4 papers related to Clifford algebra NN

Note: Communicated by Nek Valous, National Center for Tumor Diseases (NCT), Heidelberg.

Title: Global Exponential Stability of Neutral-Type Octonion-Valued Neural Networks with Time-Varying Delays
Journal: Neurocomputing, Available online 9 May 2018, In Press, Accepted Manuscript
Authors: Călin-Adrian Popa
Link: https://www.sciencedirect.com/science/article/pii/S0925231218305320
Octonion-valued neural networks (OVNNs) are a type of neural networks for which the states and weights are octonions. The octonion algebra is the only normed division algebra that can be defined over the reals, besides the complex and quaternion algebras. Being nonassociative, it clearly doesn’t belong to the Clifford algebras category, which are all associative. In this paper, sufficient conditions for the global exponential stability of neutral-type OVNNs with time-varying delays are formulated, by considering two types of Lipschitz conditions that must be satisfied by the octonion-valued activation functions. To avoid the nonassociativity of the octonions and the noncommutativity of the quaternions, the OVNNs model is decomposed into 4 complex-valued systems, using the Cayley–Dickson construction. By using Lyapunov–Krasovskii functionals with double, triple, and quadruple integral terms, the free weighting matrix method, and simple, double, and triple Jensen inequalities, the stability criteria are formulated in terms of complex-valued linear matrix inequalities. Two numerical examples are provided in order to demonstrate the effectiveness and feasibility of the theoretical results.

Title: Global Mittag-Leffler stability and synchronization analysis of fractional-order quaternion-valued neural networks with linear threshold neurons
Journal: Neural Networks, Volume 105, September 2018, Pages 88-103
Authors: Xujun Yang, Chuandong Li, Qiankun Song, Jiyang Chen, Junjian Huang
Link: https://www.sciencedirect.com/science/article/pii/S0893608018301370
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.

Title: On the Dynamics of Hopfield Neural Networks on Unit Quaternions
Journal: IEEE Transactions on Neural Networks and Learning Systems, Volume: 29, Issue: 6, June 2018
Authors: Marcos Eduardo Valle, Fidelis Zanetti de Castro
Link: https://ieeexplore.ieee.org/document/7920339/
In this paper, we first address the dynamics of the elegant multivalued quaternionic Hopfield neural network (MV-QHNN) proposed by Minemoto et al. Contrary to what was expected, we show that the MV-QHNN, as well as one of its variation, does not always come to rest at an equilibrium state under the usual conditions. In fact, we provide simple examples in which the network yields a periodic sequence of quaternionic state vectors. Afterward, we turn our attention to the continuous-valued quaternionic Hopfield neural network (CV-QHNN), which can be derived from the MV-QHNN by means of a limit process. The CV-QHNN can be implemented more easily than the MV-QHNN model. Furthermore, the asynchronous CV-QHNN always settles down into an equilibrium state under the usual conditions. Theoretical issues are all illustrated by examples in this paper.

Title: The global exponential pseudo almost periodic synchronization of quaternion-valued cellular neural networks with time-varying delays
Journal: Neurocomputing, Volume 303, 16 August 2018, Pages 75-87
Authors: Yongkun Li, Bing Li, Sisheng Yao, Lianglin Xiong
Link: https://www.sciencedirect.com/science/article/pii/S0925231218304788
In this paper, we consider the problem of the pseudo almost periodic synchronization of quaternion-valued cellular neural networks (QVCNNs) with time-varying delays. Firstly, we use the Banach fixed point theorem to obtain the existence of pseudo almost periodic solutions of QVCNNs with time varying delays. Then, by designing a novel and very general nonlinear state-feedback controller, and constructing suitable Lyapunov functions, we obtain that the drive-response structure of QVCNNs with pseudo almost periodic coefficients can realize the global exponential synchronization. Our results are completely new. Finally, a numerical example is given to illustrate the feasibility of our results.

Source: Email from N. Velous, nek.valous_AT_nct-heidelberg.de, 23/05/2018 23:32.


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