Compact Matrix Quantum Group Equivariant Neural Networks

We derive the existence of a new type of neural network, called a compact matrix quantum group equivariant neural network, that learns from data that has an underlying quantum symmetry. We apply the Woronowicz formulation of Tannaka-Krein duality to characterise the weight matrices that appear in these neural networks for any easy compact matrix quantum group. We show that compact matrix quantum group equivariant neural networks contain, as a subclass, all compact matrix group equivariant neural networks. Moreover, we obtain characterisations of the weight matrices for many compact matrix group equivariant neural networks that have not previously appeared in the machine learning literature.