Stability Conditions of Bicomplex-Valued Hopfield Neural Networks

2021 ◽  
pp. 1-11
Author(s):  
Masaki Kobayashi
Keyword(s):  
Neural Networks ◽  
Hopfield Neural Network ◽  
Number System ◽  
Hopfield Neural Networks ◽  
Stability Conditions ◽  
Noise Tolerance ◽  
Hypercomplex Numbers ◽  
Bicomplex Numbers ◽  
The Stability ◽  
Projection Rule

Hopfield neural networks have been extended using hypercomplex numbers. The algebra of bicomplex numbers, also referred to as commutative quaternions, is a number system of dimension 4. Since the multiplication is commutative, many notions and theories of linear algebra, such as determinant, are available, unlike quaternions. A bicomplex-valued Hopfield neural network (BHNN) has been proposed as a multistate neural associative memory. However, the stability conditions have been insufficient for the projection rule. In this work, the stability conditions are extended and applied to improvement of the projection rule. The computer simulations suggest improved noise tolerance.

Hyperbolic-Valued Hopfield Neural Networks in Synchronous Mode

2020 ◽  
Vol 32 (9) ◽  
pp. 1685-1696
Author(s):  
Masaki Kobayashi
Keyword(s):  
Neural Networks ◽  
Hopfield Neural Networks ◽  
Stability Conditions ◽  
Noise Tolerance ◽  
Asynchronous Mode ◽  
High Noise ◽  
Synchronous Mode ◽  
The Stability ◽  
Complex Valued ◽  
Projection Rule

For most multistate Hopfield neural networks, the stability conditions in asynchronous mode are known, whereas those in synchronous mode are not. If they were to converge in synchronous mode, recall would be accelerated by parallel processing. Complex-valued Hopfield neural networks (CHNNs) with a projection rule do not converge in synchronous mode. In this work, we provide stability conditions for hyperbolic Hopfield neural networks (HHNNs) in synchronous mode instead of CHNNs. HHNNs provide better noise tolerance than CHNNs. In addition, the stability conditions are applied to the projection rule, and HHNNs with a projection rule converge in synchronous mode. By computer simulations, we find that the projection rule for HHNNs in synchronous mode maintains a high noise tolerance.

Bicomplex Projection Rule for Complex-Valued Hopfield Neural Networks

2020 ◽  
Vol 32 (11) ◽  
pp. 2237-2248
Author(s):  
Masaki Kobayashi
Keyword(s):  
Neural Networks ◽  
Computer Simulations ◽  
Hopfield Neural Network ◽  
Activation Function ◽  
Hopfield Neural Networks ◽  
Noise Tolerance ◽  
Synchronous Mode ◽  
Memory Resources ◽  
Complex Valued ◽  
Projection Rule

A complex-valued Hopfield neural network (CHNN) with a multistate activation function is a multistate model of neural associative memory. The weight parameters need a lot of memory resources. Twin-multistate activation functions were introduced to quaternion- and bicomplex-valued Hopfield neural networks. Since their architectures are much more complicated than that of CHNN, the architecture should be simplified. In this work, the number of weight parameters is reduced by bicomplex projection rule for CHNNs, which is given by the decomposition of bicomplex-valued Hopfield neural networks. Computer simulations support that the noise tolerance of CHNN with a bicomplex projection rule is equal to or even better than that of quaternion- and bicomplex-valued Hopfield neural networks. By computer simulations, we find that the projection rule for hyperbolic-valued Hopfield neural networks in synchronous mode maintains a high noise tolerance.

Storage Capacity of Quaternion-Valued Hopfield Neural Networks with Dual Connections

2021 ◽  
pp. 1-15
Author(s):  
Masaki Kobayashi
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Stochastic Analysis ◽  
Storage Capacity ◽  
Hopfield Neural Network ◽  
Activation Function ◽  
Hopfield Neural Networks ◽  
Noise Tolerance ◽  
Complex Valued ◽  
Projection Rule

Abstract A complex-valued Hopfield neural network (CHNN) is a multistate Hopfield model. A quaternion-valued Hopfield neural network (QHNN) with a twin-multistate activation function was proposed to reduce the number of weight parameters of CHNN. Dual connections (DCs) are introduced to the QHNNs to improve the noise tolerance. The DCs take advantage of the noncommutativity of quaternions and consist of two weights between neurons. A QHNN with DCs provides much better noise tolerance than a CHNN. Although a CHNN and a QHNN with DCs have the samenumber of weight parameters, the storage capacity of projection rule for QHNNs with DCs is half of that for CHNNs and equals that of conventional QHNNs. The small storage capacity of QHNNs with DCs is caused by projection rule, not the architecture. In this work, the ebbian rule is introduced and proved by stochastic analysis that the storage capacity of a QHNN with DCs is 0.8 times as many as that of a CHNN.

