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$\texttt{bispectrum}$: Selective $G$-Bispectra Made Practical

NeurIPS 2026 · Architectures and efficiency › CNN design · approach: Group convolution theory

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Presents bispectrum, an open-source PyTorch library of selective G-bispectra for seven group/domain pairs, plus an augmented Theta(L^2) selective SO(3) invariant on S^2, and evaluates bispectral pooling on PCam, OrganMNIST3D and Spherical MNIST.

Problem

Full G-bispectra are complete invariants but cost O(|G|^2) or O(L^3) coefficients, and existing implementations are scattered across group-specific code, limiting their use as pooling layers in deep learning.

Approach

A unified differentiable PyTorch library implementing selective G-bispectra (O(|G|) coefficients for finite groups, disk bispectrum for planar rotations) and a new augmented selective SO(3) invariant on S^2 with Theta(L^2) entries, used as drop-in invariant pooling layers.

Key results

• On Spherical MNIST, the SO3onS2 bispectrum+MLP reaches 95.0% (NR/NR) with rotation robustness ~1e-4, versus 78% for the power-spectrum MLP.
• On PCam at 1% data, the bispectrum reaches 0.912 AUC versus 0.873 for NormReLU and 0.876 for Fourier-ELU.
• On OrganMNIST3D at 5% data, the bispectrum reaches 33.2% vs 18.9% for max pool; at full data max pool (73.0%) matches the bispectrum (72.6%).

Surprising findings

• At high capacity (6M params) max pooling overtakes the bispectrum on OrganMNIST3D.
• The augmented SO(3) invariant's completeness is only empirically supported via reconstruction; the authors state a formal completeness proof remains open.

Limitations

• Completeness of the SO3onS2 augmented invariant is empirical only, no formal proof.
• Inversion is not shipped as a library API; conditioning at large bandlimits unanalyzed.
• Not compared against e3nn, MACE or wavelet scattering.

Experimental setup

Datasets: PatchCamelyon OrganMNIST3D Spherical MNIST

Benchmarks: PatchCamelyon OrganMNIST3D Spherical MNIST

Metrics: AUC accuracy rotation robustness (std over rotations) forward latency (ms)

Baselines: norm pooling gated pooling Fourier-ELU pooling max pooling augmented CNN power spectrum MLP Cohen spherical CNN Regular SE(3) convolutions

Builds on: selective G-bispectrum disk bispectrum escnn e3nn DenseNet

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• The Selective $G$-Bispectrum and its Inversion: Applications to $G$-Invariant Networks (NeurIPS 2024 · similarity 0.94)
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