Logarithmic divergences from optimal transport and R\'enyi geometry

TL;DR

This paper introduces a unified family of divergences from optimal transport duality, revealing their geometric properties and connections to exponential families and Rényi divergences, with implications for statistical manifold geometry.

Contribution

It defines the $L^{(eta)}$-divergences, explores their geometric structure, and characterizes manifolds with constant curvature related to these divergences.

Findings

Induced geometries are dually projectively flat with constant sectional curvatures.

The $L^{(eta)}$-divergences generalize Bregman and Pal's $L$-divergences.

A generalized Pythagorean theorem holds for these geometries.

Abstract

Divergences, also known as contrast functions, are distance-like quantities defined on manifolds of non-negative or probability measures. Using the duality in optimal transport, we introduce and study the one-parameter family of $L^{(\pm \alpha)}$-divergences. It includes the Bregman divergence corresponding to the Euclidean quadratic cost, and the $L$-divergence introduced by Pal and the author in connection with portfolio theory and a logarithmic cost function. They admit natural generalizations of exponential family that are closely related to the $\alpha$-family and $q$-exponential family. In particular, the $L^{(\pm \alpha)}$-divergences of the corresponding potential functions are R\'{e}nyi divergences. Using this unified framework we prove that the induced geometries are dually projectively flat with constant sectional curvatures, and a generalized Pythagorean theorem holds true. Conversely, we show that if a statistical manifold is dually projectively flat with constant curvature $\pm \alpha$ with $\alpha > 0$, then it is locally induced by an $L^{(\mp \alpha)}$-divergence. We define in this context a canonical divergence which extends the one for dually flat manifolds.