Lie Neurons: Adjoint-Equivariant Neural Networks for Semisimple Lie Algebras
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910475305877504 |
|---|---|
| author | Lin, Tzu-Yuan Zhu, Minghan Ghaffari, Maani |
| author_facet | Lin, Tzu-Yuan Zhu, Minghan Ghaffari, Maani |
| contents | This paper proposes an equivariant neural network that takes data in any semi-simple Lie algebra as input. The corresponding group acts on the Lie algebra as adjoint operations, making our proposed network adjoint-equivariant. Our framework generalizes the Vector Neurons, a simple $\mathrm{SO}(3)$-equivariant network, from 3-D Euclidean space to Lie algebra spaces, building upon the invariance property of the Killing form. Furthermore, we propose novel Lie bracket layers and geometric channel mixing layers that extend the modeling capacity. Experiments are conducted for the $\mathfrak{so}(3)$, $\mathfrak{sl}(3)$, and $\mathfrak{sp}(4)$ Lie algebras on various tasks, including fitting equivariant and invariant functions, learning system dynamics, point cloud registration, and homography-based shape classification. Our proposed equivariant network shows wide applicability and competitive performance in various domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_04521 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lie Neurons: Adjoint-Equivariant Neural Networks for Semisimple Lie Algebras Lin, Tzu-Yuan Zhu, Minghan Ghaffari, Maani Machine Learning Artificial Intelligence This paper proposes an equivariant neural network that takes data in any semi-simple Lie algebra as input. The corresponding group acts on the Lie algebra as adjoint operations, making our proposed network adjoint-equivariant. Our framework generalizes the Vector Neurons, a simple $\mathrm{SO}(3)$-equivariant network, from 3-D Euclidean space to Lie algebra spaces, building upon the invariance property of the Killing form. Furthermore, we propose novel Lie bracket layers and geometric channel mixing layers that extend the modeling capacity. Experiments are conducted for the $\mathfrak{so}(3)$, $\mathfrak{sl}(3)$, and $\mathfrak{sp}(4)$ Lie algebras on various tasks, including fitting equivariant and invariant functions, learning system dynamics, point cloud registration, and homography-based shape classification. Our proposed equivariant network shows wide applicability and competitive performance in various domains. |
| title | Lie Neurons: Adjoint-Equivariant Neural Networks for Semisimple Lie Algebras |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2310.04521 |