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A Probabilistic Approach to Learning the Degree of Equivariance in Steerable CNNs
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Learns a per-layer likelihood distribution over the symmetry group via Fourier coefficients to set the degree of equivariance in steerable CNNs, yielding partially equivariant SCNNs.
Problem
Fixed equivariance constraints in steerable CNNs can be overconstraining when symmetries are unknown or vary across features and layers, and existing partial-equivariance methods are either infeasible for continuous groups or lack an interpretable degree-of-equivariance parameterisation.
Approach
Replace the uniform group averaging in the steerable kernel projection with a learnable likelihood distribution over the group, parameterised by Fourier coefficients (with softmax normalisation, alignment loss to the identity, KL-divergence between consecutive layers, and optional bandlimiting), so the learnt likelihood directly reads out the degree of equivariance.
Key results
- On DDMNIST C4-equivariant networks, the proposed PSCNN improves over restricted SCNN and RPP on most symmetries (e.g., SO(2) network on SO(2) digit symmetries reaches 0.901 vs 0.887 restricted SCNN).
- On MedMNIST Synapse with SO(3), the method reaches 0.770 accuracy versus 0.738 for the invariant SCNN and 0.695 for RPP.
- On Smoke (domain test) the method reports RMSE 0.57, the best among compared models, while using 3.12M parameters.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| learn partial equivariance with a likelihood distribution | SCNN on Synapse · Synapse | accuracy: 0.738 -> 0.770 | Helped |
| learn partial equivariance with a likelihood distribution | SCNN on Organ · Organ | accuracy: 0.936 -> 0.902 | Hurt |
| learn partial equivariance with a likelihood distribution | SCNN on Smoke · Smoke | RMSE: 0.80 -> 0.77 (Future); 0.67 -> 0.57 (Domain) | Helped |
| learn partial equivariance with a likelihood distribution | SCNN on JetFlow · JetFlow | RMSE: 0.17 -> 0.15 (Future); 0.16 -> 0.17 (Domain) | Mixed |
| increase bandlimit from L=0 to L=1 (ablation) | PSCNN (O(2)) · SO(2)-DDMNIST | accuracy: 0.469 -> 0.894 | Helped |
| increase bandlimit from L=0 to L=2 (ablation) | PSCNN (O(2)) · O(2)-DDMNIST | accuracy: 0.402 -> 0.819 | Helped |
| increase bandlimit from L=2 to L=3 (ablation) | PSCNN (O(2)) · SO(2)-DDMNIST | accuracy: 0.894 -> 0.889 | Hurt |
| increase bandlimit from L=2 to L=3 (ablation) | PSCNN (O(2)) · O(2)-DDMNIST | accuracy: 0.819 -> 0.817 | Hurt |
Surprising findings
- Fully invariant O(2) SCNNs are unable to distinguish digit reversals, and the partially equivariant model removes these confusions.
- With little data the method struggles on the non-symmetric Organ dataset, where RPP and CNN outperform it.
Limitations
- Cannot model partial translation equivariance; limited to compact groups.
- Uses significantly more parameters than regular SCNNs, and the steerable basis must be recomputed each forward pass.
- Requires alignment and KL hyperparameters, adding tuning overhead.
Experimental setup
Datasets: DDMNIST MedMNIST Organ MedMNIST Synapse MedMNIST Nodule JetFlow Smoke
Benchmarks: DDMNIST MedMNIST Organ MedMNIST Synapse MedMNIST Nodule JetFlow Smoke
Metrics: test accuracy RMSE
Baselines: SCNN with group restriction RPP RSteer CNN e2cnn Lift RGroup Combo CLCNN MLP
Builds on: steerable CNNs residual pathway priors escnn partial G-CNNs
Closest papers by meaning
- Scale-Equivariant Steerable Networks (ICLR 2020 · similarity 0.91)
- Relaxing Equivariance Constraints with Non-stationary Continuous Filters (NeurIPS 2022 · similarity 0.91)
- Meta-learning Symmetries by Reparameterization (ICLR 2021 · similarity 0.91)
- A Characterization Theorem for Equivariant Networks with Point-wise Activations (ICLR 2024 · similarity 0.90)
- Probing Equivariance and Symmetry Breaking in Convolutional Networks (NeurIPS 2025 · similarity 0.89)
- A Tale of Two Symmetries: Exploring the Loss Landscape of Equivariant Models (NeurIPS 2025 · similarity 0.88)
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