamics for the Fokker–Planck equation: stochastic thermodynamic links between information geometry and optimal transport

“…Because entropy production for the Fokker-Planck equation can be discussed from the viewpoint of both information geometry and optimal transport theory, these relations provide links between information geometry and optimal transport theory. Our proposed geometrical framework for non-equilibrium thermodynamics, namely geometric thermodynamics, offers a new perspective on links between information geometry and optimal transport theory [46][47][48][49] and the unification of nonequilibrium thermodynamic geometry.…”

λ-Deformation: A Canonical Framework for Statistical Manifolds of Constant Curvature

“…Indeed, we have the analogous reference–representation biduality (see [ 24 , 25 ]) that is characteristic of Bregman divergence and canonical divergence for dually flat spaces, that is ( 8 ). See [ 38 ] for the reference–representation biduality of a general c-divergence (which includes both the Bregman and logarithmic divergences) based on optimal transport.…”

When optimal transport meets information geometry

“…Wong and Yang [83] considered the framework of a c-divergence (based on [64] and [81]) and considered the graph of the optimal transport as a (possibly disconnected) statistical manifold equipped with the c-divergence which is induced by the cost function and the pair of Kantorovich potentials. In this work, they show that the dual connections ∇ and ∇ * are the projections of the Levi-Civita connection of the pseudo-Riemannian metric (15) on the product space.…”