# Logarithmic divergences: geometry and interpretation of curvature

## TL;DR

This paper introduces the logarithmic $L^{(eta)}$-divergence, linking it to optimal transport and geometric structures, and interprets its curvature in the context of statistical manifolds with constant sectional curvature.

## Contribution

It establishes the geometric equivalence of the logarithmic divergence to conformal transformations and affine immersions, revealing its role as a canonical divergence in curved statistical manifolds.

## Key findings

- Logarithmic divergence is equivalent to a conformal transformation of Bregman divergence.
- The divergence corresponds to a statistical manifold with constant sectional curvature.
- Provides a geometric interpretation of curvature in terms of primal and dual geodesics.

## Abstract

We study the logarithmic $L^{(\alpha)}$-divergence which extrapolates the Bregman divergence and corresponds to solutions to novel optimal transport problems. We show that this logarithmic divergence is equivalent to a conformal transformation of the Bregman divergence, and, via an explicit affine immersion, is equivalent to Kurose's geometric divergence. In particular, the $L^{(\alpha)}$-divergence is a canonical divergence of a statistical manifold with constant sectional curvature $-\alpha$. For such a manifold, we give a geometric interpretation of its sectional curvature in terms of how the divergence between a pair of primal and dual geodesics differ from the dually flat case. Further results can be found in our follow-up paper [27] which uncovers a novel relation between optimal transport and information geometry.