Computing representations for Lie algebraic networks

Noah Shutty, Casimir Wierzynski
Proceedings of the 1st NeurIPS Workshop on Symmetry and Geometry in Neural Representations, PMLR 197:1-21, 2023.

Abstract

Recent work has constructed neural networks that are equivariant to continuous symmetry groups such as 2D and 3D rotations. This is accomplished using explicit {\it Lie group representations} to derive the equivariant kernels and nonlinearities. We present three contributions motivated by frontier applications of equivariance beyond rotations and translations. First, we relax the requirement for explicit Lie group representations with a novel algorithm that finds representations of arbitrary Lie groups given only the {\it structure constants} of the associated Lie algebra. Second, we provide a self-contained method and software for building Lie group-equivariant neural networks using these representations. Third, we contribute a novel benchmark dataset for classifying objects from relativistic point clouds, and apply our methods to construct the first object-tracking model equivariant to the Poincaré group.

Cite this Paper

BibTeX
@InProceedings{pmlr-v197-shutty23a, title = {Computing representations for Lie algebraic networks}, author = {Shutty, Noah and Wierzynski, Casimir}, booktitle = {Proceedings of the 1st NeurIPS Workshop on Symmetry and Geometry in Neural Representations}, pages = {1--21}, year = {2023}, editor = {Sanborn, Sophia and Shewmake, Christian and Azeglio, Simone and Di Bernardo, Arianna and Miolane, Nina}, volume = {197}, series = {Proceedings of Machine Learning Research}, month = {03 Dec}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v197/shutty23a/shutty23a.pdf}, url = {https://proceedings.mlr.press/v197/shutty23a.html}, abstract = {Recent work has constructed neural networks that are equivariant to continuous symmetry groups such as 2D and 3D rotations. This is accomplished using explicit {\it Lie group representations} to derive the equivariant kernels and nonlinearities. We present three contributions motivated by frontier applications of equivariance beyond rotations and translations. First, we relax the requirement for explicit Lie group representations with a novel algorithm that finds representations of arbitrary Lie groups given only the {\it structure constants} of the associated Lie algebra. Second, we provide a self-contained method and software for building Lie group-equivariant neural networks using these representations. Third, we contribute a novel benchmark dataset for classifying objects from relativistic point clouds, and apply our methods to construct the first object-tracking model equivariant to the Poincaré group.} }
Endnote
%0 Conference Paper %T Computing representations for Lie algebraic networks %A Noah Shutty %A Casimir Wierzynski %B Proceedings of the 1st NeurIPS Workshop on Symmetry and Geometry in Neural Representations %C Proceedings of Machine Learning Research %D 2023 %E Sophia Sanborn %E Christian Shewmake %E Simone Azeglio %E Arianna Di Bernardo %E Nina Miolane %F pmlr-v197-shutty23a %I PMLR %P 1--21 %U https://proceedings.mlr.press/v197/shutty23a.html %V 197 %X Recent work has constructed neural networks that are equivariant to continuous symmetry groups such as 2D and 3D rotations. This is accomplished using explicit {\it Lie group representations} to derive the equivariant kernels and nonlinearities. We present three contributions motivated by frontier applications of equivariance beyond rotations and translations. First, we relax the requirement for explicit Lie group representations with a novel algorithm that finds representations of arbitrary Lie groups given only the {\it structure constants} of the associated Lie algebra. Second, we provide a self-contained method and software for building Lie group-equivariant neural networks using these representations. Third, we contribute a novel benchmark dataset for classifying objects from relativistic point clouds, and apply our methods to construct the first object-tracking model equivariant to the Poincaré group.
APA
Shutty, N. & Wierzynski, C.. (2023). Computing representations for Lie algebraic networks. Proceedings of the 1st NeurIPS Workshop on Symmetry and Geometry in Neural Representations, in Proceedings of Machine Learning Research 197:1-21 Available from https://proceedings.mlr.press/v197/shutty23a.html.

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