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Open AccessFeature PaperArticle

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

1
Key Technology Domain PCC (Processing, Control & Cognition) Representative, Thales Land & Air Systems, Voie Pierre-Gilles de Gennes, F91470 Limours, France
2
Centre National de la Recherche Scientifique (CNRS), Le Laboratoire de Météorologie Dynamique (LMD), Ecole Normale Supérieure, 75005 Paris, France
*
Author to whom correspondence should be addressed.
Entropy 2020, 22(5), 498; https://doi.org/10.3390/e22050498
Received: 23 March 2020 / Revised: 17 April 2020 / Accepted: 21 April 2020 / Published: 25 April 2020
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 with Souriau symplectic and polysymplectic models, such as the Lie-Poisson equation with cocycle. View Full-Text
Keywords: 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 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
MDPI and ACS Style

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. https://doi.org/10.3390/e22050498

AMA Style

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(5):498. https://doi.org/10.3390/e22050498

Chicago/Turabian Style

Barbaresco, Frédéric; Gay-Balmaz, François. 2020. "Lie Group Cohomology and (Multi)Symplectic Integrators: New Geometric Tools for Lie Group Machine Learning Based on Souriau Geometric Statistical Mechanics" Entropy 22, no. 5: 498. https://doi.org/10.3390/e22050498

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