Neural Discovery of Permutation Subgroups

TL;DR

This paper introduces a method to discover hidden permutation subgroups within the symmetric group by learning invariant functions and transformations, with theoretical guarantees and practical experiments demonstrating effectiveness.

Contribution

It presents a novel approach to identify permutation subgroups from data, extending previous invariant network methods to discover unknown subgroups like cyclic and dihedral groups.

Findings

Successfully discovers subgroups of type S_k, cyclic, and dihedral groups.

Theoretical proofs establish conditions under which subgroups can be identified.

Numerical experiments validate the method on image-digit sum and polynomial regression tasks.

Abstract

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.