A $2$-closed permutation group with nonuniversal first-order invariant networks
Year: 2026
Type: preprint
Abstract: We give a counterexample to a conjecture of Maron--Fetaya--Segol--Lipman on the universality of certain neural networks. More precisely let $G\cong S_6$ act on the $15$ unordered pairs of a six-element set. We show that this permutation group is $2$-closed, but every first-order $G$-invariant neural network, with arbitrary activation function, identifies the edge indicator vectors of a six-cycle and two disjoint triangle graphs. Since these vectors can be separated by a $G$-invariant cubic polynomial, we get that the latter cannot be approximated arbitrarily well by such neural networks.
Author: Pal, Ambrus
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225815894
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