Neural Group Actions

We introduce an algorithm for designing Neural Group Actions, collections of deep neural network architectures which model symmetric transformations satisfying the laws of a given finite group. This generalizes involutive neural networks $\mathcal{N}$, which satisfy $\mathcal{N}(\mathcal{N}(x))=x$ for any data $x$, the group law of $\mathbb{Z}_2$. We show how to optionally enforce an additional constraint that the group action be volume-preserving. We conjecture, by analogy to a universality result for involutive neural networks, that generative models built from Neural Group Actions are universal approximators for collections of probabilistic transitions adhering to the group laws. We demonstrate experimentally that a Neural Group Action for the quaternion group $Q_8$ can learn how a set of nonuniversal quantum gates satisfying the $Q_8$ group laws act on single qubit quantum states.