Quantum-Inspired Complex-Valued Multidirectional Associative Memory Naoki Masuyama
Chu Kiang Loa
Faculty of Computer Science and Information Technology, University of Malaya, 50603 Kuala Lumpur, MALAYSIA Email:
[email protected]
Faculty of Computer Science and Information Technology, University of Malaya, 50603 Kuala Lumpur, MALAYSIA Email: e-mail:
[email protected]
Abstract-Complex-valued neuron is one of the significant and
by a projection matrix based on least mean squared error minimization [9], and the two stages learning algorithm called Quick Learning that are applied Hebbian learning and pseudo relaxation learning [10]. One of the successful approaches is to apply the concept of quantum mechanics to neural networks [11]-[13]. In 2000's, Rigatos et al. has proposed Quantum Inspired Hopfield Associative Memory (QHAM) [14]. This model demonstrates quantum information processing in neural structures results in an exponential increase in storage capac ity and can explain the extensive memory and inferencing capabilities of humans. It is applied a fuzzy inference to weight matrix, that defined by an one-shot learning as Hebbian learning, to satisfy parallelism and unitarity. However, QHAM is limited as an auto-association. To perform the "one to many" association, we have developed Quantum-Inspired Mulitidi rectional Associative Memory (QMAM) with maintaining the features of Quantum-Inspired model. In regard as events in the real world, information rep resentation using binary or bipolar state is insufficient. In 1970's, the concept of complex-valued neuron model has introduced by Aizenberg [15]. One of the significant advan tages in complex-valued neural network is it can be handled the higher dimensions than real-valued neural networks. The fundamentals of multi-states associative memory is introduced by Noest [16]. Jankowski et al. have introduced complex valued hopfield associative memory with neurons processing a complex-valued discrete activation function [17]. Lee et al. have introduced bidirectional model [18]. Multidirectional model has implemented by Kobayashi et al. [19]. Conven tionally, several types of algorithmic improvements have been proposed; Lee et al. has shown that the real-valued projection rule can be generalized to complex domain such that the weight matrix can be designed by using a simple and effective method [20]. Based on real-valued Quick Learning, complex valued Quick Learning is applied [21]. Furthermore, based on various physiological experiments, chaotic behavior of the real neurons has observed. In particular, it is considered chaos plays an important roles in memory and learning of human brain. Several researches have introduced a chaotic associative memory to imitate the human brain functions [22], [23]. Though these models show the superior abilities, the complexity of the model structures are greatly increased.
effective innovation in artificial neural networks. It is able to deal with multi-valued pattern and oscillator models. With regard as associative memory, several types of complex-valued artificial neural associative memories have developed, and confirmed its superior abilities. Conventionally, we have developed Quantum Inspired Mulitidirectional Associative Memory (QMAM). This model demonstrates quantum information processing in neural structures results in an exponential increase in storage capacity and can explain the extensive memory and inferencing capa bilities of humans. This model is applied a fuzzy inference to weight matrix to satisfy parallelism and unitarity. In this paper, we introduce Quantum-Inspired Complex-Valued Multidi rectional Associative Memory (QCMAM) to handle multi-valued information. In addition, the mathematical proofs of parallelism and unitarity for a complex-valued model are presented. The simulation experiments show that QCMAM has superior abilities comparing with conventional model.
Index Terms-Associative Memory; Quantum-Inspired Com puting; Complex-Valued Neural Networks;
I.
INTRODUCTION
In terms of human-human communication, associative mem ory is one of the important functions. It can deal with continuity and relevance between information, or events [1]. It is useful in various scenes in human activities, such as retrieve the memories, and recalling the related information. The importance of relevance and continuity of information are discussed in fields of psychological and cognitive sciences [2], [3]. In past decades, to mimic above useful functions, various types of artificial neural associative memories have proposed. In early 1980's, Hopfield has proposed an auto-associative memory model to store and recall information [4]. One of the limitations of auto-associative memory is it cannot be recalled from the input information to the different information. To overcome this problem, Kosko and Hagiwara et al. have proposed a hetero-association models that are called Bidirec tional Associative Memory (BAM) [5] and Multidirectional Associative Memory (MAM) [6], respectively. However, these fundamental models are suffered from low storage capacity and poor recall reliability. Conventionally, several methods have proposed to improve the abilities of models; as exam ples of model structural improvement, it has added dummy neurons [7] and hidden layers [8]. From the point of view of the algorithmic improvements, there is a weight learning
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In this paper, we introduce Quantum-Inspired Complex Valued Multidirectional Associative Memory (QCMAM) based on QMAM. We present the mathematical proofs that the features of Quantum-Inspired model as parallelism and unitarity are satisfied in weight matrices, that is defined by Hebbian learning, of QCMAM. The paper is divided as follows; Section II describes the fundamentals of QCMAM. Section ill presents the similarity between fuzzy inference and quantum mechanics, and presents the mathematical proofs of superposition and unitarity in proposed model. In section I V, it will be presented simulation experiments of proposed models in terms of the memory capacity and noise tolerance comparing with conventional model. II.
FUNDAMENTALS OF
1m
Fig. 1: Discrete Complex Unit Circle
•
a-th layer to ,B-th layer
WiJ
QUANTUM-INSPIRED
COMPLEX-VALUED MULTIDIRECTIONAL ASSOCIATIVE MEMORY
Here, it will be presented the equations of Quantum-Inspired Complex-Valued Multidirectional Associative Memory (QC MAM) between the a-th layer and the ,B-th layer. • a-th layer to ,B-th layer
•
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y(k) = � � W*x(k) i=l Jy k) = ¢ (Yj(k)) 'J
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(1)
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(2)
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exp(j27rnjq),
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If Arg ¢Z ( )=
Yj(k)
{exp(jf7rn/q) } I
