Navigation through Non-Compact Symmetric Spaces: a mathematical perspective on Cartan Neural Networks
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866913955241263104 |
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| author | Fré, Pietro Giuseppe Milanesio, Federico Sanguinetti, Guido Santoro, Matteo |
| author_facet | Fré, Pietro Giuseppe Milanesio, Federico Sanguinetti, Guido Santoro, Matteo |
| contents | Recent work has identified non-compact symmetric spaces U/H as a promising class of homogeneous manifolds to develop a geometrically consistent theory of neural networks. An initial implementation of these concepts has been presented in a twin paper under the moniker of Cartan Neural Networks, showing both the feasibility and the performance of these geometric concepts in a machine learning context. The current paper expands on the mathematical structures underpinning Cartan Neural Networks, detailing the geometric properties of the layers and how the maps between layers interact with such structures to make Cartan Neural Networks covariant and geometrically interpretable. Together, these twin papers constitute a first step towards a fully geometrically interpretable theory of neural networks exploiting group-theoretic structures |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_16871 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Navigation through Non-Compact Symmetric Spaces: a mathematical perspective on Cartan Neural Networks Fré, Pietro Giuseppe Milanesio, Federico Sanguinetti, Guido Santoro, Matteo Machine Learning High Energy Physics - Theory Recent work has identified non-compact symmetric spaces U/H as a promising class of homogeneous manifolds to develop a geometrically consistent theory of neural networks. An initial implementation of these concepts has been presented in a twin paper under the moniker of Cartan Neural Networks, showing both the feasibility and the performance of these geometric concepts in a machine learning context. The current paper expands on the mathematical structures underpinning Cartan Neural Networks, detailing the geometric properties of the layers and how the maps between layers interact with such structures to make Cartan Neural Networks covariant and geometrically interpretable. Together, these twin papers constitute a first step towards a fully geometrically interpretable theory of neural networks exploiting group-theoretic structures |
| title | Navigation through Non-Compact Symmetric Spaces: a mathematical perspective on Cartan Neural Networks |
| topic | Machine Learning High Energy Physics - Theory |
| url | https://arxiv.org/abs/2507.16871 |