Complex-valued Hopfield Neural Networks Research Articles

A direct analysis method to state bounding for high‐order quaternion Hopfield neural networks with mixed delays and bounded disturbances

The main research in this paper is the state bounding problem of high‐order quaternion Hopfield neural networks (HOQHNNs) with time‐varying discrete delays, unbounded distributed delays, and bounded disturbances. In the research process, firstly, the considered HOQHNNs are decomposed into four equivalent real‐valued Hopfield neural networks (RHNNs) to solve the problem that quaternion multiplication is not commutative, and then, a sufficient condition related to time delays is given to ensure that the state trajectories that are contained in or globally exponentially convergent into a cuboid. Thereby, the corresponding reachable set estimation is obtained. Numerical examples are provided to illustrate validity of theoretical results. At present, the research on HOQHNNs with mixed time delays adopts Lyapunov–Krasovskii functional (LKF) method, which needs to not only choose the appropriate LKF but also take into account both conservatism and computational complexity. Besides, there is no literature to study the state bounding of HOQHNNs with mixed delays. The method suggested in the present text has three merits: (1) According to the definition of state bounding straightway, it has no use any LKF, which avoids massive calculations and solutions of high‐dimensional matrix inequalities; (2) it is available not only to HOQHNNs but also to RHNNs and complex‐valued Hopfield neural networks (CHNNs); and (3) make up for the blank of state bounding research of HOQHNNs.

Noise Robust Projection Rule for Klein Hopfield Neural Networks

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.

Two-Level Complex-Valued Hopfield Neural Networks.

In multistate neural associative memories, some neurons have small noise and the others have large noise. If we know which neurons have small noise, the noise tolerance could be improved. In this brief, we provide a novel method to reinforce neurons with small noise and apply our new method to images with the Gaussian noise. A complex-valued multistate neuron is decomposed to two neurons, referred to as high and low neurons. For the Gaussian noise, the high neurons are expected to have small noise. The noise tolerance is improved by reinforcement of high neurons. The computer simulations support the efficiency of reinforced neurons.

Complex-valued Hopfield neural networks with real weights in synchronous mode

A complex-valued Hopfield neural network with a multistate activation function converges in asynchronous mode, whereas it converges or is trapped at a cycle of length two in synchronous mode. For parallel processing, convergence in synchronous mode is necessary. In particular, a complex-valued Hopfield neural network (CHNN) is the most widely used multistate Hopfield model, and it is desirable that a CHNN converges in synchronous mode. In this work, a real-weight CHNN (RWCHNN) is proposed. An RWCHNN restricts the weights of a CHNN to real numbers and has half weight parameters of a CHNN. We provide the stability conditions of an RWCHNN in synchronous mode and prove that the projection rule satisfies them. The RWCHNN is the first model such that it converges in synchronous mode and has half weight parameters of a CHNN. Computer simulations show that the average update count until convergence is far less than those of the hypercomplex-valued Hopfield neural networks with the same number of weight parameters of an RWCHNN.

Bicomplex Projection Rule for Complex-Valued Hopfield Neural Networks.

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.

Hyperbolic-Valued Hopfield Neural Networks in Synchronous Mode.

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.

Split Quaternion-Valued Twin-Multistate Hopfield Neural Networks

Complex-valued Hopfield neural networks have been extended to 4-dimensional models using quaternions and commutative quaternions. The algebra of split quaternions is another 4-dimensional hypercomplex number system. In this work, a split quaternion-valued Hopfield neural network is defined. By the computer simulations using the twin-multistate activation function and projection rule, we evaluate the noise tolerance.

Hopfield neural networks using Klein four-group

Complex-valued Hopfield neural networks have been extended to several high-dimensional Hopfield neural networks (HDHNNs) using hypercomplex numbers. However, the extensions by hypercomplex numbers are limited. For example, the dimensions of Clifford and Cayley-Dickson algebras are power of two. Group theory provides more various algebras since the orders of groups are any. In this work, Klein four-group is introduced to HDHNNs. It is an abelian group whose order is 4. A Klein Hopfield neural network employs the twin-multistate activation function. We compare the noise tolerance with that of quaternion-valued and commutative quaternion-valued twin-multistate Hopfield neural networks by computer simulations.

