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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2510.23288 |
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| _version_ | 1866909870991605760 |
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| author | Li, Daiyuan Arya, Shreya Ghrist, Robert |
| author_facet | Li, Daiyuan Arya, Shreya Ghrist, Robert |
| contents | Most equivariant neural networks rely on a single global symmetry, limiting their use in domains where symmetries are instead local. We introduce Torsor CNNs, a framework for learning on graphs with local symmetries encoded as edge potentials -- group-valued transformations between neighboring coordinate frames. We establish that this geometric construction is fundamentally equivalent to the classical group synchronization problem, yielding: (1) a Torsor Convolutional Layer that is provably equivariant to local changes in coordinate frames, and (2) the frustration loss -- a standalone geometric regularizer that encourages locally equivariant representations when added to any NN's training objective. The Torsor CNN framework unifies and generalizes several architectures -- including classical CNNs and Gauge CNNs on manifolds -- by operating on arbitrary graphs without requiring a global coordinate system or smooth manifold structure. We establish the mathematical foundations of this framework and demonstrate its applicability to multi-view 3D recognition, where relative camera poses naturally define the required edge potentials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_23288 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Learning from Frustration: Torsor CNNs on Graphs Li, Daiyuan Arya, Shreya Ghrist, Robert Machine Learning Algebraic Topology es: 68T07, 22E70 Most equivariant neural networks rely on a single global symmetry, limiting their use in domains where symmetries are instead local. We introduce Torsor CNNs, a framework for learning on graphs with local symmetries encoded as edge potentials -- group-valued transformations between neighboring coordinate frames. We establish that this geometric construction is fundamentally equivalent to the classical group synchronization problem, yielding: (1) a Torsor Convolutional Layer that is provably equivariant to local changes in coordinate frames, and (2) the frustration loss -- a standalone geometric regularizer that encourages locally equivariant representations when added to any NN's training objective. The Torsor CNN framework unifies and generalizes several architectures -- including classical CNNs and Gauge CNNs on manifolds -- by operating on arbitrary graphs without requiring a global coordinate system or smooth manifold structure. We establish the mathematical foundations of this framework and demonstrate its applicability to multi-view 3D recognition, where relative camera poses naturally define the required edge potentials. |
| title | Learning from Frustration: Torsor CNNs on Graphs |
| topic | Machine Learning Algebraic Topology es: 68T07, 22E70 |
| url | https://arxiv.org/abs/2510.23288 |