Neural Discovery of Permutation Subgroups
Pavan Karjol Rohan Kashyap Prathosh A P
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.