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Improving Equivariant Networks with Probabilistic Symmetry Breaking
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Proposes SymPE, which breaks symmetries in equivariant networks by concatenating a randomly sampled canonicalization group element (applied to a learned vector) as positional encoding, with theory showing it represents any equivariant conditional distribution.
Problem
Equivariant networks cannot break symmetries: outputs must have at least the self-symmetries of self-symmetric inputs, which hurts prediction on symmetric inputs and generative models that decode from symmetric latent spaces.
Approach
Represent equivariant conditional distributions as equivariant functions of the input and a random group element sampled from an inversion kernel (randomized canonicalization); implement it as a learned, group-transformed positional encoding concatenated to the input of any equivariant network.
Key results
- On QM9 with explicit hydrogens, DiGress + SymPE reaches NLL 30.3 versus 129.7 for DiGress with Laplacian features, with molecular stability 82.5 vs 79.8.
- In graph autoencoding, SymPE achieves 0.77% reconstruction error versus 3.9% for no symmetry breaking and 0.98% for Laplacian features.
- For Ising ground-state prediction, SymPE achieves significantly lower energy than baselines on the OOD test set and is only slightly worse than relaxed group convolutions in distribution.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Apply SymPE positional encodings | EGNN graph autoencoder · Graph autoencoding | BCE: 10.0 -> 3.7 | Helped |
| Apply SymPE positional encodings | EGNN graph autoencoder · Graph autoencoding | % Error: 3.9 -> 0.77 | Helped |
| Apply SymPE positional encodings rather than Laplacian features | EGNN graph autoencoder · Graph autoencoding | % Error: 0.98 -> 0.77 | Helped |
| Apply SymPE to DiGress | DiGress · QM9 explicit hydrogens | NLL: 129.7 -> 30.3 | Helped |
| Apply SymPE to DiGress | DiGress · QM9 explicit hydrogens | Mol. stability: 79.8 -> 82.5 | Helped |
| Replace SymPE with standard-normal noise (ablation) | DiGress · QM9 explicit hydrogens | NLL: 30.3 -> 126.5 | Hurt |
| Apply SymPE to node features only, omitting adjacency-matrix encodings (ablation) | DiGress · QM9 explicit hydrogens | NLL: 30.3 -> 128.8 | Hurt |
| Apply SymPE to node features only, omitting adjacency-matrix encodings (ablation) | DiGress · QM9 explicit hydrogens | Mol. stability: 82.5 -> 83.9 | Helped |
Surprising findings
- Concatenating i.i.d. Gaussian noise as symmetry breaking did not give a comparable likelihood gain, suggesting the model learns to ignore uninformative noise.
- Breaking symmetry only on node features (not the adjacency matrix) gave no significant likelihood improvement on QM9.
Limitations
- Requires a canonicalization method and a way to randomize its outputs, which may be unavailable for uncommon groups.
- For graphs, sorting-based canonicalization does not guarantee sampling from an exact inversion kernel.
- Partial symmetry breaking is not treated.
Experimental setup
Datasets: QM9 MOSES Erdos-Renyi random graphs
Benchmarks: QM9 with explicit hydrogens MOSES graph autoencoding reconstruction error Ising ground-state energy (in-distribution and OOD)
Metrics: negative log-likelihood validity uniqueness atomic stability molecular stability binary cross-entropy % reconstruction error average energy
Baselines: No SB Noise Uniform Laplacian positional encoding DiGress ConGress G-CNN MLP relaxed group convolutions
Builds on: Bloem-Reddy and Teh probabilistic symmetries DiGress relaxed group convolutions symmetry breaking sets EGNN
Closest papers by meaning
- A Tale of Two Symmetries: Exploring the Loss Landscape of Equivariant Models (NeurIPS 2025 · similarity 0.92)
- On the hardness of learning under symmetries (ICLR 2024 · similarity 0.91)
- Learning (Approximately) Equivariant Networks via Constrained Optimization (NeurIPS 2025 · similarity 0.90)
- Learning symmetries via weight-sharing with doubly stochastic tensors (NeurIPS 2024 · similarity 0.90)
- On Universality Classes of Equivariant Networks (NeurIPS 2025 · similarity 0.90)
- Are High-Degree Representations Really Unnecessary in Equivariant Graph Neural Networks? (NeurIPS 2024 · similarity 0.90)
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