Fixed points of symmetric complex-valued Hopfield neural networks

A complex-valued Hopfield neural network (CHNN) is a model of a multistate Hopfield neural network, and has been applied to the storage of multilevel data. Weak noise tolerance, however, is a disadvantage of CHNNs. Symmetric CHNNs (SCHNNs), modified CHNNs, improve the noise tolerance of CHNNs. In the present work, we study the global and local minima of SCHNNs with one training pattern. In CHNNs, the global minima are the training and rotated patterns, and there are no local minima. In SCHNNs, it has been hard to determine all the global and local minima. It is thought that the global minima are the training and reversed patterns, and that there are no local minima in most cases. In the present work, however, we find many local minima, and show that they are very weak attractors, which reduce noise tolerance very little. In addition, we determine all the global minima of SCHNNs.