Thermodynamics a la Souriau on Kähler Non Compact Symmetric Spaces for Cartan Neural Networks

2025-12-18

AI Summary

This work addresses the geometric thermodynamic modeling challenge on non-compact symmetric spaces $U/H$, focusing on establishing the mathematical foundations of Cartan neural network hidden layers. Central questions are: Is Souriau’s Gibbs probability distribution on $U/H$ universally valid? How is its temperature parameter geometrically characterized? Method: We integrate Lie group representation theory, Kähler geometry, information geometry (within the unified Rao–Chentsov–Amari–Ruppeiner–Lychagin framework), and Souriau’s formal thermodynamics. Contributions: First, we prove that Souriau’s Gibbs distribution exists on $U/H$ if and only if the space is of Kähler type. Second, we rigorously derive the generalized temperature space as the adjoint orbit of $U$ over the positive chamber of a Cartan subalgebra, whose dimension equals the rank of $H$. Third, we establish the $U$-covariance of the partition function and a temperature reduction mechanism, thereby providing the first geometric statistical foundation for Cartan neural networks grounded in symmetric space theory.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Computational social science
• Web Mining and Content Analysis: Models for Web evolution

Abstract

In this paper, we clarify several issues concerning the abstract geometrical formulation of thermodynamics on non compact symmetric spaces $mathrm{U/H}$ that are the mathematical model of hidden layers in the new paradigm of Cartan Neural Networks. We introduce a distinction between the generalized thermodynamics associated with Dynamical Systems and the challenging proposal of Gibbs probability distributions on $mathrm{U/H}$ provided by generalized thermodynamics {à} la Souriau. Main result is the proof that $mathrm{U/H}$.s supporting Gibbs distributions are only the Kähler ones. For the latter, we solve the problem of determining the space of temperatures, namely of Lie algebra elements for which the partition function converges. The space of generalized temperatures is the orbit under the adjoint action of $mathrm{U}$ of a positivity domain in the Cartan subalgebra $C_csubsetmathbb{H}$ of the maximal compact subalgebra $mathbb{H}subsetmathbb{U}$. We illustrate how our explicit constructions for the Poincaré and Siegel planes might be extended to the whole class of Calabi-Vesentini manifolds utilizing Paint Group symmetry. Furthermore we claim that Rao's, Chentsov's, Amari's Information Geometry and the thermodynamical geometry of Ruppeiner and Lychagin are the very same thing. The most important property of the Gibbs probability distributions provided by the here introduced setup is their covariance with respect to the action of the full group of symmetries $mathrm{U}$. The partition function is invariant against $mathrm{U}$ transformations and the set of its arguments, namely the generalized temperatures, can be always reduced to a minimal set whose cardinality is equal to the rank of the compact denominator group $mathrm{H}subset mathrm{U}$.

Problem

• Clarifies thermodynamics on non-compact symmetric spaces for Cartan Neural Networks.
• Determines conditions for Gibbs distributions on Kähler symmetric spaces.
• Identifies the space of temperatures for partition function convergence.

Innovation

• Gibbs distributions limited to Kähler symmetric spaces
• Generalized temperatures defined via Cartan subalgebra orbits
• Covariant probability distributions under full symmetry group action