Brauer’s Group Equivariant Neural Networks

Edward Pearce-Crump
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:27461-27482, 2023.

Abstract

We provide a full characterisation of all of the possible group equivariant neural networks whose layers are some tensor power of Rn for three symmetry groups that are missing from the machine learning literature: O(n), the orthogonal group; SO(n), the special orthogonal group; and Sp(n), the symplectic group. In particular, we find a spanning set of matrices for the learnable, linear, equivariant layer functions between such tensor power spaces in the standard basis of Rn when the group is O(n) or SO(n), and in the symplectic basis of Rn when the group is Sp(n).

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-pearce-crump23a, title = {Brauer’s Group Equivariant Neural Networks}, author = {Pearce-Crump, Edward}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {27461--27482}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/pearce-crump23a/pearce-crump23a.pdf}, url = {https://proceedings.mlr.press/v202/pearce-crump23a.html}, abstract = {We provide a full characterisation of all of the possible group equivariant neural networks whose layers are some tensor power of $\mathbb{R}^{n}$ for three symmetry groups that are missing from the machine learning literature: $O(n)$, the orthogonal group; $SO(n)$, the special orthogonal group; and $Sp(n)$, the symplectic group. In particular, we find a spanning set of matrices for the learnable, linear, equivariant layer functions between such tensor power spaces in the standard basis of $\mathbb{R}^{n}$ when the group is $O(n)$ or $SO(n)$, and in the symplectic basis of $\mathbb{R}^{n}$ when the group is $Sp(n)$.} }
Endnote
%0 Conference Paper %T Brauer’s Group Equivariant Neural Networks %A Edward Pearce-Crump %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-pearce-crump23a %I PMLR %P 27461--27482 %U https://proceedings.mlr.press/v202/pearce-crump23a.html %V 202 %X We provide a full characterisation of all of the possible group equivariant neural networks whose layers are some tensor power of $\mathbb{R}^{n}$ for three symmetry groups that are missing from the machine learning literature: $O(n)$, the orthogonal group; $SO(n)$, the special orthogonal group; and $Sp(n)$, the symplectic group. In particular, we find a spanning set of matrices for the learnable, linear, equivariant layer functions between such tensor power spaces in the standard basis of $\mathbb{R}^{n}$ when the group is $O(n)$ or $SO(n)$, and in the symplectic basis of $\mathbb{R}^{n}$ when the group is $Sp(n)$.
APA
Pearce-Crump, E.. (2023). Brauer’s Group Equivariant Neural Networks. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:27461-27482 Available from https://proceedings.mlr.press/v202/pearce-crump23a.html.

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