Lie Neurons: Adjoint-Equivariant Neural Networks for Semisimple Lie Algebras

Tzu-Yuan Lin, Minghan Zhu, Maani Ghaffari
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:30529-30545, 2024.

Abstract

This paper proposes an equivariant neural network that takes data in any finite-dimensional 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-lin24aa, title = {Lie Neurons: Adjoint-Equivariant Neural Networks for Semisimple Lie Algebras}, author = {Lin, Tzu-Yuan and Zhu, Minghan and Ghaffari, Maani}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {30529--30545}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/lin24aa/lin24aa.pdf}, url = {https://proceedings.mlr.press/v235/lin24aa.html}, abstract = {This paper proposes an equivariant neural network that takes data in any finite-dimensional 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.} }
Endnote
%0 Conference Paper %T Lie Neurons: Adjoint-Equivariant Neural Networks for Semisimple Lie Algebras %A Tzu-Yuan Lin %A Minghan Zhu %A Maani Ghaffari %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-lin24aa %I PMLR %P 30529--30545 %U https://proceedings.mlr.press/v235/lin24aa.html %V 235 %X This paper proposes an equivariant neural network that takes data in any finite-dimensional 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.
APA
Lin, T., Zhu, M. & Ghaffari, M.. (2024). Lie Neurons: Adjoint-Equivariant Neural Networks for Semisimple Lie Algebras. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:30529-30545 Available from https://proceedings.mlr.press/v235/lin24aa.html.

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