Graph Automorphism Group Equivariant Neural Networks

Edward Pearce-Crump, William Knottenbelt
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:40051-40077, 2024.

Abstract

Permutation equivariant neural networks are typically used to learn from data that lives on a graph. However, for any graph $G$ that has $n$ vertices, using the symmetric group $S_n$ as its group of symmetries does not take into account the relations that exist between the vertices. Given that the actual group of symmetries is the automorphism group Aut$(G)$, we show how to construct neural networks that are equivariant to Aut$(G)$ by obtaining a full characterisation of the learnable, linear, Aut$(G)$-equivariant functions between layers that are some tensor power of $\mathbb{R}^{n}$. In particular, we find a spanning set of matrices for these layer functions in the standard basis of $\mathbb{R}^{n}$. This result has important consequences for learning from data whose group of symmetries is a finite group because a theorem by Frucht (1938) showed that any finite group is isomorphic to the automorphism group of a graph.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-pearce-crump24a, title = {Graph Automorphism Group Equivariant Neural Networks}, author = {Pearce-Crump, Edward and Knottenbelt, William}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {40051--40077}, 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/pearce-crump24a/pearce-crump24a.pdf}, url = {https://proceedings.mlr.press/v235/pearce-crump24a.html}, abstract = {Permutation equivariant neural networks are typically used to learn from data that lives on a graph. However, for any graph $G$ that has $n$ vertices, using the symmetric group $S_n$ as its group of symmetries does not take into account the relations that exist between the vertices. Given that the actual group of symmetries is the automorphism group Aut$(G)$, we show how to construct neural networks that are equivariant to Aut$(G)$ by obtaining a full characterisation of the learnable, linear, Aut$(G)$-equivariant functions between layers that are some tensor power of $\mathbb{R}^{n}$. In particular, we find a spanning set of matrices for these layer functions in the standard basis of $\mathbb{R}^{n}$. This result has important consequences for learning from data whose group of symmetries is a finite group because a theorem by Frucht (1938) showed that any finite group is isomorphic to the automorphism group of a graph.} }
Endnote
%0 Conference Paper %T Graph Automorphism Group Equivariant Neural Networks %A Edward Pearce-Crump %A William Knottenbelt %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-pearce-crump24a %I PMLR %P 40051--40077 %U https://proceedings.mlr.press/v235/pearce-crump24a.html %V 235 %X Permutation equivariant neural networks are typically used to learn from data that lives on a graph. However, for any graph $G$ that has $n$ vertices, using the symmetric group $S_n$ as its group of symmetries does not take into account the relations that exist between the vertices. Given that the actual group of symmetries is the automorphism group Aut$(G)$, we show how to construct neural networks that are equivariant to Aut$(G)$ by obtaining a full characterisation of the learnable, linear, Aut$(G)$-equivariant functions between layers that are some tensor power of $\mathbb{R}^{n}$. In particular, we find a spanning set of matrices for these layer functions in the standard basis of $\mathbb{R}^{n}$. This result has important consequences for learning from data whose group of symmetries is a finite group because a theorem by Frucht (1938) showed that any finite group is isomorphic to the automorphism group of a graph.
APA
Pearce-Crump, E. & Knottenbelt, W.. (2024). Graph Automorphism Group Equivariant Neural Networks. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:40051-40077 Available from https://proceedings.mlr.press/v235/pearce-crump24a.html.

Related Material