arXiv:2509.09088v3 Announce Type: replace Abstract: We study the Riemannian geometry of the Deep Linear Network (DLN) as a foundation for a thermodynamic description of the learning process. The main tools are the use of group actions to analyze overparametrization and the use of Riemannian submersion from the space of parameters to the space of observables. The foliation of the balanced manifold in the parameter space by group orbits is used to define and compute a Boltzmann entropy. We also show that the Riemannian geometry on the space of observables defined in [2] is obtained by Riemannian
This paper leverages advanced mathematical concepts to deepen the theoretical understanding of deep learning, aligning with a broader academic push for more robust AI foundations.
Understanding the fundamental physics and thermodynamics of deep learning could lead to more efficient, predictable, and scalable AI systems, moving beyond empirical breakthroughs.
This research provides new theoretical tools for analyzing neural networks, potentially guiding future architectural designs and optimizing training processes.
- · AI researchers
- · Hyperscalers
- · Academic institutions
- · Companies relying solely on empirical 'black box' AI development
Improved theoretical models for deep linear networks.
Development of more energy-efficient and robust AI training algorithms guided by thermodynamic principles.
Potential for new AI hardware architectures designed with these thermodynamic and geometric insights in mind.
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