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Probing Equivariance and Symmetry Breaking in Convolutional Networks
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Studies when group-equivariance versus translation-only or pose-informed symmetry breaking helps on point-cloud tasks, using a unified regular group-convolution architecture (Rapidash) across QM9, ShapeNet, and CMU motion.
Problem
Whether and when explicit structural priors such as SE(3) group-equivariance help versus less constrained or symmetry-broken models, given conflicting evidence in the literature and no controlled comparisons.
Approach
Introduces Rapidash, a unified regular group convolution architecture over R^3 x S^2 (built on PsiNITA) whose variants toggle equivariance strength, input geometric types (scalars vs vectors), and pose-informed symmetry breaking, enabling matched comparisons between SE(3), T(3), and non-equivariant models.
Key results
- On QM9 molecule generation, SE(3)-equivariant Rapidash variants reach about 93% molecule stability and 90.8% discovery, exceeding reported EGNN-EDM (90.7% stability, 89.5% discovery) results.
- On CMU motion prediction, the best Rapidash variant (SO(3) effective equivariance with vector inputs and symmetry breaking) reaches MSE 4.88e-2, versus CSMPN 7.55e-2.
- Increasing channel capacity of R^3 (T(3)) models yields little gain, and T(3) models do not close the gap to R^3 x S^2 SE(3) models on QM9 property prediction.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use an R³ × S² base instead of an R³-only base (ablation) | Rapidash with SE(3) equivariance · QM9 property prediction (μ) | MAE (×10^-3): 17.41_{\pm 0.37} -> 10.39_{\pm 0.33} | Helped |
| Add scalar coordinate features (ablation) | Rapidash with SE(3) equivariance · QM9 property prediction (α) | MAE (×10^-3): 42.20_{\pm 1.17} -> 40.21_{\pm 0.56} | Helped |
| Add scalar coordinate features (ablation) | Rapidash with SE(3) equivariance · QM9 property prediction (ε_HOMO) | MAE (×10^-3): 19.30_{\pm 1.02} -> 18.69_{\pm 0.20} | Helped |
| Add scalar coordinate features (ablation) | Rapidash with SE(3) equivariance · QM9 molecule generation | molecule stability (%): 93.12_{\pm 0.28} -> 92.71_{\pm 0.51} | Hurt |
| Add scalar coordinate features (ablation) | Rapidash with SE(3) equivariance · QM9 molecule generation | discovery (%): 90.78_{\pm 0.11} -> 91.54_{\pm 0.34} | Helped |
| Add a global frame for symmetry breaking (ablation) | Rapidash with R³ × S² and SE(3) equivariance · ShapeNet3D part segmentation, aligned | mIoU: 84.27_{\pm 0.19} -> 85.44_{\pm 0.03} | Helped |
| Add a global frame for symmetry breaking (ablation) | Rapidash with R³ × S² and SE(3) equivariance · ShapeNet3D part segmentation, rotated | mIoU: 84.17_{\pm 0.08} -> 85.39_{\pm 0.11} | Helped |
| Add velocity vector inputs (ablation) | Rapidash with R³ × S² and SE(3) equivariance · CMU motion prediction | MSE (×10^-2): 5.43_{\pm 0.15} -> 4.77_{\pm 0.11} | Helped |
| Add a global frame for symmetry breaking (ablation) | Rapidash with R³ × S² and SE(3) equivariance, velocity vectors · CMU motion prediction | MSE (×10^-2): 4.77_{\pm 0.11} -> 4.69_{\pm 0.10} | Helped |
| Increase representation capacity in R³ models (ablation) | Rapidash · ShapeNet3D part segmentation | mIoU | No clear effect |
| Use stronger equivariance (ablation) | Rapidash · ShapeNet data efficiency | training samples required to converge | Helped |
| Use R³ × S² SE(3) models instead of R³ T₃ models (ablation) | Rapidash · QM9 property prediction | MAE | Helped |
Surprising findings
- Scaling representation capacity of translation-equivariant R^3 models does not close the performance gap to more constrained SE(3) models.
- On aligned ShapeNet data, T(3)-equivariant R^3 models perform comparably to the richer SE(3) variants, despite weaker constraints.
Limitations
- Modest computational budget limits exploration of scaling laws, especially for R^3 models.
- Scope limited to convolutional architectures on point clouds; transformers and other modalities not studied.
- Discretizing S^2 makes equivariance approximate up to grid resolution.
Experimental setup
Datasets: QM9 ShapeNet CMU Motion Capture
Benchmarks: QM9 ShapeNet part segmentation ShapeNet shape generation CMU motion capture
Metrics: MAE atom stability molecule stability discovery mIoU 1-NNA MSE
Baselines: EGNN DimeNet++ SE(3)-Transformer EDM PsiNITA MuDiff PointNeXt CSMPN
Builds on: PsiNITA ConvNeXt SymPE
Closest papers by meaning
- A Tale of Two Symmetries: Exploring the Loss Landscape of Equivariant Models (NeurIPS 2025 · similarity 0.91)
- Variational Partial Group Convolutions for Input-Aware Partial Equivariance of Rotations and Color-Shifts (ICML 2024 · similarity 0.90)
- Improving Equivariant Networks with Probabilistic Symmetry Breaking (ICLR 2025 · similarity 0.90)
- Learning symmetries via weight-sharing with doubly stochastic tensors (NeurIPS 2024 · similarity 0.90)
- A Probabilistic Approach to Learning the Degree of Equivariance in Steerable CNNs (ICML 2024 · similarity 0.89)
- Equivariance by Contrast: Identifiable Equivariant Embeddings from Unlabeled Finite Group Actions (NeurIPS 2025 · similarity 0.89)
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