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Jinde Cao - One of the best experts on this subject based on the ideXlab platform.
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Novel Inequalities to Global Mittag-Leffler Synchronization and Stability Analysis of Fractional-Order Quaternion-Valued Neural Networks[-.3pc].
IEEE transactions on neural networks and learning systems, 2020Co-Authors: Jianying Xiao, Jinde Cao, Shiping Wen, Jun Cheng, Ruimei Zhang, Shouming ZhongAbstract:This article is concerned with the problem of the global Mittag-Leffler synchronization and stability for fractional-order Quaternion-valued neural networks (FOQVNNs). The systems of FOQVNNs, which contain either general activation functions or linear threshold ones, are successfully established. Meanwhile, two distinct methods, such as separation and nonseparation, have been employed to solve the transformation of the studied systems of FOQVNNs, which dissatisfy the commutativity of Quaternion Multiplication. Moreover, two novel inequalities are deduced based on the general parameters. Compared with the existing inequalities, the new inequalities have their unique superiorities because they can make full use of the additional parameters. Due to the Lyapunov theory, two novel Lyapunov-Krasovskii functionals (LKFs) can be easily constructed. The novelty of LKFs comes from a wider range of parameters, which can be involved in the construction of LKFs. Furthermore, mainly based on the new inequalities and LKFs, more multiple and more flexible criteria are efficiently obtained for the discussed problem. Finally, four numerical examples are given to demonstrate the related effectiveness and availability of the derived criteria.
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Novel fixed-time stabilization of Quaternion-valued BAMNNs with disturbances and time-varying coefficients
AIMS Mathematics, 2020Co-Authors: Ruoyu Wei, Jinde Cao, Jurgen KurthsAbstract:In this paper, with the Quaternion number and time-varying coefficients introduced into traditional BAMNNs, the model of Quaternion-valued BAMNNs are formulated. For the first time, fixed-time stabilization of time-varying Quaternion-valued BAMNNs is investigated. A novel fixedtime control method is adopted, in which the choice of the Lyapunov function is more general than in most previous results. To cope with the noncommutativity of the Quaternion Multiplication, two different fixed-time control methods are provided, a decomposition method and a non-decomposition method. Furthermore, to reduce the control strength and improve control efficiency, an adaptive fixed-time control strategy is proposed. Lastly, numerical examples are presented to demonstrate the effectiveness of the theoretical results.
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Exponential input-to-state stability of Quaternion-valued neural networks with time delay
Applied Mathematics and Computation, 2019Co-Authors: Haibo Bao, Jinde CaoAbstract:Abstract This paper debated the exponential input-to-state stability (EITSS) of the solution for a kind of Quaternion-valued neural networks (QVNNs) with time delay. It fills the blank of QVNN in the aspect of input-to-state stability (ITSS). In virtue of the Quaternion Multiplication is not suitable for commutative law, QVNN is ordinarily resolved into four real-valued neural networks (RVNNs). Making use of a novel Lyapunov–Krasovskii function and some inequalities, we obtain a little sufficient conditions to assure the considered system is EITSS. Finally, by means of two examples, it is certified that the calculation results in this paper are fine.
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Stability Analysis of Quaternion-Valued Neural Networks: Decomposition and Direct Approaches
IEEE transactions on neural networks and learning systems, 2017Co-Authors: Yang Liu, Dandan Zhang, Jungang Lou, Jinde CaoAbstract:In this paper, we investigate the global stability of Quaternion-valued neural networks (QVNNs) with time-varying delays. On one hand, in order to avoid the noncommutativity of Quaternion Multiplication, the QVNN is decomposed into four real-valued systems based on Hamilton rules: $ij=-ji=k,~jk=-kj=i$ , $ki=-ik=j$ , $i^{2}=j^{2}=k^{2}=ijk=-1$ . With the Lyapunov function method, some criteria are, respectively, presented to ensure the global $\mu $ -stability and power stability of the delayed QVNN. On the other hand, by considering the noncommutativity of Quaternion Multiplication and time-varying delays, the QVNN is investigated directly by the techniques of the Lyapunov-Krasovskii functional and the linear matrix inequality (LMI) where Quaternion self-conjugate matrices and Quaternion positive definite matrices are used. Some new sufficient conditions in the form of Quaternion-valued LMI are, respectively, established for the global $\mu $ -stability and exponential stability of the considered QVNN. Besides, some assumptions are presented for the two different methods, which can help to choose Quaternion-valued activation functions. Finally, two numerical examples are given to show the feasibility and the effectiveness of the main results.
