Complex-valued Multidirectional Associative Memory

Abstract

Hopfield model is a representative associative memory. It was improved to Bidirectional Associative Memory(BAM) by Kosko and Multidirectional Associative Memory(MAM) by Hagiwara. They have two layers or multilayers. Since they have symmetric connections between layers, they ensure to converge. MAM can deal with multiples of many patterns, such as (x 1 , x 2 , · · · ), where x m is the pattern on layer-m.Noest, Hirose and Nemoto proposed complex-valued Hopfield model. Lee proposed complex-valued Bidirectional Associative Memory. Zemel proved the rotation invariance of complex-valued Hopfield model. It means that the rotated pattern also stored.In this paper, the complex-valued Multidirectional Associative Memory is proposed. The rotation invariance is also proved. Moreover it is shown by computer simulation that the differences of angles of given patterns are automatically reduced.At first we define complex-valued Multidirectional Associative Memory. Then we define the energy function of network. By using energy function, we prove that the network ensures to converge.Next, we define the learning law and show the characteristic of recall process. The characteristic means that the differences of angles of given patterns are automatically reduced. Especially we prove the following theorem. In case that only a multiple of patterns is stored, if patterns with different angles are given to each layer, the differences are automatically reduced.Finally, we invest that the differences of angles influence the noise robustness. It reduce the noise robustness, because input to each layer become small. We show that by computer simulations.

Attractors accompanied with a training pattern of multivalued hopfield neural networks

“…However, we do not know if only the rotated patterns are stable. Kobayashi proved that only the rotated patterns are attractors for the phasor models of multidirectional associative memories . In this paper, we reveal all attractors when complex‐valued and rotor Hopfield neural networks learn a 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.…”

Chaotic complex‐valued bipartite auto‐associative memory with a periodic activation function

“…Hagiwara proposed a multidirectional associative memory (MAM) consisting of multilayers . Further extensions of a MAM have been proposed . High‐dimensional associative memories have also been proposed .…”