Universally Invariant Learning in Equivariant GNNs
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arXiv
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| Main Authors: | , , , , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912648360099840 |
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| author | Cen, Jiacheng Li, Anyi Lin, Ning Xu, Tingyang Rong, Yu Zhao, Deli Wang, Zihe Huang, Wenbing |
| author_facet | Cen, Jiacheng Li, Anyi Lin, Ning Xu, Tingyang Rong, Yu Zhao, Deli Wang, Zihe Huang, Wenbing |
| contents | Equivariant Graph Neural Networks (GNNs) have demonstrated significant success across various applications. To achieve completeness -- that is, the universal approximation property over the space of equivariant functions -- the network must effectively capture the intricate multi-body interactions among different nodes. Prior methods attain this via deeper architectures, augmented body orders, or increased degrees of steerable features, often at high computational cost and without polynomial-time solutions. In this work, we present a theoretically grounded framework for constructing complete equivariant GNNs that is both efficient and practical. We prove that a complete equivariant GNN can be achieved through two key components: 1) a complete scalar function, referred to as the canonical form of the geometric graph; and 2) a full-rank steerable basis set. Leveraging this finding, we propose an efficient algorithm for constructing complete equivariant GNNs based on two common models: EGNN and TFN. Empirical results demonstrate that our model demonstrates superior completeness and excellent performance with only a few layers, thereby significantly reducing computational overhead while maintaining strong practical efficacy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_13169 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Universally Invariant Learning in Equivariant GNNs Cen, Jiacheng Li, Anyi Lin, Ning Xu, Tingyang Rong, Yu Zhao, Deli Wang, Zihe Huang, Wenbing Machine Learning Equivariant Graph Neural Networks (GNNs) have demonstrated significant success across various applications. To achieve completeness -- that is, the universal approximation property over the space of equivariant functions -- the network must effectively capture the intricate multi-body interactions among different nodes. Prior methods attain this via deeper architectures, augmented body orders, or increased degrees of steerable features, often at high computational cost and without polynomial-time solutions. In this work, we present a theoretically grounded framework for constructing complete equivariant GNNs that is both efficient and practical. We prove that a complete equivariant GNN can be achieved through two key components: 1) a complete scalar function, referred to as the canonical form of the geometric graph; and 2) a full-rank steerable basis set. Leveraging this finding, we propose an efficient algorithm for constructing complete equivariant GNNs based on two common models: EGNN and TFN. Empirical results demonstrate that our model demonstrates superior completeness and excellent performance with only a few layers, thereby significantly reducing computational overhead while maintaining strong practical efficacy. |
| title | Universally Invariant Learning in Equivariant GNNs |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2510.13169 |