Compact Matrix Quantum Group Equivariant Neural Networks

Edward Pearce-Crump
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:48563-48594, 2025.

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 $\mathcal{C}^{\ast}$-algebras, since the $\mathcal{C}^{\ast}$-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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-pearce-crump25a, title = {Compact Matrix Quantum Group Equivariant Neural Networks}, author = {Pearce-Crump, Edward}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {48563--48594}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/pearce-crump25a/pearce-crump25a.pdf}, url = {https://proceedings.mlr.press/v267/pearce-crump25a.html}, 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 $\mathcal{C}^{\ast}$-algebras, since the $\mathcal{C}^{\ast}$-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.} }
Endnote
%0 Conference Paper %T Compact Matrix Quantum Group Equivariant Neural Networks %A Edward Pearce-Crump %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-pearce-crump25a %I PMLR %P 48563--48594 %U https://proceedings.mlr.press/v267/pearce-crump25a.html %V 267 %X 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 $\mathcal{C}^{\ast}$-algebras, since the $\mathcal{C}^{\ast}$-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.
APA
Pearce-Crump, E.. (2025). Compact Matrix Quantum Group Equivariant Neural Networks. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:48563-48594 Available from https://proceedings.mlr.press/v267/pearce-crump25a.html.

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