# The Selective G-Bispectrum and its Inversion: Applications to G-Invariant Networks

arXiv: https://arxiv.org/abs/2407.07655 · HTML: https://mlsanity.com/paper/2407.07655/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 36 references that are in the map.
- First posted on arXiv 10 Jul 2024.

## Builds on (most cited first, 8 of 8)

- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Towards a Definition of Disentangled Representations](https://mlsanity.com/paper/1812.02230/) (2018)
- [EMNIST: an extension of MNIST to handwritten letters](https://mlsanity.com/paper/1702.05373/) (2017)
- [Bispectral Neural Networks](https://mlsanity.com/paper/2209.03416/) (2022)
- [A General Framework for Robust G-Invariance in G-Equivariant Networks](https://mlsanity.com/paper/2310.18564/) (2023)
- [A Spectral Method for Stable Bispectrum Inversion with Application to Multireference Alignment](https://mlsanity.com/paper/1802.10493/) (2018)

## Built on by (most cited first, 6 of 6)

- [Selective Disk Bispectrum: A Complete and Rotation Invariant Image Descriptor](https://mlsanity.com/paper/2511.19706/) (2025)
- [Exact Symmetry as Algebra: Group-Structured Tensors that Enforce Physical Selection Rules](https://mlsanity.com/paper/2605.20440/) (2026)
- [Rotation-Invariant Spherical Watermarking via Third-Order SO(3) Representation Coupling](https://mlsanity.com/paper/2605.26702/) (2026)
- [Bispectral OT: Dataset Comparison using Symmetry-Aware Optimal Transport](https://mlsanity.com/paper/2509.20678/) (2025)
- [bispectrum: Selective G-Bispectra Made Practical](https://mlsanity.com/paper/2605.07270/) (2026)
- [Varifold Moment Invariants for Sustainable and Explainable Contour Feature Extraction](https://mlsanity.com/paper/2606.07333/) (2026)

## Abstract (arXiv)

An important problem in signal processing and deep learning is to achieve textitinvariance to nuisance factors not relevant for the task. Since many of these factors are describable as the action of a group G (e.g. rotations, translations, scalings), we want methods to be G-invariant. The G-Bispectrum extracts every characteristic of a given signal up to group action: for example, the shape of an object in an image, but not its orientation. Consequently, the G-Bispectrum has been incorporated into deep neural network architectures as a computational primitive for G-invariancetextemdash akin to a pooling mechanism, but with greater selectivity and robustness. However, the computational cost of the G-Bispectrum (O(|G|2), with |G| the size of the group) has limited its widespread adoption. Here, we show that the G-Bispectrum computation contains redundancies that can be reduced into a textitselective G-Bispectrum with O(|G|) complexity. We prove desirable mathematical properties of the selective G-Bispectrum and demonstrate how its integration in neural networks enhances accuracy and robustness compared to traditional approaches, while enjoying considerable speeds-up compared to the full G-Bispectrum.