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Open AccessFeature PaperArticle

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

Department of Advanced Radar Concepts, Thales Land Air Systems, Voie Pierre-Gilles de Gennes, 91470 Limours, France
Entropy 2018, 20(11), 840; https://doi.org/10.3390/e20110840
Received: 9 August 2018 / Revised: 23 September 2018 / Accepted: 9 October 2018 / Published: 2 November 2018
We introduce poly-symplectic extension of Souriau Lie groups thermodynamics based on higher-order model of statistical physics introduced by 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 surveys, …), the 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 Ingarden. We use a 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 moment 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. We also introduce a more synthetic Koszul definition of Fisher Metric, based on the Souriau model, that we name Souriau-Fisher metric. This Lie groups thermodynamics is the bedrock for Lie group machine learning providing a full covariant maximum entropy Gibbs density based on representation theory (symplectic structure of coadjoint orbits for Souriau non-equivariant model associated to a class of co-homology). View Full-Text
Keywords: higher order thermodynamics; Lie groups thermodynamics; homogeneous manifold; poly-symplectic manifold; dynamical systems; non-equivariant cohomology; Lie group machine learning; Souriau-Fisher metric higher order thermodynamics; Lie groups thermodynamics; homogeneous manifold; poly-symplectic manifold; dynamical systems; non-equivariant cohomology; Lie group machine learning; Souriau-Fisher metric
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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.

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