Neural Discovery of Permutation Subgroups
Pavan Karjol, Rohan Kashyap, Prathosh AP
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.
BibTeX
@InProceedings{pmlr-v206-karjol23a,
title = {Neural Discovery of Permutation Subgroups},
author = {Karjol, Pavan and Kashyap, Rohan and {AP}, Prathosh},
booktitle = {Proceedings of The 26th International Conference on Artificial Intelligence and Statistics},
pages = {4668--4678},
year = {2023},
editor = {Ruiz, Francisco and Dy, Jennifer and van de Meent, Jan-Willem},
volume = {206},
series = {Proceedings of Machine Learning Research},
month = {25--27 Apr},
publisher = {PMLR},
pdf = {https://proceedings.mlr.press/v206/karjol23a/karjol23a.pdf},
url = {https://proceedings.mlr.press/v206/karjol23a.html},
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.}
}