On the performance of Quaternionic Bidirectional Auto-Associative Memory

Minemoto, Toshifumi; Isokawa, Teijiro; Matsui, Nobuyuki; Kobayashi, Masaki; Nishimura, Haruhiko

doi:10.1109/ijcnn.2015.7280692

Cited by 7 publications

(4 citation statements)

References 14 publications

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“…Quaternionic Hopfield neural networks have been studied by several researchers. Isokawa et al proposed multi-state quaternionic Hopfield neural networks and applied them to store color images [27][28][29][30]. Kobayashi proposed hybrid quaternionic Hopfield neural networks with a split activation function [31,32] [33].…”

Section: Introductionmentioning

confidence: 99%

Hyperbolic Hopfield neural networks with four‐state neurons

Kobayashi

2016

IEEJ Transactions Elec Engng

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In recent years, applications of neural networks with Clifford algebra have become widespread. Clifford algebra is also referred to as geometric algebra and is useful in dealing with geometric objects. Hopfield neural networks with Clifford algebra, such as complex numbers and quaternions, have been proposed. However, it has been difficult to construct Hopfield neural networks by Clifford algebra with positive part of the signature, such as hyperbolic numbers. Hyperbolic numbers are useful algebra to deal with hyperbolic geometry. Kuroe proposed hyperbolic Hopfield neural networks and provided their continuous activation functions and stability conditions. However, the learning algorithm has not been provided. In this paper, we provide two quantized activation functions and the primitive learning algorithm satisfying the stability condition. We also perform computer simulations and compare the activation functions.

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Section: Introductionmentioning

confidence: 99%

Hyperbolic Hopfield neural networks with four‐state neurons

Kobayashi

2016

IEEJ Transactions Elec Engng

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“…Furthermore, we pointed out exactly which were the wrong assumptions used to show the convergence of sequence defined by the MV-QHNN model os Isokawa et al [37,40]. We believe this is an important theoretical issue because MV-QHNNs have been used as the basis for many other quaternionic auto-AM models [64]. Moreover, we expect to instigate further research on the dynamics of multivalued quaternionic recurrent neural networks.…”

Section: Discussionmentioning

confidence: 86%

“…In this chapter, we focus on quaternionic Hopfield neural networks (QHNNs), a topic that has been extensively investigated in the last years [37,39,40,45,46,50,64,65,69,87,88].…”

Section: Quaternion-valued Hopfield Neural Networkmentioning

confidence: 99%

“…Briefly, using the phase-angle representation of a quaternion, Isokawa and collaborators defined a quaternionic multivalued signum function which generalizes the complex-valued signum function of Jankowski et al [42]. Furthermore, an improved version of the quaternionic multivalued signum function has been proposed recently by Minemoto and collaborators [64,65].…”

Section: Multivalued Quaternionic Hopfield Neural Networkmentioning

confidence: 99%

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Stability analysis of hypercomplex-valued discrete-time Hopfield-type neural networks

Castro¹

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ResumoA rede neural de Hopfield é uma das mais importantes redes neurais recorrentes, concebida, inicialmente, para o armazenamento e recuperação de padrões bipolares. Nesta tese de doutorado, analisamos a estabilidade de algumas de suas generalizações em domínios complexos e quaterniônicos. Introduzimos dois modelos complexos e um modelo quaterniônico que sempre geram sequências convergentes para um estado estacionário, admitindo-se modo de atualização assíncrona e as condições usuais sobre a matriz de pesos sinápticos, isto é, matriz Hermitiana com elementos da diagonal principal não negativos. No tocante aos modelos quaterniônicos, implementamos memórias autoassociativas visando o armazenamento e a recordação de padrões sintéticos bem como de imagens coloridas reais. Estudamos a capacidade de armazenamento e a tolerância a ruído desses modelos. Por fim, introduzimos novos sistemas de números hipercomplexos e uma ampla família de funções de ativação com valores nesses sistemas. Usando as funções pertencentes a essa família, definimos uma ampla classe de redes hipercomplexas do tipo Hopfield. Essas redes sempre geram sequências convergentes para um estado estacionário, dado qualquer estado inicial. Nesta tese, resultados teóricos são obtidos e demonstrados, os quais propiciam um locus para o desenvolvimento de outros modelos hipercomplexos e o seu uso em várias aplicações.

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Rotational invariance of quaternionic hopfield neural networks

Kobayashi

2016

IEEJ Transactions Elec Engng

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High-dimensional neural networks have been studied by many researchers for their flexible representation. Quaternionic Hopfield neural networks (QHNNs) are one of them. Quaternions have the inherent property of non-commutative multiplication. Connection weights act differently from left and right in QHNNs, and we can have left QHNNs (LQHNNs) and right QHNNs (RQHNNs). Hybrid QHNNs (HQHNNs), which are the compound models of LQHNNs and RQHNNs, have been proposed, and their excellent noise tolerance has been shown through computer simulations. It has been pointed out that the improvement in their noise tolerance is related to the avoidance of rotational invariance. Thus, it is very important to study rotational invariance. In this paper, we investigate the rotational invariance of QHNNs. Rotational invariance in LQHNNs and RQHNNs is different. We define the left and right-rotated vectors, which are fixed in RQHNNs and LQHNNs, respectively. From the standpoint of HQHNNs, the common rotated vectors are important. We also study the common rotated vectors and provide a couple of exceptional examples.

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On the performance of Quaternionic Bidirectional Auto-Associative Memory

Cited by 7 publications

References 14 publications

Hyperbolic Hopfield neural networks with four‐state neurons

Hyperbolic Hopfield neural networks with four‐state neurons

Stability analysis of hypercomplex-valued discrete-time Hopfield-type neural networks

Rotational invariance of quaternionic hopfield neural networks

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