Souriau-Casimir Lie Groups Thermodynamics and Machine Learning

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).

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Lie Group Machine Learning and Gibbs Density on Poincaré Unit Disk from Souriau Lie Groups Thermodynamics and SU(1,1) Coadjoint Orbits

Lecture Notes in Computer Science - Geometric Science of Information ◽

10.1007/978-3-030-26980-7_17 ◽

2019 ◽

pp. 157-170

Author(s):

Frédéric Barbaresco

Keyword(s):

Machine Learning ◽

Lie Groups ◽

Unit Disk ◽

Lie Group ◽

Coadjoint Orbits

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Mind wandering as data augmentation: How mental travel supports abstraction

Behavioral and Brain Sciences ◽

10.1017/s0140525x1900311x ◽

2020 ◽

Vol 43 ◽

Author(s):

Myrthe Faber

Keyword(s):

Machine Learning ◽

Data Augmentation ◽

Mental Content ◽

Mind Wandering ◽

Theoretical Framework ◽

Important Addition

Abstract Gilead et al. state that abstraction supports mental travel, and that mental travel critically relies on abstraction. I propose an important addition to this theoretical framework, namely that mental travel might also support abstraction. Specifically, I argue that spontaneous mental travel (mind wandering), much like data augmentation in machine learning, provides variability in mental content and context necessary for abstraction.

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