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Variational Partial Group Convolutions for Input-Aware Partial Equivariance of Rotations and Color-Shifts
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Proposes Variational Partial G-CNN (VP G-CNN), which makes the output group-element distribution of group convolutions input-conditioned and trains it with a variational (ELBO) objective, plus a stable straight-through discrete distribution for color-shift groups.
Problem
Standard G-CNNs enforce full equivariance to a whole group, and Partial G-CNN uses a fixed partial equivariance level for all inputs; discrete group variants (color-shift H_m) also suffer unstable training with Gumbel-Softmax.
Approach
Condition the output group-element distribution q(u|f) on the previous layer's features, train it as an approximate posterior via ELBO with a KL prior toward uniform (full equivariance), use reparametrization for SO(2) (learned width theta) and a new softmax-threshold straight-through distribution for discrete H_m.
Key results
- On Flowers102 with T(2)xH_3, VP CEResNet reaches 69.4% test accuracy vs 68.0% for CEResNet and 67.2% for Partial CEResNet.
- On long-tailed ColorMNIST with H_6, VP G-CNN achieves 88.4% vs 88.7% for fully equivariant CEConv and 63.5% for Partial G-CNN.
- On CIFAR10 with SE(2) and 8 elements, VP achieves 87.6% vs 86.8% for the fully equivariant baseline and 87.3% for Partial G-CNN.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use input-aware partial group convolution instead of full equivariance | SE(2)-CNN on CIFAR10 · CIFAR10 | accuracy: 83.9 ± 0.3 -> 85.1 ± 0.4 | Helped |
| Use input-aware partial group convolution instead of Partial G-CNN | SE(2)-CNN with 4 elements · CIFAR10 | accuracy: 85.1 ± 0.6 -> 85.1 ± 0.4 | No clear effect |
| Use input-aware partial group convolution instead of Partial G-CNN | SE(2)-CNN with 8 elements · CIFAR10 | accuracy: 87.3 ± 0.4 -> 87.6 ± 0.2 | No clear effect |
| Use input-aware partial convolution instead of Partial G-CNN | Color-equivariant CNN with 3 elements · ColorMNIST | accuracy: 61.4 ± 0.8 -> 85.4 ± 1.0 | Helped |
| Use input-aware partial convolution instead of full color equivariance | Color-equivariant CNN with 3 elements · ColorMNIST | accuracy: 87.1 ± 0.1 -> 85.4 ± 1.0 | Hurt |
| Use input-aware partial convolution instead of Partial G-CNN | Color-equivariant CNN with 6 elements · ColorMNIST | accuracy: 63.5 ± 0.6 -> 88.4 ± 1.1 | Helped |
| Use input-aware partial convolution instead of Partial G-CNN | Color-equivariant CNN with 3 elements · Flowers102 | accuracy: 67.2 ± 1.5 -> 69.4 ± 0.6 | Helped |
| Use input-aware partial convolution instead of Partial G-CNN | Color-equivariant CNN with 6 elements · Flowers102 | accuracy: 66.8 ± 0.7 -> 69.3 ± 0.4 | Helped |
| Use input-aware partial convolution instead of full color equivariance | Color-equivariant CNN with 3 elements · Flowers102 | accuracy: 68.0 ± 0.5 -> 69.4 ± 0.6 | Helped |
| Use input-aware partial convolution instead of full color equivariance | Color-equivariant CNN with 6 elements · Flowers102 | accuracy: 65.0 ± 0.7 -> 69.3 ± 0.4 | Helped |
| Use variational input-aware partial equivariance | VP G-CNN · Flowers102 | NLL: 0.0171 -> 0.0121 | Helped |
| Use variational input-aware partial equivariance | VP G-CNN · Flowers102 | BS: 0.0042 -> 0.0035 | Helped |
Surprising findings
- Partial G-CNN with Gumbel-Softmax performs very poorly on ColorMNIST (61.4% and 63.5%) due to unstable discrete-group training.
- On ColorMNIST (fully equivariant task), the fully equivariant CEConv baseline with 6 elements still slightly outperforms VP G-CNN (88.7 vs 88.4), so the gain is not uniform.
Limitations
- Only SO(2) and discrete color-shift H_m groups are instantiated; other groups are left to future work.
- Partial equivariance is applied only to a subset of layers, and the gains are modest on CIFAR10 where the equivariance level differences are small.
- Some results (e.g., ColorMNIST with H_6) are not better than the fully equivariant baseline.
Experimental setup
Datasets: MNIST MNIST67-180 CIFAR10 CIFAR100 ColorMNIST (long-tailed) Flowers102
Benchmarks: MNIST67-180 CIFAR10 CIFAR100 long-tailed ColorMNIST Flowers102
Metrics: test accuracy negative log-likelihood Brier score
Baselines: ResNet (T(2)-CNN) G-CNN Partial G-CNN InstaAug AdaAug CEConv
Builds on: Partial G-CNN G-CNN CEConv Augerino InstaAug
Closest papers by meaning
- Learning Partial Equivariances From Data (NeurIPS 2022 · similarity 0.95)
- Group Equivariant Convolutional Networks (ICML 2016 · similarity 0.92)
- Implicit Bias of Linear Equivariant Networks (ICML 2022 · similarity 0.91)
- Probing Equivariance and Symmetry Breaking in Convolutional Networks (NeurIPS 2025 · similarity 0.90)
- Learning Color Equivariant Representations (ICLR 2025 · similarity 0.90)
- Group Downsampling with Equivariant Anti-aliasing (ICLR 2025 · similarity 0.89)
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