Barbaresco, F. Higher Order Geometric Theory of Information and Heat Based on Poly-Symplectic Geometry of Souriau Lie Groups Thermodynamics and Their Contextures: The Bedrock for Lie Group Machine Learning. Entropy2018, 20, 840.
Barbaresco, F. Higher Order Geometric Theory of Information and Heat Based on Poly-Symplectic Geometry of Souriau Lie Groups Thermodynamics and Their Contextures: The Bedrock for Lie Group Machine Learning. Entropy 2018, 20, 840.
Barbaresco, F. Higher Order Geometric Theory of Information and Heat Based on Poly-Symplectic Geometry of Souriau Lie Groups Thermodynamics and Their Contextures: The Bedrock for Lie Group Machine Learning. Entropy2018, 20, 840.
Barbaresco, F. Higher Order Geometric Theory of Information and Heat Based on Poly-Symplectic Geometry of Souriau Lie Groups Thermodynamics and Their Contextures: The Bedrock for Lie Group Machine Learning. Entropy 2018, 20, 840.
Abstract
We introduce poly-symplectic extension of Souriau Lie groups Thermodynamics based on higher-order model of statistical physics introduced by R.S. Ingarden. This extended model could be used for small data analytics and Machine Learning on Lie groups. Souriau Geometric Theory of Heat is well adapted to describe density of probability (Maximum Entropy Gibbs density) of data living on groups or on homogeneous manifolds. For Small Data Analytics (Rarified Gases , sparse statistical survey,…), density of maximum entropy should consider Higher Order Moments constraints (Gibbs density is not only defined by first moment but fluctuations request 2nd order and higher moments) as introduced by R.S. Ingarden. We use Poly-sympletic model introduced by Christian Günther, replacing the symplectic form by a vector-valued form. The poly-symplectic approach generalizes the Noether theorem, the existence of momentum mappings, the Lie algebra structure of the space of currents, the (non-)equivariant cohomology and the classification of G-homogeneous systems. The formalism is covariant, i.e. no special coordinates or coordinate systems on the parameter space are used to construct the Hamiltonian equations. We underline the contextures of these models, and the process to build these generic structures.
Keywords
Higher Order Thermodynamics; Lie Groups Thermodynamics; Homogeneous Manifold; Poly-Symplectic Manifold; Dynamical Systems; Non-equivariant Cohomology
Subject
Physical Sciences, Mathematical Physics
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
