Neural Discovery of Permutation Subgroups
This addresses a limitation in invariant networks for machine learning, where subgroup knowledge is often assumed, but it is incremental as it builds on existing invariant network frameworks.
The paper tackles the problem of discovering unknown permutation subgroups from data, rather than assuming they are known, and shows that subgroups like symmetric, cyclic, and dihedral types can be discovered by learning an invariant function and a linear transformation, with validation through numerical experiments on tasks such as image-digit sum and symmetric polynomial regression.
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.