Three Dimensional Cyclic-Group-Ring-Valued Hopfield Networks

Abstract

Hopfield networks have been extended to two dimensional multistate models such as complex-valued Hopfield networks. They have been applied to storage of multilevel information, such as image data. Few 3D models of Hopfield networks have been proposed. First, vector product Hopfield networks (VPHNs) were proposed. However, VPHNs have two disadvantages, the resolution and learning algorithms. Since the activation function is defined based on the regular polyhedrons, the resolution factor is limited. The projection rule, which is a practical learning algorithm, is not applicable for VPHNs. Quaternionic vector product Hopfield networks (QVPHNs), which are extensions of VPHNs, are proposed to solve the second disadvantage. Nevertheless, the QVPHNs require more weight parameters than the VPHNs. Recently, Hopfield networks were extended using group theory. In this paper, cyclic-group-ring-valued Hopfield networks are introduced, such that the projection rule is applicable without increase in weight parameters. We also determine the stability conditions for projection rules. The noise tolerance and memory costs for weight parameters are considered to be the trade-off. Our computer simulations show that the CGRVHNs achieve the noise tolerance according to the memory costs.