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.

Lay Summary:

Many machine learning models improve their performance by encoding symmetries, which are typically described by groups, into their architectures. These models work well for data that lives in a classical geometric space but cannot be used to learn from data that lives in a non-commutative geometry since traditional group symmetries no longer apply.We introduce a new type of neural network that is designed to be equivariant to symmetries that are described by compact matrix quantum groups. These quantum groups generalise groups to model symmetries in certain non-commutative spaces. We prove the existence of these networks and precisely characterise the structure of their weight matrices for specific compact matrix quantum groups.Our approach makes it possible to learn from symmetries that have not been previously explored in machine learning, with potential applications in quantum physics and statistical mechanics.