Delay independent robust stability analysis of delayed fractional quaternion-valued leaky integrator echo state neural networks with QUAD condition

This paper studies the stability of fractional order (FO) continuous-time quaternion-valued Leaky Integrator Echo State Neural Networks (NN). The delay independent robust stability of NN with QUAD vector field activation functions under time varying delays is derived. The analysis follows the FO Lyapunov theorem while decomposing the NN into four real-valued systems. The existence and uniqueness of the equilibrium point are also verified by means of a contraction map on the decomposed NN. The new approach is tested in the stability analysis of the FO quaternion-valued Echo State NN whitout time delays, complex-valued Echo State and quaternion-/complex-valued Hopfield NN with/without time delays. Two numerical examples demonstrate the feasibility of the proposed method.

O(2) -Valued Hopfield Neural Networks.

In complex-valued Hopfield neural networks (CHNNs), the neuron states are complex numbers whose amplitudes are: 1) they can also be described in special orthogonal matrices of order and 2) here, we propose a new Hopfield model, the O(2) -valued Hopfield neural network [ O(2) -HNN], whose neuron states are extended to orthogonal matrices. Its neuron states are embedded in 4-D space, while those of CHNNs are embedded in 2-D space. Computer simulations were conducted to compare the noise tolerance (NT) and storage capacity (SC) of CHNNs, O(2) -HNNs, and rotor Hopfield neural networks. In terms of SC, O(2) -HNNs outperformed the others, while in NT, they outdid CHNNs.

Global Synchronization in Finite-time of Fractional-order Complexvalued Delayed Hopfield Neural Networks

This paper deals with the synchronization issue of fractional-order complex-valued Hopfield neural networks with time delay. In this paper, by means of properties of the fractional-order inequality, such as Holder inequality and Gronwall inequality, sufficient conditions are presented to guarantee the finite-time synchronization of the fractional-order complex-valued delayed neural networks when 1/2 ≤ γ < 1 and 0 < γ < 1/2. Finally, two numerical simulations are provided to show the effectiveness of the obtained results.

Generalized stability for discontinuous complex-valued Hopfield neural networks via differential inclusions.

Some dynamical behaviours of discontinuous complex-valued Hopfield neural networks are discussed in this paper. First, we introduce a method to construct the complex-valued set-valued mapping and define some basic definitions for discontinuous complex-valued differential equations. In addition, Leray-Schauder alternative theorem is used to analyse the equilibrium existence of the networks. Lastly, we present the dynamical behaviours, including global stability and convergence in measure for discontinuous complex-valued neural networks (CVNNs) via differential inclusions. The main contribution of this paper is that we extend previous studies on continuous CVNNs to discontinuous ones. Several simulations are given to substantiate the correction of the proposed results.

Storage Capacities of Twin-Multistate Quaternion Hopfield Neural Networks.

A twin-multistate quaternion Hopfield neural network (TMQHNN) is a multistate Hopfield model and can store multilevel information, such as image data. Storage capacity is an important problem of Hopfield neural networks. Jankowski et al. approximated the crosstalk terms of complex-valued Hopfield neural networks (CHNNs) by the 2-dimensional normal distributions and evaluated their storage capacities. In this work, we evaluate the storage capacities of TMQHNNs based on their idea.

Hyperbolic Hopfield neural networks with directional multistate activation function

Abstract Complex-valued Hopfield neural networks (CHNNs) have been applied to various fields, although they tend to suffer from low noise tolerance. Rotational invariance, which is an inherent property of CHNNs, reduces noise tolerance. CHNNs have been used in attempts to extend by Clifford algebra, such as hyperbolic and quaternion algebra. In this work, the directional multistate activation function is introduced to hyperbolic Hopfield neural networks (HHNNs), and the stability condition is also given. The proposed models do not have rotational invariance, and have fewer pseudomemories than CHNNs. The projection rule is also introduced to the HHNNs. In general, the diagonal elements should be eliminated from the obtained matrix for noise tolerance, although they cannot be eliminated in HHNNs. We investigate the relation between the diagonal elements and noise tolerance by computer simulations.

