The Geometry of Machine Learning Models
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916879283519488 |
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| author | Gajer, Pawel Ravel, Jacques |
| author_facet | Gajer, Pawel Ravel, Jacques |
| contents | This paper presents a mathematical framework for analyzing machine learning
models through the geometry of their induced partitions. By representing
partitions as Riemannian simplicial complexes, we capture not only adjacency
relationships but also geometric properties including cell volumes, volumes of
faces where cells meet, and dihedral angles between adjacent cells. For neural
networks, we introduce a differential forms approach that tracks geometric
structure through layers via pullback operations, making computations
tractable by focusing on data-containing cells. The framework enables
geometric regularization that directly penalizes problematic spatial
configurations and provides new tools for model refinement through extended
Laplacians and simplicial splines. We also explore how data distribution
induces effective geometric curvature in model partitions, developing discrete
curvature measures for vertices that quantify local geometric complexity and
statistical Ricci curvature for edges that captures pairwise relationships
between cells. While focused on mathematical foundations, this geometric
perspective offers new approaches to model interpretation, regularization, and
diagnostic tools for understanding learning dynamics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_02080 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Geometry of Machine Learning Models Gajer, Pawel Ravel, Jacques Machine Learning This paper presents a mathematical framework for analyzing machine learning models through the geometry of their induced partitions. By representing partitions as Riemannian simplicial complexes, we capture not only adjacency relationships but also geometric properties including cell volumes, volumes of faces where cells meet, and dihedral angles between adjacent cells. For neural networks, we introduce a differential forms approach that tracks geometric structure through layers via pullback operations, making computations tractable by focusing on data-containing cells. The framework enables geometric regularization that directly penalizes problematic spatial configurations and provides new tools for model refinement through extended Laplacians and simplicial splines. We also explore how data distribution induces effective geometric curvature in model partitions, developing discrete curvature measures for vertices that quantify local geometric complexity and statistical Ricci curvature for edges that captures pairwise relationships between cells. While focused on mathematical foundations, this geometric perspective offers new approaches to model interpretation, regularization, and diagnostic tools for understanding learning dynamics. |
| title | The Geometry of Machine Learning Models |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2508.02080 |