Graph Automorphism Group Equivariant Neural Networks
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866910460726476800 |
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| author | Pearce-Crump, Edward Knottenbelt, William J. |
| author_facet | Pearce-Crump, Edward Knottenbelt, William J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_07810 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Graph Automorphism Group Equivariant Neural Networks Pearce-Crump, Edward Knottenbelt, William J. Machine Learning Combinatorics Representation Theory 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. |
| title | Graph Automorphism Group Equivariant Neural Networks |
| topic | Machine Learning Combinatorics Representation Theory |
| url | https://arxiv.org/abs/2307.07810 |