Group ring-valued Hopfield networks

Abstract

The Hopfield networks have been successfully applied to associative memories for binary data. The complex-valued Hopfield networks extend the classic Hopfield networks for multilevel data. Several researchers have been attempting further extensions by Clifford algebra. However, the dimensions of Clifford algebras are limited to power of two. More flexible algebraic families are necessary for extensions of high-dimensional Hopfield networks. In the present paper, we propose high-dimensional Hopfield networks using group rings, referred to as group ring-valued Hopfield networks (GRVHNs). The GRVHN is a broad family which provides high-dimensional Hopfield networks of any dimensions. Moreover, it is also shown that many conventional high-dimensional Hopfield networks are included in the GRVHNs.