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

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.

Lie Group Statistics and Lie Group Machine Learning Based on Souriau Lie Groups Thermodynamics & Koszul-Souriau-Fisher Metric: New Entropy Definition as Generalized Casimir Invariant Function in Coadjoint Representation

“…It opens the door to new generalization of Maximum Entropy method and first of all computation of “Gaussian densities” for any Lie group. Applications of this new property is not developed in this paper but in a twin paper in the same special issue [ 17 ]. We refert to M. Gromov papers to consider more geometric structures of Entropy [ 30 , 31 ].…”

“…Trofimov who have also deeply studied Casimir functions (but in case of null cohomology) and have developed the following equation that we can write for Entropy in null cohomology case with a representation of Lie algebras defined on basis in . We refer to a twin paper [ 17 ] developing consequences of this new definition of Entropy as an invariant Casimir function. In this twin paper, we study the associated Euler-Poincaré equation and the stochastic extension based on a new Stratonovich differential equation for the stochastic process given by the following relation by mean of Souriau’s symplectic cocycle .…”

“…This allows to extend the general Lie algebraic approach developed in [ 77 , 78 ] for Casimir dissipation, to take into account of a cocycle, and to a wider class of dissipation. Paper [ 17 ] will exploit this new Casimir equation in case of non-null cohomology.…”

“…We show that we can also introduce an Euler-Poincaré equation and its stochastic variant to study other open problems in statistics and thermodynamics. The application of this Casimir characterization, which is demonstrated in this article, are developed in another twin article published in the same special issue with François Gay-Balmaz [ 17 ].…”

Pyramid Transform of Manifold Data via Subdivision Operators

“…The manifold expresses both a global nonlinear structure with constrained, high-dimensional elements. The employment of manifold values as data models raises the demand for computational methods to address fundamental tasks like integration, interpolation, and regression, which become challenging under the manifold setting, see, e.g., [1,2,24,41]. We focus on constructing a multiscale representation for manifold values using a fast pyramid transform.Multiscale transforms are standard tools in signal and image processing that enables a hierarchical analysis of an object mathematically.…”