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Schur Nets: exploiting local structure for equivariance in higher order graph neural networks
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Proposes Schur layers, which build automorphism-equivariant linear maps for subgraphs directly from eigenspaces of the subgraph Laplacian (no group theory), and plugs them into P-tensor higher-order MPNNs for molecular property prediction.
Problem
Higher-order GNNs over subgraphs need local operations equivariant to each subgraph's automorphism group, but explicitly computing automorphism groups and their irreducible representations for arbitrary subgraphs is generally infeasible.
Approach
Use eigenspaces of the combinatorial Laplacian of a subgraph as invariant subspaces; any linear map that acts as a scalar on each eigenspace (weight matrix per eigenspace) is automorphism-equivariant, giving a simple EVD-based basis without computing irreps.
Key results
- Schur-Net reaches 0.064 ± 0.002 test MAE on ZINC-12K, second only to N2-GNN (0.059) and better than Linmaps/P-tensors (0.071).
- Schur-Net achieves 81.6 ± 0.295 ROC-AUC on OGB-MOLHIV, the best among the compared baselines (CIN 80.94, P-tensors 80.76).
- Replacing Linmaps with Schur layers improves ZINC-12K test MAE from 0.074 to 0.070 under the overlap ≥1 message-passing scheme.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace Linmaps with Schur-Net | higher-order GNN · ZINC-12K | test MAE: 0.071 ± 0.004 -> 0.064 ± 0.002; N²-GNN: 0.059 ± 0.002 | Mixed |
| replace CIN with Schur-Net | higher-order GNN · OGB-HIV | ROC-AUC (%): 80.94 ± 0.57 -> 81.6 ± 0.295 | Helped |
| replace Linmaps with Schur layer | P-tensors on ZINC-12K, overlap ≥1 · ZINC-12K | test MAE: 0.074 ± 0.008 -> 0.070 ± 0.006 | Helped |
| replace Linmaps with Schur layer | P-tensors on ZINC-12K, overlap ≥2 · ZINC-12K | test MAE: 0.074 ± 0.005 -> 0.071 ± 0.003 | Helped |
| apply Schur layer directly to subgraph features (ablation) | Schur-Net · ZINC-12K | test MAE: 0.071 ± 0.004 -> 0.070 ± 0.005 | Helped |
| apply Schur layer to aggregated subgraph representation (ablation) | Schur-Net · ZINC-12K | test MAE: 0.071 ± 0.004 -> 0.068 ± 0.002 | Helped |
| apply Schur layer to incoming messages and original cycle representation (ablation) | Schur-Net · ZINC-12K | test MAE: 0.071 ± 0.004 -> 0.064 ± 0.002 | Helped |
| use Schur layer in place of the two-layer MLP (ablation) | Linmaps architecture · ZINC-12K | validation MAE: 0.087 -> 0.081 | Helped |
| use Schur layer to learn a new feature (ablation) | Linmaps architecture · ZINC-12K | validation MAE: 0.087 -> 0.076 | Helped |
| add cycles with up to three branches (ablation) | Schur-Net on 5,6 cycles · ZINC-12K | validation MAE: 0.113 ± 0.008 -> 0.105 ± 0.001 | Helped |
| replace Linmaps with Schur layer | P-tensors · Proteins | accuracy: 74.7 ± 3.8 -> 75.4 ± 4.8 | No clear effect |
| replace Linmaps with Schur layer | P-tensors · MUTAG | accuracy: 89.9 ± 5.5 -> 90.94 ± 4.7 | No clear effect |
Surprising findings
- Schur layers are not more effective when used in place of the MLP; they work better as a source of new features.
- Adding more channels beyond the number of distinct eigenvalues gives no further gain.
Limitations
- The spectral construction does not give the full set of automorphism-equivariant maps; it ignores isomorphic subspaces, so it can miss equivariant maps (gap quantified for small examples).
- Experiments only cover simple subgraphs (cycles, edges) with first-order activations.
- Only first-order activations were used in the benchmark experiments.
Experimental setup
Datasets: ZINC-12K OGB-MOLHIV TUdatasets (Proteins, MUTAG, PTC_MR, NCI1)
Benchmarks: ZINC-12K OGB-MOLHIV TUdatasets
Metrics: MAE ROC-AUC classification accuracy
Baselines: Linmaps GCN GIN GINE PNA HIMP N2-GNN CIN P-tensors DS-GNN DSS-GNN GNN-AK+
Builds on: P-tensors Autobahn CIN Deep Sets
Closest papers by meaning
- Homomorphism Expressivity of Spectral Invariant Graph Neural Networks (ICLR 2025 · similarity 0.91)
- Building powerful and equivariant graph neural networks with structural message-passing (NeurIPS 2020 · similarity 0.91)
- Spectral Graph Neural Networks are Incomplete on Graphs with a Simple Spectrum (NeurIPS 2025 · similarity 0.91)
- Are High-Degree Representations Really Unnecessary in Equivariant Graph Neural Networks? (NeurIPS 2024 · similarity 0.90)
- On the Expressive Power of Spectral Invariant Graph Neural Networks (ICML 2024 · similarity 0.90)
- A Flexible, Equivariant Framework for Subgraph GNNs via Graph Products and Graph Coarsening (NeurIPS 2024 · similarity 0.89)
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