Attractors accompanied with a training pattern of multivalued hopfield neural networks

Abstract

Recently, multivalued Hopfield neural networks, such as complex-valued Hopfield neural networks, and their applications have been studied by many researchers. In their application, low noise robustness is an important problem. Too many attractors accompanied with training patterns damage the noise robustness. Rotated patterns are well-known attractors of complex-valued Hopfield neural networks. In the present work, we reveal that any other patterns are never attractors in complex-valued Hopfield neural networks with a training pattern. We also prove a similar result for rotor Hopfield neural network. In addition, we investigate the relation between noise robustness and the number of attractors.

Dual‐numbered Hopfield neural networks

“…From the result for the case of P = 1, we can evaluate how many fixed points there are for a training pattern. It is known that a CHNN has exactly four global minima for a training pattern . An HHNN also has four global minima for a training pattern .…”

“…This problem becomes fatal in CHNNs and QHNNs. In CHNNs and QHNNs, the rotated patterns of training patterns are stored . It is referred to as rotational invariance.…”

Hyperbolic Hopfield neural networks with four‐state neurons

“…In the real-valued Hopfield neural networks, for a training pattern, the reversed pattern is also stable. In the complex-valued K -state Hopfield neural networks, there exist K stable states, referred to as the rotated patterns, for one training pattern [36,37]. In HHNNs, we expect four stable patterns to exist for one training pattern.…”

Rotational invariance of quaternionic hopfield neural networks

“…In the case of the CHNNs with one training vector, the fixed vectors are completely determined . For the training vector y , e iθ y is the vector whose components are uniformly rotated.…”

“…This is referred to as rotational invariance. In the case of one training vector, it is known that we can obtain all fixed vectors by rotating the training vector . Rotational invariance makes many fixed vectors and deteriorates the noise tolerance.…”