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The Selective $G$-Bispectrum and its Inversion: Applications to $G$-Invariant Networks
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Shows the G-Bispectrum has redundancies that allow a selective subset with O(|G|) coefficients that is still complete (invertible) for commutative, dihedral, and octahedral groups, and uses it as a G-invariant pooling layer in G-CNNs on rotated MNIST/EMNIST.
Problem
The G-Bispectrum is a complete G-invariant that is useful for G-CNN invariant pooling, but its O(|G|^2) space and time cost limits adoption in deep learning.
Approach
An algorithm selects a subset of O(|G|) G-bispectral coefficients (pairs of irreps generating all irreps) that remains complete, with proofs of invertibility for commutative groups, dihedral groups, and octahedral/full octahedral groups, and uses it as a cheaper complete invariant pooling layer.
Key results
- On SO(2)-MNIST with C8 and 24 filters, the selective G-Bispectrum reaches 0.95 accuracy vs 0.96 for G-TC and 0.96 for Max G-pooling at matched parameters.
- On O(2)-MNIST with D8, selective G-Bispectrum gets 0.93 vs 0.78 for Max G-pooling and 0.96 for G-TC.
- Training on C_{2^7} with 60000 images takes 247 seconds for the selective G-Bispectrum vs 1465 seconds for G-TC (FFT available).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace Max G-pooling with Selective G-Bispectrum | G-CNN · SO(2)-MNIST with 2 filters | accuracy: 63% -> 96% | Helped |
| Replace Max G-pooling with Selective G-Bispectrum | G-CNN · SO(2)-MNIST (C8) | accuracy: 0.96 -> 0.95 | No clear effect |
| Replace Max G-pooling with Selective G-Bispectrum | G-CNN · O(2)-MNIST (D8) | accuracy: 0.78 -> 0.93 | Helped |
| Replace Max G-pooling with Selective G-Bispectrum | G-CNN · SO(2)-EMNIST (C8) | accuracy: 0.76 -> 0.77 | Helped |
| Replace Max G-pooling with Selective G-Bispectrum | G-CNN · O(2)-EMNIST (D8) | accuracy: 0.71 -> 0.74 | Helped |
| Replace G-TC with Selective G-Bispectrum | G-CNN · SO(2)-MNIST (C_{2^7}, 60000 images) | training time: 1465 seconds -> 247 seconds | Helped |
| Replace G-TC with Selective G-Bispectrum | G-CNN · SO(2)-MNIST (C8) | accuracy: 0.96 -> 0.95 | Hurt |
| Replace G-TC with Selective G-Bispectrum | G-CNN · O(2)-MNIST (D8) | accuracy: 0.96 -> 0.93 | Hurt |
| Replace G-TC with Selective G-Bispectrum | G-CNN · SO(2)-EMNIST (C8) | accuracy: 0.80 -> 0.77 | Hurt |
| Replace G-TC with Selective G-Bispectrum | G-CNN · O(2)-EMNIST (D8) | accuracy: 0.79 -> 0.74 | Hurt |
| Increase the number of filters | G-CNN with Max G-pooling · Rotated MNIST and rotated-reflected MNIST | accuracy | Helped |
| Reduce the number of filters to 2 | G-CNN with Max G-pooling · Rotated MNIST and rotated-reflected MNIST | accuracy: less than 60% | Hurt |
Surprising findings
- Despite being complete, the selective G-Bispectrum achieves slightly lower accuracy than the G-TC, attributed to the limits of practical MLPs.
- Max G-pooling performs well with many filters, but drops below 60% with 2 filters while complete invariants stay above 85%.
Limitations
- Selective G-Bispectrum achieves slightly lower accuracy than G-TC in experiments
- Experiments limited to MNIST/EMNIST with simple G-CNN architectures, not state-of-the-art accuracy
- Completeness proven only for commutative, dihedral, and octahedral groups; general finite groups not covered
Experimental setup
Datasets: MNIST EMNIST SO(2)-MNIST O(2)-MNIST SO(2)-EMNIST O(2)-EMNIST
Benchmarks: SO(2)-MNIST O(2)-MNIST SO(2)-EMNIST O(2)-EMNIST
Metrics: validation accuracy training time parameter count
Baselines: Max G-pooling Avg G-pooling G-TC pooling full G-Bispectrum
Builds on: G-CNN G-triple correlation G-bispectrum escnn
Closest papers by meaning
- $\texttt{bispectrum}$: Selective $G$-Bispectra Made Practical (NeurIPS 2026 · similarity 0.94)
- Lie Group Decompositions for Equivariant Neural Networks (ICLR 2024 · similarity 0.84)
- Group Downsampling with Equivariant Anti-aliasing (ICLR 2025 · similarity 0.84)
- On Making $SE(2)$-Invariant Networks Optimal (NeurIPS 2026 · similarity 0.83)
- Variational Partial Group Convolutions for Input-Aware Partial Equivariance of Rotations and Color-Shifts (ICML 2024 · similarity 0.83)
- A Renormalization Group Framework for Scale-Invariant Feature Learning in Deep Neural Networks (Student Abstract) (AAAI 2025 · similarity 0.82)
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