Towards a Canonical Divergence within Information Geometry

TL;DR

This paper extends classical Riemannian geodesic results to Information Geometry, introducing a new canonical divergence that recovers dual structures and generalizes existing divergences in various geometric contexts.

Contribution

It proposes a novel canonical divergence in Information Geometry that generalizes and unifies existing divergences, applicable to dually convex sets and dual structures.

Findings

The new divergence recovers known divergences in special cases.

It characterizes geodesic rays via tangent vectors in the pseudo-distance framework.

The divergence aligns with Ay and Amari's in self-dual and flat cases.

Abstract

In Riemannian geometry geodesics are integral curves of the Riemannian distance gradient. We extend this classical result to the framework of Information Geometry. In particular, we prove that the rays of level-sets defined by a pseudo-distance are generated by the sum of two tangent vectors. By relying on these vectors, we propose a novel definition of a canonical divergence and its dual function. We prove that the new divergence allows to recover a given dual structure $(\mathrm{g},\nabla,\nabla^*)$ of {a dually convex set on} a smooth manifold $\mathrm{M}$. Additionally, we show that this divergence coincides with the canonical divergence proposed by Ay and Amari in the case of: (a) self-duality, (b) dual flatness, (c) statistical geometric analogue of the concept of symmetric spaces in Riemannian geometry. For a dually convex set, the case (c) leads to a further comparison of the new divergence with the one introduced by Henmi and Kobayashi.