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Information geometry on groupoids: the case of singular metrics

Katarzyna Grabowska, Janusz Grabowski, Marek Kus, Giuseppe Marmo

TL;DR

The case when the two-form is degenerate is studied and it is shown how in sufficiently regular cases one reduces it to a pseudometric structures.

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.

Information geometry on groupoids: the case of singular metrics

TL;DR

The case when the two-form is degenerate is studied and it is shown how in sufficiently regular cases one reduces it to a pseudometric structures.

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.

Paper Structure

This paper contains 8 sections, 12 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Information geometry
  3. Lie groupoids and Lie algebroids
  4. Lie algebroid connections
  5. Contrast functions on Lie groupoids
  6. Singular contrast functions
  7. Reduction to metric contrast function
  8. Concluding Remarks

Key Result

Proposition 3.1

There exists a unique Lie bracket $\left[.,. \right]$ on the space of sections $\operatorname{Sec}\left(\mathrm{ Lie}(\mathcal{G})\right)$, such that The bundle $\mathrm{ Lie}(\mathcal{G})\rightarrow M$ with the bracket $\left[.,. \right]$ and map $\alpha$ is a Lie algebroid. For $f\in\mathcal{C}^\infty(M)$ we define $f^L=\mathsf{s}^\ast f$ and $f^R=\mathsf{t}^\ast f$, then

Theorems & Definitions (27)

  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Example 3.3
  • Proposition 3.4: grabowska19
  • Remark 3.5
  • Theorem 3.6: grabowska19
  • Proposition 4.1
  • Remark 4.2
  • Proposition 4.3
  • ...and 17 more