# Brauer's Group Equivariant Neural Networks

arXiv: https://arxiv.org/abs/2212.08630 · HTML: https://mlsanity.com/paper/2212.08630/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 40 references that are in the map.
- First posted on arXiv 16 Dec 2022.

## Builds on (most cited first, 14 of 14)

- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [Lorentz Group Equivariant Neural Network for Particle Physics](https://mlsanity.com/paper/2006.04780/) (2020)
- [Deep Models of Interactions Across Sets](https://mlsanity.com/paper/1803.02879/) (2018)
- [What is an equivariant neural network?](https://mlsanity.com/paper/2205.07362/) (2022)
- [Universal Approximation Theorem for Equivariant Maps by Group CNNs](https://mlsanity.com/paper/2012.13882/) (2020)
- [A Simple and Universal Rotation Equivariant Point-cloud Network](https://mlsanity.com/paper/2203.01216/) (2022)

## Built on by (most cited first, 17 of 17)

- [Improving Equivariant Model Training via Constraint Relaxation](https://mlsanity.com/paper/2408.13242/) (2024)
- [Learning on LoRAs: GL-Equivariant Processing of Low-Rank Weight Spaces for Large Finetuned Models](https://mlsanity.com/paper/2410.04207/) (2024)
- [Expressive Sign Equivariant Networks for Spectral Geometric Learning](https://mlsanity.com/paper/2312.02339/) (2023)
- [Fast computation of permutation equivariant layers with the partition algebra](https://mlsanity.com/paper/2303.06208/) (2023)
- [Symmetry-Based Structured Matrices for Efficient Approximately Equivariant Networks](https://mlsanity.com/paper/2409.11772/) (2024)
- [High-Rank Irreducible Cartesian Tensor Decomposition and Bases of Equivariant Spaces](https://mlsanity.com/paper/2412.18263/) (2024)
- [How Jellyfish Characterise Alternating Group Equivariant Neural Networks](https://mlsanity.com/paper/2301.10152/) (2023)
- [Graph Automorphism Group Equivariant Neural Networks](https://mlsanity.com/paper/2307.07810/) (2023)
- [Tensor learning with orthogonal, Lorentz, and symplectic symmetries](https://mlsanity.com/paper/2406.01552/) (2024)
- [Categorification of Group Equivariant Neural Networks](https://mlsanity.com/paper/2304.14144/) (2023)
- [Revisiting Multi-Permutation Equivariance through the Lens of Irreducible Representations](https://mlsanity.com/paper/2410.06665/) (2024)
- [An Algorithm for Computing with Brauer's Group Equivariant Neural Network Layers](https://mlsanity.com/paper/2304.14165/) (2023)
- [Compact Matrix Quantum Group Equivariant Neural Networks](https://mlsanity.com/paper/2311.06358/) (2023)
- [Precoder Learning for Weighted Sum Rate Maximization](https://mlsanity.com/paper/2503.04497/) (2025)
- [Permutation Equivariant Neural Networks for Symmetric Tensors](https://mlsanity.com/paper/2503.11276/) (2025)
- [Constructing 3D Rotational Invariance and Equivariance with Symmetric Tensor Networks](https://mlsanity.com/paper/2508.12596/) (2025)
- [Learning Group Actions In Disentangled Latent Image Representations](https://mlsanity.com/paper/2512.04015/) (2025)

## Abstract (arXiv)

We provide a full characterisation of all of the possible group equivariant neural networks whose layers are some tensor power of ℝn for three symmetry groups that are missing from the machine learning literature: O(n), the orthogonal group; SO(n), the special orthogonal group; and Sp(n), the symplectic group. In particular, we find a spanning set of matrices for the learnable, linear, equivariant layer functions between such tensor power spaces in the standard basis of ℝn when the group is O(n) or SO(n), and in the symplectic basis of ℝn when the group is Sp(n).