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Decomposition approach to the stability of recurrent neural networks with asynchronous time delays in Quaternion field.
Neural networks : the official journal of the International Neural Network Society, 2017Co-Authors: Dandan Zhang, Yang Liu, Kit Ian Kou, Jinde CaoAbstract:Abstract In this paper, the global exponential stability for recurrent neural networks (QVNNs) with asynchronous time delays is investigated in Quaternion field. Due to the non-commutativity of Quaternion Multiplication resulting from Hamilton rules: i j = − j i = k , j k = − k j = i , k i = − i k = j , i j k = i 2 = j 2 = k 2 = − 1 , the QVNN is decomposed into four real-valued systems, which are studied separately. The exponential convergence is proved directly accompanied with the existence and uniqueness of the equilibrium point to the consider systems. Combining with the generalized ∞ -norm and Cauchy convergence property in the Quaternion field, some sufficient conditions to guarantee the stability are established without using any Lyapunov–Krasovskii functional and linear matrix inequality. Finally, a numerical example is given to demonstrate the effectiveness of the results.
Yuan Yan Tang - One of the best experts on this subject based on the ideXlab platform.
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Robust Sparse Representation in Quaternion Space.
IEEE transactions on image processing : a publication of the IEEE Signal Processing Society, 2021Co-Authors: Yulong Wang, Kit Ian Kou, Cuiming Zou, Yuan Yan TangAbstract:Sparse representation has achieved great success across various fields including signal processing, machine learning and computer vision. However, most existing sparse representation methods are confined to the real valued data. This largely limit their applicability to the Quaternion valued data, which has been widely used in numerous applications such as color image processing. Another critical issue is that their performance may be severely hampered due to the data noise or outliers in practice. To tackle the problems above, in this work we propose a robust Quaternion valued sparse representation (RQVSR) method in a fully Quaternion valued setting. To handle the Quaternion noises, we first define a new robust estimator referred as Quaternion Welsch estimator to measure the Quaternion residual error. Compared to the conventional Quaternion mean square error, it can largely suppress the impact of large data corruption and outliers. To implement RQVSR, we have overcome the difficulties raised by the noncommutativity of Quaternion Multiplication and developed an effective algorithm by leveraging the half-quadratic theory and the alternating direction method of multipliers framework. The experimental results show the effectiveness and robustness of the proposed method for Quaternion sparse signal recovery and color image reconstruction.
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Quaternion block sparse representation for signal recovery and classification
Signal Processing, 2021Co-Authors: Cuiming Zou, Kit Ian Kou, Yulong Wang, Yuan Yan TangAbstract:Abstract This paper presents a Quaternion block sparse representation (QBSR) method for structural sparse signal recovery in Quaternion space. Due to the noncommutativity of Quaternion Multiplication, conventional optimization algorithms originally designed for real-valued and complex-valued signal recovery problems are not applicable to the optimization problem of QBSR. To combat this problem, we leverage several Quaternion operators and devise an effective algorithm for QBSR within the ADMM (Alternating Direction Method of Multipliers) framework. The second contribution of this work is to develop a QBSR based classifier referred to as QBSRC for Quaternion data classification with application to color face recognition. Compared with real-valued representation based classifiers handling multiple color channels of a color image independently, QBSRC treats a color image as a Quaternion signal and represents it in a holistic manner. The third contribution is to provide the theoretical analysis of QBSRC and rigorously prove that QBSRC is guaranteed to succeed in classification of any new test sample under appropriate condition. Experiments on both synthetic and real-world datasets demonstrate the effectiveness of the proposed method for Quaternion signal recovery and classification.
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Cauchy greedy algorithm for robust sparse recovery and multiclass classification
Signal Processing, 2019Co-Authors: Yulong Wang, Cuiming Zou, Yuan Yan Tang, Zhaowei ShangAbstract:Abstract Greedy algorithms have attracted considerable interest for sparse signal recovery (SSR) due to their appealing efficiency and performance recently. However, conventional greedy algorithms utilize the l2 norm based loss function and suffer from severe performance degradation in the presence of gross corruption and outliers. Furthermore, they cannot be directly applied to the recovery of Quaternion sparse signals due to the noncommutativity of Quaternion Multiplication. To alleviate these problems, we propose a robust greedy algorithm referred as Cauchy matching pursuit (CauchyMP) for SSR and extend it for Quaternion SSR. By leveraging the Cauchy estimator and generalizing it to the Quaternion space to measure the residual error, our method can robustly recover the sparse signal in both real and Quaternion space from noisy data corrupted by various severe noises and outliers. To tackle the resulting Quaternion optimization problem, we develop an efficient half-quadratic optimization algorithm by introducing two Quaternion operators. In addition, we have also devised a CauchyMP based classifier termed CauchyMPC for robust multiclass classification. The experiments on both synthetic and real-world datasets validate the efficacy and robustness of the proposed methods for SSR, block SSR, Quaternion SSR and multiclass classification.
