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Group Downsampling with Equivariant Anti-aliasing
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Generalizes uniform downsampling and anti-aliasing from sequences to finite groups via subgroup selection on Cayley graphs and an equivariant anti-aliasing projection, validated on rotated MNIST and CIFAR-10 G-CNNs.
Problem
Downsampling signals on finite groups (for group equivariant networks like G-CNNs) is ill-defined: there is no notion of "every R-th element", the discarded set may not be a subgroup, and existing group subsampling lacks proper anti-aliasing that preserves equivariance.
Approach
Form a subgroup by removing a generator's edges on the Cayley graph and adding its R-th power, define a subgroup sampling theorem via bandlimitedness in the Fourier domain, and learn an equivariant anti-aliasing projection by optimizing an equivariance objective plus a smoothness (Laplacian) term under a perfect-reconstruction constraint.
Key results
- On rotated MNIST (SO(2)), subsampling rate 2 with anti-aliasing reduces equivariance error from 0.056 to 0.049 and raises Acc_orbit to 0.8358 with 194K parameters versus 323K for the full model.
- On CIFAR-10 (SO(2)), rate 2 with anti-aliasing reaches Acc_orbit 0.4279 vs 0.4096 without anti-aliasing, and L_equi drops from 0.398 to 0.336.
- Anti-aliasing achieves zero reconstruction error up to numerical precision for bandlimited signals on dihedral and cyclic groups.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add equivariant anti-aliasing to R=2 subgroup downsampling (ablation) | G-CNN · Rotated MNIST, SO(2) | Acc_orbit: 0.8106 -> 0.8358 | Helped |
| add equivariant anti-aliasing to R=2 subgroup downsampling (ablation) | G-CNN · Rotated MNIST, O(2) | Acc_orbit: 0.5660 -> 0.5749 | Helped |
| add equivariant anti-aliasing to R=2 subgroup downsampling (ablation) | G-CNN · CIFAR-10, SO(2) | Acc_orbit: 0.4096 -> 0.4279 | Helped |
| add equivariant anti-aliasing to R=2 subgroup downsampling (ablation) | G-CNN · CIFAR-10, O(2) | Acc_orbit: 0.3310 -> 0.3274 | Hurt |
| replace C12 and C6 with D12 and D6 using proposed subgroup selection (ablation) | 3-layer equivariant CNN · Rotated MNIST, SO(2), subgroup schedule 1,2,2 | Acc_orbit: 0.6128 -> 0.5489 | Hurt |
| replace C6 subgroups with D6 subgroups using proposed subgroup selection (ablation) | 3-layer equivariant CNN · Rotated MNIST, SO(2), subgroup schedule 1,4,1 | Acc_orbit: 0.4596 -> 0.4860 | Helped |
| replace C14 subgroups with D14 subgroups using proposed subgroup selection (ablation) | 3-layer equivariant CNN · Rotated MNIST, SO(2), subgroup schedule 1,2,1 | Acc_orbit: 0.5191 -> 0.5792 | Helped |
| add equivariant anti-aliasing (ablation) | G-CNN with equivariant index selection · Rotated MNIST, SO(2), R=2 | Acc_orbit: 0.8225 -> 0.8297 | Helped |
| add equivariant anti-aliasing (ablation) | G-CNN · STL-10, SO(2), R=2 | ACC_orbit: 0.35 -> 0.37 | Helped |
Surprising findings
- Higher subsampling rates (e.g., R=4) on rotated MNIST increase clean accuracy while sharply lowering orbit and local accuracy and raising equivariance error.
- Anti-aliasing does not uniformly help: on some O(2) settings (e.g., MNIST R=3 with equivariant index selection) accuracy slightly drops.
Limitations
- Experiments limited to small-scale datasets and models; authors describe the paper as theoretical.
- Only finite (discrete) groups are supported; continuous groups are approximated by C_24 and D_24.
- Subgroup selection heuristic has worst-case quadratic complexity in Cayley graph edges and the optimal subgroup choice is task-dependent.
Experimental setup
Datasets: Rotated MNIST CIFAR-10 STL-10
Benchmarks: Rotated MNIST CIFAR-10 STL-10
Metrics: accuracy on unaugmented test set (Acc_no_aug) accuracy on locally augmented test set (Acc_loc) accuracy on full orbit (Acc_orbit) equivariance error (L_equi) reconstruction error
Baselines: full equivariant G-CNN (no subsampling) naive subgroup subsampling without anti-aliasing equivariant index selection
Builds on: G-CNN Shift-equivariant anti-aliasing (BlurPool) graph signal processing sampling theory Reynolds operator equivariance constraints
Closest papers by meaning
- Group Equivariant Subsampling (NeurIPS 2021 · similarity 0.90)
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- Variational Partial Group Convolutions for Input-Aware Partial Equivariance of Rotations and Color-Shifts (ICML 2024 · similarity 0.89)
- Implicit Bias of Linear Equivariant Networks (ICML 2022 · similarity 0.89)
- Continuous Rotation Group Equivariant Network Inspired by Neural Population Coding (AAAI 2024 · similarity 0.87)
- Lie Group Decompositions for Equivariant Neural Networks (ICLR 2024 · similarity 0.86)
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