Stability and Hopf bifurcation analysis of fractional-order complex-valued neural networks with time delays

This paper considers a class of fractional-order complex-valued Hopfield neural networks (CVHNNs) with time delay for analyzing the dynamic behaviors such as local asymptotic stability and Hopf bifurcation. In the case of a neural network with hub and ring structure, the stability of the equilibrium state is investigated by analyzing the eigenvalue of the corresponding characteristic matrix for the hub and ring structured fractional-order time delay models using a Laplace transformation for the Caputo-fractional derivatives. Some sufficient conditions are established to guarantee the uniqueness of the equilibrium point. In addition, conditions for the occurrence of a Hopf bifurcation are also presented. Finally, numerical examples are given to demonstrate the effectiveness of the derived results.

Quaternionic Hopfield neural networks with twin-multistate activation function

In the present work, a Hopfield neural network with a new quaternionic activation function, referred to as a twin-multistate activation function, is proposed. The multistate activation function has been used in complex-valued Hopfield neural networks (CHNNs). It is useful for representing multilevel information, and the CHNNs with a multistate activation function have been applied to the storage of multilevel data, such as gray-scale images. A twin-multistate activation function consists of two multistate activation functions. Quaternionic Hopfield neural networks (QHNNs) with a twin-multistate activation function can take the place of the CHNNs with a multistate activation function. The QHNNs require half the number of connection parameters of CHNNs. Projection rule is a fast learning algorithm, and is suitable under restricted computational power. However, it requires full-connection, and the sparse connections are not allowed. Projection rule is also available for the QHNNs with a twin-multistate activation function. When the memory resource and computational power are restricted, the QHNNs with a twin-multistate activation function is useful. In the present work, the conventional network topology is only considered. To take advantage of non-commutativity of quaternions, several researchers proposed network topology for QHNNs, such as dual connections. In future, the QHNNs with a twin-multistate activation function will be extended to those with such network topology.

Fixed points of symmetric complex-valued Hopfield neural networks

Abstract A complex-valued Hopfield neural network (CHNN) is a model of a multistate Hopfield neural network, and has been applied to the storage of multilevel data. Weak noise tolerance, however, is a disadvantage of CHNNs. Symmetric CHNNs (SCHNNs), modified CHNNs, improve the noise tolerance of CHNNs. In the present work, we study the global and local minima of SCHNNs with one training pattern. In CHNNs, the global minima are the training and rotated patterns, and there are no local minima. In SCHNNs, it has been hard to determine all the global and local minima. It is thought that the global minima are the training and reversed patterns, and that there are no local minima in most cases. In the present work, however, we find many local minima, and show that they are very weak attractors, which reduce noise tolerance very little. In addition, we determine all the global minima of SCHNNs.

Symmetric quaternionic Hopfield neural networks

Quaternion algebra is an extension of complex number system and has the property of non-commutativity. Quaternionic Hopfield neural networks (QHNNs) are extensions of complex-valued Hopfield neural networks (CHNNs) using quaternions, and several models have been proposed. Both the CHNNs and QHNNs have low noise tolerance due to rotational invariance. Recently, a novel CHNN model, a symmetric CHNN (SCHNN), has been proposed to improve noise tolerance of CHNNs. In the present work, this scheme is extended to the QHNNs. The proposed model is referred to as a symmetric QHNN (SQHNN). We show that the SQHNNs improve the noise tolerance by computer simulations.

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

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.

Pseudomemories of two‐dimensional multistate hopfield neural networks

Complex‐valued Hopfield neural networks (CVHNNs) are available for storage of multilevel data, such as gray‐scale images. Such networks have low noise tolerance. This is a severe problem for their applications. To improve the noise tolerance, we have to study pseudomemories. In the case of one training pattern, CVHNNs have only rotated patterns as pseudomemories. There are many rotated patterns. This is considered the reason why CVHNNs have low noise tolerance. In the present paper, we investigate the pseudomemories of two‐dimensional multistate Hopfield neural networks, including the complex‐valued ones, with multiple training patterns. Computer simulations show that there are many pseudomemories other than the rotated patterns. © 2016 Institute of Electrical Engineers of Japan. Published by John Wiley &amp; Sons, Inc.