Qiankun Song - One of the best experts on this subject based on the ideXlab platform.
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Effects of State-Dependent Impulses on Robust Exponential Stability of Quaternion-Valued Neural Networks Under Parametric Uncertainty
IEEE transactions on neural networks and learning systems, 2018Co-Authors: Xujun Yang, Qiankun Song, Junjian HuangAbstract:This paper addresses the state-dependent impulsive effects on robust exponential stability of Quaternion-valued neural networks (QVNNs) with parametric uncertainties. In view of the noncommutativity of Quaternion Multiplication, we have to separate the concerned Quaternion-valued models into four real-valued parts. Then, several assumptions ensuring every solution of the separated state-dependent impulsive neural networks intersects each of the discontinuous surface exactly once are proposed. In the meantime, by applying the $B$ -equivalent method, the addressed state-dependent impulsive models are reduced to fixed-time ones, and the latter can be regarded as the comparative systems of the former. For the subsequent analysis, we proposed a novel norm inequality of block matrix, which can be utilized to analyze the same stability properties of the separated state-dependent impulsive models and the reduced ones efficaciously. Afterward, several sufficient conditions are well presented to guarantee the robust exponential stability of the origin of the considered models; it is worth mentioning that two cases of addressed models are analyzed concretely, that is, models with exponential stable continuous subsystems and destabilizing impulses, and models with unstable continuous subsystems and stabilizing impulses. In addition, an application case corresponding to the stability problem of models with unstable continuous subsystems and stabilizing impulses for state-dependent impulse control to robust exponential synchronization of QVNNs is considered summarily. Finally, some numerical examples are proffered to illustrate the effectiveness and correctness of the obtained results.
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global mittag leffler stability and synchronization analysis of fractional order Quaternion valued neural networks with linear threshold neurons
Neural Networks, 2018Co-Authors: Xujun Yang, Qiankun Song, Jiyang Chen, Junjian HuangAbstract:This paper talks about the stability and synchronization problems of fractional-order Quaternion-valued neural networks (FQVNNs) with linear threshold neurons. On account of the non-commutativity of Quaternion Multiplication resulting from Hamilton rules, the FQVNN models are separated into four real-valued neural network (RVNN) models. Consequently, the dynamic analysis of FQVNNs can be realized by investigating the real-valued ones. Based on the method of M-matrix, the existence and uniqueness of the equilibrium point of the FQVNNs are obtained without detailed proof. Afterwards, several sufficient criteria ensuring the global Mittag-Leffler stability for the unique equilibrium point of the FQVNNs are derived by applying the Lyapunov direct method, the theory of fractional differential equation, the theory of matrix eigenvalue, and some inequality techniques. In the meanwhile, global Mittag-Leffler synchronization for the drive-response models of the addressed FQVNNs are investigated explicitly. Finally, simulation examples are designed to verify the feasibility and availability of the theoretical results.
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Global µ-stability of Quaternion-valued neural networks with mixed time-varying delays
Neurocomputing, 2018Co-Authors: Xingxing You, Qiankun Song, Jing Liang, Yurong Liu, Fuad E. AlsaadiAbstract:Abstract In this paper, the problem of global µ-stability for Quaternion-valued neural networks with time-varying delays and unbounded distributed delays is investigated. To avoid the non-commutativity of Quaternion Multiplication, the Quaternion-valued neural networks is decomposed into two complex-valued systems. By employing the homomorphic mapping principle, a sufficient condition for the existence and uniqueness of equilibrium point of the considered Quaternion-valued neural networks is proposed in the form of linear matrix inequality (LMI) in complex-valued domain. Further, the appropriate Lyapunov–Krasovkii functional is constructed in the Hermitian quadratic form, and sufficient condition to ensure the global µ-stability of the equilibrium point is obtained by using inequality technique. Finally, two numerical examples with simulations are provided to verify the effectiveness of the obtained results.
