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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.09088 |
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| _version_ | 1866914583861526528 |
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| author | Menon, Govind Yu, Tianmin |
| author_facet | Menon, Govind Yu, Tianmin |
| contents | 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 submersion of the balanced manifold. The main technical step is an explicit construction of an orthonormal basis for the tangent space of the balanced manifold using the theory of Jacobi matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09088 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An entropy formula for the Deep Linear Network Menon, Govind Yu, Tianmin Machine Learning Differential Geometry Dynamical Systems 68T07, 49J40, 94A17, 60B20 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 submersion of the balanced manifold. The main technical step is an explicit construction of an orthonormal basis for the tangent space of the balanced manifold using the theory of Jacobi matrices. |
| title | An entropy formula for the Deep Linear Network |
| topic | Machine Learning Differential Geometry Dynamical Systems 68T07, 49J40, 94A17, 60B20 |
| url | https://arxiv.org/abs/2509.09088 |