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.

翻译：我们提出了一类新型神经网络的存在性，称为紧致矩阵量子群等变神经网络，该类网络能从具有底层量子对称性的数据中学习。我们应用Woronowicz形式的Tannaka-Krein对偶理论，刻画了任意简单紧致矩阵量子群中此类神经网络所涉及的权重矩阵。研究表明，紧致矩阵量子群等变神经网络将紧致矩阵群等变神经网络作为其子类包含在内。此外，我们还获得了此前机器学习文献中未曾出现的多种紧致矩阵群等变神经网络权重矩阵的完整刻画。