Neural Discovery of Permutation Subgroups

We consider the problem of discovering subgroup $H$ of permutation group $S_n$. Unlike the traditional $H$-invariant networks wherein $H$ is assumed to be known, we present a method to discover the underlying subgroup, given that it satisfies certain conditions. Our results show that one could discover any subgroup of type $S_k (k \leq n)$ by learning an $S_n$-invariant function and a linear transformation. We also prove similar results for cyclic and dihedral subgroups. Finally, we provide a general theorem that can be extended to discover other subgroups of $S_n$. We also demonstrate the applicability of our results through numerical experiments on image-digit sum and symmetric polynomial regression tasks.