Image Reconstruction by the Complex-Valued Neural Networks

2009 ◽  
pp. 236-255 ◽  
Author(s):  
Donq-Liang Lee
Keyword(s):  
Neural Networks ◽  
Computer Simulation ◽  
Generalized Inverse ◽  
Initial State ◽  
Inverse Technique ◽  
Projection Geometry ◽  
The Stability ◽  
Complex Valued ◽  
Projection Rule ◽  
Mode A

New design methods for the complex-valued multistate Hopfield associative memories (CVHAMs) are presented. The author of this chapter shows that the well-known projection rule can be generalized to complex domain such that the weight matrix of the CVHAM can be designed by using the generalized inverse technique. The stability of the presented CVHAM is analyzed by using energy function approach which shows that in synchronous update mode a CVHAM is guaranteed to converge to a fixed point from any given initial state. Moreover, the projection geometry of the generalized projection rule is discussed. In order to enhance the recall capability, a strategy of eliminating the spurious memories is reported. Next, a generalized intraconnected bidirectional associative memory (GIBAM) is introduced. A GIBAM is a complex generalization of the intraconnected BAM (IBAM). Lee shows that the design of the GIBAM can also be accomplished by using the generalized inverse technique. Finally, the validity and the performance of the introduced methods are investigated by computer simulation.

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.

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.

scholarly journals Exponential stability for delayed complex-valued neural networks with reaction-diffusion terms

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Xiaohui Xu ◽  
Jibin Yang ◽  
Quan Xu ◽  
Yanhai Xu ◽  
Shulei Sun
Keyword(s):  
Neural Networks ◽  
Exponential Stability ◽  
Reaction Diffusion ◽  
Mixed Delays ◽  
Lyapunov Function Method ◽  
Matrix Properties ◽  
The Stability ◽  
Homeomorphism Mapping ◽  
Complex Valued ◽  
Positive Effect

AbstractIn this study, we investigate reaction-diffusion complex-valued neural networks with mixed delays. The mixed delays include both time-varying and infinite distributed delays. Criteria are derived to ensure the existence, uniqueness, and exponential stability of the equilibrium state of the addressed system on the basis of the M-matrix properties and homeomorphism mapping theories as well as the vector Lyapunov function method. The results demonstrate the positive effect of reaction-diffusion on the stability, which further improves the existing conditions. Finally, the analysis of several examples is compared to the present results to verify the correctness and reduced conservatism of the primary results.

scholarly journals Robust Exponential Stability of Switched Complex-Valued Neural Networks with Interval Parameter Uncertainties and Impulses

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Xiaohui Xu ◽  
Huanbin Xue ◽  
Yiqiang Peng ◽  
Quan Xu ◽  
Jibin Yang
Keyword(s):  
Neural Networks ◽  
Exponential Stability ◽  
Matrix Theory ◽  
Sufficient Conditions ◽  
Average Dwell Time ◽  
Interval Parameter ◽  
Parameter Uncertainties ◽  
Robust Exponential Stability ◽  
The Stability ◽  
Complex Valued

In this paper, dynamic behavior analysis has been discussed for a class of switched complex-valued neural networks with interval parameter uncertainties and impulse disturbance. Sufficient conditions for guaranteeing the existence, uniqueness, and global robust exponential stability of the equilibrium point have been obtained by using the homomorphism mapping theorem, the scalar Lyapunov function method, the average dwell time method, and M-matrix theory. Since there is no result concerning the stability problem of switched neural networks defined in complex number domain, the stability results we describe in this paper generalize the existing ones. The effectiveness of the proposed results is illustrated by a numerical example.

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.

Nonlinear Measure Approach for the Stability Analysis of Complex-Valued Neural Networks

2015 ◽  
Vol 44 (2) ◽  
pp. 539-554 ◽  
Author(s):  
Weiqiang Gong ◽  
Jinling Liang ◽  
Congjun Zhang ◽  
Jinde Cao
Keyword(s):  
Neural Networks ◽  
Stability Analysis ◽  
The Stability ◽  
Complex Valued ◽  
Nonlinear Measure

scholarly journals Complex-Valued Associative Memories with Projection and Iterative Learning Rules

2018 ◽  
Vol 8 (3) ◽  
pp. 237-249 ◽  
Author(s):  
Teijiro Isokawa ◽  
Hiroki Yamamoto ◽  
Haruhiko Nishimura ◽  
Takayuki Yumoto ◽  
Naotake Kamiura ◽  
...  
Keyword(s):  
Neural Network ◽  
Learning Rule ◽  
Hopfield Neural Network ◽  
Iterative Learning ◽  
Associative Memories ◽  
Learning Rules ◽  
The Stability ◽  
Complex Valued ◽  
Projection Rule ◽  
Learning Schemes

AbstractIn this paper, we investigate the stability of patterns embedded as the associative memory distributed on the complex-valued Hopfield neural network, in which the neuron states are encoded by the phase values on a unit circle of complex plane. As learning schemes for embedding patterns onto the network, projection rule and iterative learning rule are formally expanded to the complex-valued case. The retrieval of patterns embedded by iterative learning rule is demonstrated and the stability for embedded patterns is quantitatively investigated.

Sign in / Sign up

Export Citation Format

Share Document