Compact Matrix Quantum Group Equivariant Neural Networks
This work addresses a theoretical gap for researchers in quantum machine learning and non-commutative geometry, but it is incremental as it extends existing equivariant network concepts to a specialized domain.
The paper tackles the limitation of group equivariant neural networks in handling data from non-commutative geometries by introducing compact matrix quantum group equivariant neural networks, resulting in new characterizations of equivariant weight matrices for certain compact matrix groups.
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.