Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Lie Group Cohomology and (Multi)Symplectic Integrators : New Geometric Tools for Lie Group Machine Learning based on Souriau Geometric Statistical Mechanics

Version 1 : Received: 23 March 2020 / Approved: 31 March 2020 / Online: 31 March 2020 (10:28:16 CEST)

A peer-reviewed article of this Preprint also exists.

Barbaresco, F.; Gay-Balmaz, F. Lie Group Cohomology and (Multi)Symplectic Integrators: New Geometric Tools for Lie Group Machine Learning Based on Souriau Geometric Statistical Mechanics. Entropy 2020, 22, 498. Barbaresco, F.; Gay-Balmaz, F. Lie Group Cohomology and (Multi)Symplectic Integrators: New Geometric Tools for Lie Group Machine Learning Based on Souriau Geometric Statistical Mechanics. Entropy 2020, 22, 498.

Journal reference: Entropy 2020, 22, 498
DOI: 10.3390/e22050498

Abstract

In this paper we describe and exploit a geometric framework for Gibbs probability densities and the associated concepts in statistical mechanics, which unifies several earlier works on the subject, including Souriau's symplectic model of statistical mechanics, its polysymplectic extension, Koszul model, and approaches developed in quantum information geometry. We emphasize the role of equivariance with respect to Lie group actions and the role of several concepts from geometric mechanics, such as momentum maps, Casimir functions, coadjoint orbits, and Lie-Poisson brackets with cocycles, as unifying structures appearing in various applications of this framework to information geometry and machine learning. For instance, we discuss the expression of the Fisher metric in presence of equivariance and we exploit the property of the entropy of the Souriau model as a Casimir function to apply a geometric model for energy preserving entropy production. We illustrate this framework with several examples including multivariate Gaussian probability densities, and the Bogoliubov-Kubo-Mori metric as a quantum version of the Fisher metric for quantum information on coadjoint orbits. We exploit this geometric setting and Lie group equivariance to present symplectic and multisymplectic variational Lie group integration schemes for some of the equations associated to Souriau symplectic and polysymplectic models, such as the Lie-Poisson equation with cocycle.

Subject Areas

Momentum Maps; Cocycles; Lie Group Actions; Coadjoint Orbits; Variational Integrators; (Multi)symplectic Integrators; Fisher Metric; Gibbs Probability Density; Entropy; Lie Group Machine Learning; Casimir Functions

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