# Compact Matrix Quantum Group Equivariant Neural Networks

arXiv: https://arxiv.org/abs/2311.06358 · HTML: https://mlsanity.com/paper/2311.06358/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 24 of its 80 references that are in the map.
- First posted on arXiv 10 Nov 2023.

## Builds on (most cited first, 24 of 24)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [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)
- [On the Universality of Invariant Networks](https://mlsanity.com/paper/1901.09342/) (2019)
- [Equivariance Through Parameter-Sharing](https://mlsanity.com/paper/1702.08389/) (2017)
- [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)
- [On Learning Sets of Symmetric Elements](https://mlsanity.com/paper/2002.08599/) (2020)
- [Clifford Group Equivariant Neural Networks](https://mlsanity.com/paper/2305.11141/) (2023)
- [Universal Equivariant Multilayer Perceptrons](https://mlsanity.com/paper/2002.02912/) (2020)
- [Geometric Clifford Algebra Networks](https://mlsanity.com/paper/2302.06594/) (2023)
- [SE(3)-Equivariant Attention Networks for Shape Reconstruction in Function Space](https://mlsanity.com/paper/2204.02394/) (2022)
- [What is an equivariant neural network?](https://mlsanity.com/paper/2205.07362/) (2022)
- [Brauer's Group Equivariant Neural Networks](https://mlsanity.com/paper/2212.08630/) (2022)
- [C^*-algebra Net: A New Approach Generalizing Neural Network Parameters to C^*-algebra](https://mlsanity.com/paper/2206.09513/) (2022)
- [Connecting Permutation Equivariant Neural Networks and Partition Diagrams](https://mlsanity.com/paper/2212.08648/) (2022)
- [Fast computation of permutation equivariant layers with the partition algebra](https://mlsanity.com/paper/2303.06208/) (2023)
- [How Jellyfish Characterise Alternating Group Equivariant Neural Networks](https://mlsanity.com/paper/2301.10152/) (2023)
- [Noncommutative C^*-algebra Net: Learning Neural Networks with Powerful Product Structure in C^*-algebra](https://mlsanity.com/paper/2302.01191/) (2023)
- [Categorification of Group Equivariant Neural Networks](https://mlsanity.com/paper/2304.14144/) (2023)

## Abstract (arXiv)

Group equivariant neural networks have proven effective in modelling a wide range of tasks where the data lives in a classical geometric space and exhibits well-defined group symmetries. However, these networks are not suitable for learning from data that lives in a non-commutative geometry, described formally by non-commutative C*-algebras, since the C*-algebra of continuous functions on a compact matrix group is commutative. To address this limitation, we derive the existence of a new type of equivariant neural network, called compact matrix quantum group equivariant neural networks, which encode symmetries that are described by compact matrix quantum groups. We characterise the weight matrices that appear in these neural networks for the easy compact matrix quantum groups, which are defined by set partitions. As a result, we obtain new characterisations of equivariant weight matrices for some compact matrix groups that have not appeared previously in the machine learning literature.