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 (x1, x2,…), where xm 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.