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On the Fourier analysis in the SO(3) space : the EquiLoPO Network
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Proposes EquiLoPO, a ResNet-style SO(3)-equivariant network for 3D volumes that uses Wigner-matrix Fourier convolution and a local polynomial ReLU approximation in the Wigner space, evaluated on MedMNIST3D.
Problem
Rotation-equivariant deep learning for volumetric data: group convolutions are limited to discrete rotations, while steerable networks constrain filter structure; prior SO(3) activations are global.
Approach
Treat Wigner-matrix (irrep) coefficients as Fourier coefficients of SO(3) functions, perform a 6D roto-translational convolution with unconstrained trainable filter coefficients, and define a local ReLU-approximating quadratic activation in the Wigner space (with adaptive, constant, or trainable coefficients) that maps function values pointwise and preserves analytical equivariance.
Key results
- Local adaptive activation EquiLoPO reaches AUC 0.984 and ACC 0.940 on Synapse, and AUC 0.968 on Vessel, outperforming listed baselines on most datasets.
- On Organ, accuracy lags baselines (local adaptive ACC 0.767 vs ILPOResNet-50 0.879), which the authors attribute to limited channels for parameter count.
- Local activations substantially outperform global activations with constant or adaptive coefficients across datasets; the ILPOResNet-50 (38k params) is the only smaller model.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use EquiLoPO with local adaptive activation instead of the strongest listed baseline | EquiLoPO ResNet-18 · Organ | ACC: 0.907 -> 0.767 | Hurt |
| Use EquiLoPO with local adaptive activation instead of the strongest listed baseline | EquiLoPO ResNet-18 · Nodule | AUC: 0.914 -> 0.916 | Helped |
| Use EquiLoPO with local adaptive activation instead of the strongest listed baseline | EquiLoPO ResNet-18 · Fracture | AUC: 0.767 -> 0.803 | Helped |
| Use EquiLoPO with local adaptive activation instead of the strongest listed baseline | EquiLoPO ResNet-18 · Vessel | AUC: 0.930 -> 0.968 | Helped |
| Use local adaptive rather than local constant polynomial coefficients (ablation) | EquiLoPO ResNet-18 · MedMNIST3D datasets except Adrenal | AUC: Organ 0.761 -> 0.977; Nodule 0.856 -> 0.916; Fracture 0.758 -> 0.803; Vessel 0.805 -> 0.968; Synapse 0.883 -> 0.984 | Helped |
| Use local adaptive rather than local constant polynomial coefficients (ablation) | EquiLoPO ResNet-18 · Adrenal | AUC: 0.896 -> 0.885 | Hurt |
| Use local activation rather than global activation with trainable coefficients (ablation) | EquiLoPO ResNet-18 · MedMNIST3D | AUC | Mixed |
| Use local activation rather than global activation with constant or adaptive coefficients (ablation) | EquiLoPO ResNet-18 · MedMNIST3D | AUC | Helped |
| Increase activation output resolution from L_out = L_in to L_out = 2L_in (ablation) | EquiLoPO with local trainable activation · MedMNIST3D | performance | Helped |
| Replace SoftMax with 2-norm in global activation (ablation) | EquiLoPO global activation · MedMNIST3D | performance | Hurt |
Surprising findings
- Global activation with SoftMax relies on a local ReLU approximation internally, and replacing SoftMax with 2-norm yields significantly inferior performance.
- Reduced output resolution (L_out = L_in) produces ambiguous input-output mapping and underperforms the full-resolution model.
Limitations
- Limited to regular voxelized volumetric data; irregular data requires significant modifications.
- Higher parameter count (418k) than the ILPOResNet-50 baseline (38k), and computational complexity scales as O(N^3 L^3 ...), increasing with expansion order.
- Evaluated only on 28x28x28 MedMNIST v2 3D datasets.
Experimental setup
Datasets: MedMNIST v2 3D
Benchmarks: MedMNIST v2 3D (Organ, Nodule, Fracture, Adrenal, Vessel, Synapse)
Metrics: AUC ACC
Baselines: ResNet-18 + 2.5D ResNet-18 + 3D ResNet-18 + ACS ResNet-50 + 3D auto-sklearn AutoKeras FPVT SE3MovFrNet Regular SE(3) convolution ILPOResNet-50
Builds on: G-CNN steerable 3D convolutions ResNet ILPOResNet
Closest papers by meaning
- General Nonlinearities in SO(2)-Equivariant CNNs (NeurIPS 2021 · similarity 0.89)
- EqGINO: Equivariant Geometry-Informed Fourier Neural Operators for 3D PDEs (ICML 2026 · similarity 0.88)
- Probing Equivariance and Symmetry Breaking in Convolutional Networks (NeurIPS 2025 · similarity 0.88)
- A Regularization-Guided Equivariant Approach for Image Restoration (CVPR 2025 · similarity 0.87)
- Equivariant Neural Tangent Kernels (ICML 2025 · similarity 0.87)
- A Tale of Two Symmetries: Exploring the Loss Landscape of Equivariant Models (NeurIPS 2025 · similarity 0.87)
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