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Are High-Degree Representations Really Unnecessary in Equivariant Graph Neural Networks?
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Shows that EGNN-style 1st-degree equivariant GNNs degenerate to zero on symmetric graphs (k-fold rotations, regular polyhedra), and proposes HEGNN, a high-degree EGNN extension that keeps scalarization via inner products of steerable features, evaluated on symmetric toy graphs, N-body, and MD17.
Problem
Whether high-degree steerable representations are necessary for equivariant GNNs; EGNN using only 1st-degree (Cartesian vector) features may lose expressivity on symmetric geometric graphs, outputting zero equivariant functions.
Approach
Group-theoretic proof that any O(3)-equivariant function restricted to a symmetric graph is constrained by the group average of the Wigner-D representation, so degree-1 (and some other degrees) features vanish on symmetric graphs; HEGNN adds high-degree steerable features updated via inner-product scalarization (no Clebsch-Gordan products), retaining EGNN's efficiency.
Key results
- On N-body, HEGNN_{l<=2} reaches MSE 0.47e-2 at 5-body vs EGNN 0.65e-2, and 0.78e-2 at 20-body vs EGNN 1.01e-2.
- On MD17, HEGNN_{l<=6} achieves best error on 6 of 8 molecules (e.g., Aspirin 9.94 vs EGNN 14.41).
- Symmetric-graph experiments confirm EGNN and HEGNN_{l=1} stay at 50% (chance) on all regular polyhedra and k-folds, while certain higher-degree HEGNN variants distinguish them.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add high-degree steerable features with maximum degree 2 | EGNN on N-body systems · 5-body | MSE (× 10⁻²): 0.65 -> 0.47 | Helped |
| Add high-degree steerable features with maximum degree 2 | EGNN on N-body systems · 20-body | MSE (× 10⁻²): 1.01 -> 0.78 | Helped |
| Add high-degree steerable features with maximum degree 3 | EGNN on N-body systems · 50-body | MSE (× 10⁻²): 1.00 -> 0.84 | Helped |
| Add high-degree steerable features with maximum degree 6 | EGNN on N-body systems · 100-body | MSE (× 10⁻²): 1.36 -> 0.86 | Helped |
| Increase maximum degree from 1 to 6 (ablation) | HEGNN on N-body systems · 5-body | MSE (× 10⁻²): 0.52 -> 0.69 | Hurt |
| Increase maximum degree from 1 to 6 (ablation) | HEGNN on N-body systems · 100-body | MSE (× 10⁻²): 1.13 -> 0.86 | Helped |
| Increase maximum degree from 1 to 6 (ablation) | HEGNN on MD17 molecules · MD17 | prediction error (× 10⁻²): Aspirin: 10.32 -> 9.94; Benzene: 62.53 -> 59.93; Ethanol: 4.63 -> 4.62; Malonaldehyde: 12.85 -> 12.85; Naphthalene: 0.38 | Mixed |
| Add degree-4 steerable features (ablation) | HEGNN on symmetric regular polyhedra · Cube and Octahedron | accuracy: 50.0 {\scriptsize ± 0.0} -> 90.0 { ±30.0} | Helped |
| Add degree-10 steerable features (ablation) | HEGNN on symmetric regular polyhedra · Octahedron | accuracy: 50.0 {\scriptsize ± 0.0} -> 95.0 { ±15.0} | Helped |
| Use only degree-2 features rather than Cartesian vectors (ablation) | HEGNN on k-fold structures · 2-fold | accuracy: 50.0 {\scriptsize ± 0.0} -> 95.0 { ±15.0} | Helped |
| Add high-degree steerable features | HEGNN on MD17 · MD17 | prediction error (× 10⁻²) | Mixed |
| Use degree-1-only Cartesian representations | EGNN on symmetric graphs · Regular polyhedra | accuracy: 50.0 {\scriptsize ± 0.0} | No clear effect |
Surprising findings
- Degree-1 equivariant features vanish identically on symmetric graphs, so EGNN cannot distinguish rotated symmetric structures at all.
- Under small noise perturbations, HEGNN shows better robustness than EGNN on tetrahedron-based symmetry breaking.
Limitations
- Number of angles needed for completeness can be extremely large, so the expressivity result is impractical to attain exactly.
- Experiments on symmetric toy graphs use limited training epochs and small trial counts, with high variance in some results.
- Pseudovector handling needs additional CG products, which the scalarization approach does not cover by default.
Experimental setup
Datasets: N-body system (simulated Coulomb charged particles) MD17 symmetric toy graphs (k-fold, regular polyhedra)
Benchmarks: N-body MD17 symmetric toy graphs (k-fold, regular polyhedra)
Metrics: MSE classification accuracy relative inference time
Baselines: EGNN GVP-GNN TFN MACE SE(3)-Transformer RF SchNet MPNN GMN Linear
Builds on: EGNN TFN e3nn SEGNN
Closest papers by meaning
- Universally Invariant Learning in Equivariant GNNs (NeurIPS 2025 · similarity 0.92)
- On the Generalization of Equivariant Graph Neural Networks (ICML 2024 · similarity 0.92)
- A Characterization Theorem for Equivariant Networks with Point-wise Activations (ICLR 2024 · similarity 0.91)
- Schur Nets: exploiting local structure for equivariance in higher order graph neural networks (NeurIPS 2024 · similarity 0.90)
- On Explaining Equivariant Graph Networks via Improved Relevance Propagation (ICML 2025 · similarity 0.90)
- Improving Equivariant Networks with Probabilistic Symmetry Breaking (ICLR 2025 · similarity 0.90)
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