Universal Collection of Euclidean Invariants between Pairs of Position-Orientations
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912461513293824 |
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| author | Bellaard, Gijs Smets, Bart M. N. Duits, Remco |
| author_facet | Bellaard, Gijs Smets, Bart M. N. Duits, Remco |
| contents | Euclidean E(3) equivariant neural networks that employ scalar fields on position-orientation space M(3) have been effectively applied to tasks such as predicting molecular dynamics and properties. To perform equivariant convolutional-like operations in these architectures one needs Euclidean invariant kernels on M(3) x M(3). In practice, a handcrafted collection of invariants is selected, and this collection is then fed into multilayer perceptrons to parametrize the kernels. We rigorously describe an optimal collection of 4 smooth scalar invariants on the whole of M(3) x M(3). With optimal we mean that the collection is independent and universal, meaning that all invariants are pertinent, and any invariant kernel is a function of them. We evaluate two collections of invariants, one universal and one not, using the PONITA neural network architecture. Our experiments show that using a collection of invariants that is universal positively impacts the accuracy of PONITA significantly. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_03299 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Universal Collection of Euclidean Invariants between Pairs of Position-Orientations Bellaard, Gijs Smets, Bart M. N. Duits, Remco Differential Geometry Machine Learning Euclidean E(3) equivariant neural networks that employ scalar fields on position-orientation space M(3) have been effectively applied to tasks such as predicting molecular dynamics and properties. To perform equivariant convolutional-like operations in these architectures one needs Euclidean invariant kernels on M(3) x M(3). In practice, a handcrafted collection of invariants is selected, and this collection is then fed into multilayer perceptrons to parametrize the kernels. We rigorously describe an optimal collection of 4 smooth scalar invariants on the whole of M(3) x M(3). With optimal we mean that the collection is independent and universal, meaning that all invariants are pertinent, and any invariant kernel is a function of them. We evaluate two collections of invariants, one universal and one not, using the PONITA neural network architecture. Our experiments show that using a collection of invariants that is universal positively impacts the accuracy of PONITA significantly. |
| title | Universal Collection of Euclidean Invariants between Pairs of Position-Orientations |
| topic | Differential Geometry Machine Learning |
| url | https://arxiv.org/abs/2504.03299 |