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

Area of Science:

• Geometric Mechanics
• Statistical Mechanics
• Information Geometry
• Quantum Information

Background:

• Existing works on statistical mechanics and information geometry lack a unified framework.
• Geometric mechanics offers powerful tools like momentum maps and Casimir functions.
• Prior approaches include Souriau's symplectic model and quantum information geometry.

Purpose of the Study:

• To present a unified geometric framework for Gibbs probability densities and statistical mechanics.
• To emphasize the role of Lie group actions and geometric mechanics concepts.
• To demonstrate applications in information geometry, machine learning, and quantum information.

Main Methods:

• Utilizing a geometric framework based on Lie group actions and equivariance.
• Applying concepts from geometric mechanics: momentum maps, Casimir functions, coadjoint orbits, Lie-Poisson brackets.
• Developing symplectic and multisymplectic variational Lie group integration schemes.

Main Results:

• Unification of several earlier works on statistical mechanics and information geometry.
• Expression of the Fisher metric in the presence of equivariance.
• Geometric model for energy-preserving entropy production using entropy as a Casimir function.
• Illustration with multivariate Gaussian densities and the Bogoliubov-Kubo-Mori metric.

Conclusions:

• The geometric framework provides a powerful and unifying approach to statistical mechanics and information geometry.
• Equivariance and geometric mechanics concepts are crucial for diverse applications.
• The framework facilitates the development of novel integration schemes for complex systems.