Noise Robust Projection Rule for Klein Hopfield Neural Networks

2021 ◽  
pp. 1-19
Author(s):  
Masaki Kobayashi
Keyword(s):  
Neural Networks ◽  
Hopfield Neural Networks ◽  
Rotational Invariance ◽  
Associative Memories ◽  
Noise Tolerance ◽  
Weak Noise ◽  
Noise Robust ◽  
The Stability ◽  
Complex Valued ◽  
Projection Rule

Multistate Hopfield models, such as complex-valued Hopfield neural networks (CHNNs), have been used as multistate neural associative memories. Quaternion-valued Hopfield neural networks (QHNNs) reduce the number of weight parameters of CHNNs. The CHNNs and QHNNs have weak noise tolerance by the inherent property of rotational invariance. Klein Hopfield neural networks (KHNNs) improve the noise tolerance by resolving rotational invariance. However, the KHNNs have another disadvantage of self-feedback, a major factor of deterioration in noise tolerance. In this work, the stability conditions of KHNNs are extended. Moreover, the projection rule for KHNNs is modified using the extended conditions. The proposed projection rule improves the noise tolerance by a reduction in self-feedback. Computer simulations support that the proposed projection rule improves the noise tolerance of KHNNs.

scholarly journals Fast Recall for Complex-Valued Hopfield Neural Networks with Projection Rules

2017 ◽  
Vol 2017 ◽  
pp. 1-6 ◽  
Author(s):  
Masaki Kobayashi
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Computer Simulations ◽  
Hopfield Neural Network ◽  
Hopfield Neural Networks ◽  
Local Minima ◽  
Noise Tolerance ◽  
Multilevel Data ◽  
Complex Valued ◽  
Network Complex

Many models of neural networks have been extended to complex-valued neural networks. A complex-valued Hopfield neural network (CHNN) is a complex-valued version of a Hopfield neural network. Complex-valued neurons can represent multistates, and CHNNs are available for the storage of multilevel data, such as gray-scale images. The CHNNs are often trapped into the local minima, and their noise tolerance is low. Lee improved the noise tolerance of the CHNNs by detecting and exiting the local minima. In the present work, we propose a new recall algorithm that eliminates the local minima. We show that our proposed recall algorithm not only accelerated the recall but also improved the noise tolerance through computer simulations.

scholarly journals Nonlinear Dynamics and Chaos in Fractional-Order Hopfield Neural Networks with Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Xia Huang ◽  
Zhen Wang ◽  
Yuxia Li
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Fractional Order ◽  
Hopfield Neural Network ◽  
Hopfield Neural Networks ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Chaotic Motions ◽  
Dynamical Behaviors ◽  
Nonlinear Dynamics And Chaos

A fractional-order two-neuron Hopfield neural network with delay is proposed based on the classic well-known Hopfield neural networks, and further, the complex dynamical behaviors of such a network are investigated. A great variety of interesting dynamical phenomena, including single-periodic, multiple-periodic, and chaotic motions, are found to exist. The existence of chaotic attractors is verified by the bifurcation diagram and phase portraits as well.

ATTRACTABILITY AND LOCATION OF EQUILIBRIUM POINT OF CELLULAR NEURAL NETWORKS WITH TIME-VARYING DELAYS

2004 ◽  
Vol 14 (05) ◽  
pp. 337-345 ◽  
Author(s):  
ZHIGANG ZENG ◽  
DE-SHUANG HUANG ◽  
ZENGFU WANG
Keyword(s):  
Neural Networks ◽  
Exponential Stability ◽  
Equilibrium Point ◽  
Global Exponential Stability ◽  
Cellular Neural Networks ◽  
Stability Conditions ◽  
Time Varying ◽  
The Stability ◽  
Theoretical Results

This paper presents new theoretical results on global exponential stability of cellular neural networks with time-varying delays. The stability conditions depend on external inputs, connection weights and delays of cellular neural networks. Using these results, global exponential stability of cellular neural networks can be derived, and the estimate for location of equilibrium point can also be obtained. Finally, the simulating results demonstrate the validity and feasibility of our proposed approach.

Average number of fixed points and attractors in Hopfield neural networks

2018 ◽  
Vol 29 (08) ◽  
pp. 1850076
Author(s):  
Jiandu Liu ◽  
Bokui Chen ◽  
Dengcheng Yan ◽  
Lei Wang
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Numerical Calculation ◽  
Fixed Points ◽  
Spin Glass ◽  
Statistical Method ◽  
Hopfield Neural Network ◽  
Hopfield Neural Networks ◽  
Hard Problem ◽  
Np Hard Problem

Calculating the exact number of fixed points and attractors of an arbitrary Hopfield neural network is a non-deterministic polynomial (NP)-hard problem. In this paper, we first calculate the average number of fixed points in such networks versus their size and threshold of neurons, in terms of a statistical method, which has been applied to the calculation of the average number of metastable states in spin glass systems. Then the same method is expanded to study the average number of attractors in such networks. The results of the calculation qualitatively agree well with the numerical calculation. The discrepancies between them are also well explained.

scholarly journals Chaotic Hopfield Neural Network Swarm Optimization and Its Application

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Yanxia Sun ◽  
Zenghui Wang ◽  
Barend Jacobus van Wyk
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Discrete Time ◽  
Optimization Algorithm ◽  
Optimization Problems ◽  
Search Behavior ◽  
Hopfield Neural Network ◽  
Hopfield Neural Networks ◽  
Swarm Optimization ◽  
Continuous State

A new neural network based optimization algorithm is proposed. The presented model is a discrete-time, continuous-state Hopfield neural network and the states of the model are updated synchronously. The proposed algorithm combines the advantages of traditional PSO, chaos and Hopfield neural networks: particles learn from their own experience and the experiences of surrounding particles, their search behavior is ergodic, and convergence of the swarm is guaranteed. The effectiveness of the proposed approach is demonstrated using simulations and typical optimization problems.

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