Neural Discovery of Permutation Subgroups
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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 ≤ 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.
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