Information geometry on groupoids: the case of singular metrics

TL;DR

This paper extends information geometry to the framework of Lie groupoids and Lie algebroids, exploring singular metrics, degeneracies, and generalized Levi-Civita connections in a unified geometric setting.

Contribution

It introduces a novel approach to contrast functions on Lie groupoids, analyzing their induced forms and pseudometric structures, and generalizes Levi-Civita connections for Riemannian foliations.

Findings

Degenerate two-forms lead to pseudometric structures.

Contrast functions induce two-forms and three-forms on Lie algebroids.

Generalization of Levi-Civita connections to Lie groupoid/Lie algebroid context.

Abstract

We use the general setting for contrast (potential) functions in statistical and information geometry provided by Lie groupoids and Lie algebroids. The contrast functions are defined on Lie groupoids and give rise to two-forms and three-forms on the corresponding Lie algebroid. We study the case when the two-form is degenerate and show how in sufficiently regular cases one reduces it to a pseudometric structures. Transversal Levi-Civita connections for Riemannian foliations are generalized to the Lie groupoid/Lie algebroid case.