Multistate vector product hopfield neural networks

Several high-dimensional models of Hopfield neural networks, such as complex-valued and quaternionic Hopfield neural networks, have been proposed. However, it has been hard to construct three-dimensional models of Hopfield neural networks. A split type of vector product Hopfield neural network (VPHNN) was proposed as a special case of ordinary Hopfield neural networks. It is easier to construct split types of Hopfield neural networks than multistate types of ones, because the split types can be often regarded as special cases of ordinary ones. In the present work, we extend the split VPHNN to the multistate VPHNN. We define its energy and a primitive learning algorithm, the Hebbian learning rule. In addition, we prove the stability of multistate VPHNNs. Furthermore, we investigated the fundamentals of multistate VPHNNs, such as the storage capacity and noise tolerance, by computer simulations.