Dandan Zhang - One of the best experts on this subject based on the ideXlab platform.
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Stability Analysis of Quaternion-Valued Neural Networks: Decomposition and Direct Approaches
IEEE transactions on neural networks and learning systems, 2017Co-Authors: Yang Liu, Dandan Zhang, Jungang Lou, Jinde CaoAbstract:In this paper, we investigate the global stability of Quaternion-valued neural networks (QVNNs) with time-varying delays. On one hand, in order to avoid the noncommutativity of Quaternion Multiplication, the QVNN is decomposed into four real-valued systems based on Hamilton rules: $ij=-ji=k,~jk=-kj=i$ , $ki=-ik=j$ , $i^{2}=j^{2}=k^{2}=ijk=-1$ . With the Lyapunov function method, some criteria are, respectively, presented to ensure the global $\mu $ -stability and power stability of the delayed QVNN. On the other hand, by considering the noncommutativity of Quaternion Multiplication and time-varying delays, the QVNN is investigated directly by the techniques of the Lyapunov-Krasovskii functional and the linear matrix inequality (LMI) where Quaternion self-conjugate matrices and Quaternion positive definite matrices are used. Some new sufficient conditions in the form of Quaternion-valued LMI are, respectively, established for the global $\mu $ -stability and exponential stability of the considered QVNN. Besides, some assumptions are presented for the two different methods, which can help to choose Quaternion-valued activation functions. Finally, two numerical examples are given to show the feasibility and the effectiveness of the main results.
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Decomposition approach to the stability of recurrent neural networks with asynchronous time delays in Quaternion field.
Neural networks : the official journal of the International Neural Network Society, 2017Co-Authors: Dandan Zhang, Yang Liu, Kit Ian Kou, Jinde CaoAbstract:Abstract In this paper, the global exponential stability for recurrent neural networks (QVNNs) with asynchronous time delays is investigated in Quaternion field. Due to the non-commutativity of Quaternion Multiplication resulting from Hamilton rules: i j = − j i = k , j k = − k j = i , k i = − i k = j , i j k = i 2 = j 2 = k 2 = − 1 , the QVNN is decomposed into four real-valued systems, which are studied separately. The exponential convergence is proved directly accompanied with the existence and uniqueness of the equilibrium point to the consider systems. Combining with the generalized ∞ -norm and Cauchy convergence property in the Quaternion field, some sufficient conditions to guarantee the stability are established without using any Lyapunov–Krasovskii functional and linear matrix inequality. Finally, a numerical example is given to demonstrate the effectiveness of the results.
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global µ stability criteria for Quaternion valued neural networks with unbounded time varying delays
Information Sciences, 2016Co-Authors: Dandan Zhang, Jianquan LuAbstract:In this paper, we first propose Quaternion-valued neural networks (QVNNs) with unbounded time-varying delays. Some sufficient conditions on the global µ-stability in the form of both complex-valued and real-valued linear matrix inequalities (LMIs) are provided by solving two difficulties. One is decomposing the QVNN into two complex-valued systems with the plural decomposition method of Quaternion, which can reduce the complexity of calculations by avoiding the non-commutativity of Quaternion Multiplication. The other is choosing the appropriate Lyapunov-Krasovskii functional in the form of Hermitian matrices, which is a big challenge. Finally, two numerical examples are provided to verify the effectiveness of the obtained results.
Fuad E. Alsaadi - One of the best experts on this subject based on the ideXlab platform.
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Global µ-stability of Quaternion-valued neural networks with mixed time-varying delays
Neurocomputing, 2018Co-Authors: Xingxing You, Qiankun Song, Jing Liang, Yurong Liu, Fuad E. AlsaadiAbstract:Abstract In this paper, the problem of global µ-stability for Quaternion-valued neural networks with time-varying delays and unbounded distributed delays is investigated. To avoid the non-commutativity of Quaternion Multiplication, the Quaternion-valued neural networks is decomposed into two complex-valued systems. By employing the homomorphic mapping principle, a sufficient condition for the existence and uniqueness of equilibrium point of the considered Quaternion-valued neural networks is proposed in the form of linear matrix inequality (LMI) in complex-valued domain. Further, the appropriate Lyapunov–Krasovkii functional is constructed in the Hermitian quadratic form, and sufficient condition to ensure the global µ-stability of the equilibrium point is obtained by using inequality technique. Finally, two numerical examples with simulations are provided to verify the effectiveness of the obtained results.
