Compact Matrix Quantum Group Equivariant Neural Networks

TL;DR

This work extends equivariant neural networks to non-commutative geometry by formulating compact matrix quantum group (CMQG) equivariant networks, underpinned by Woronowicz's Tannaka–Krein duality. It proves the existence of CMQG-equivariant architectures and fully characterises the learnable linear layers for easy CMQGs, which are defined via two-coloured partition categories and universal $C^{*}$-algebras. The framework recovers classical compact group symmetries and yields new characterisations for CMQGs such as the hyperoctahedral and bistochastic groups, as well as unitary variants, with explicit constructions at $n=2$ demonstrating practical weight-matrix forms. The practical impact lies in enabling symmetry-aware learning on data with non-commutative geometric structure, while outlining future work to extend these results to equivariant nonlinearities for real-world deployment.

Abstract

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.

Figures (2)

• Figure 1: The fifteen $(2,2)$--partition diagrams.
• Figure 2: For $n=2$, we display the images under the map $d_\pi \mapsto D_\pi$ for each of the $(2,2)$--partition diagrams.

Theorems & Definitions (90)

• Definition 3.1
• Remark 3.2
• Proposition 3.3: Appendix: Section \ref{['missingproofs']}
• Theorem 3.4: Fundamental Theorem of Compact Matrix Quantum Groups
• Example 3.5
• Lemma 4.1
• proof
• Definition 4.2
• Remark 4.3
• Definition 4.4