Complex-Valued Recurrent Correlation Neural Networks

Event-Triggered Exponential Stabilization for State-Based Switched Inertial Complex-Valued Neural Networks With Multiple Delays

This article explores the exponential stabilization issue of a class of state-based switched inertial complex-valued neural networks with multiple delays via event-triggered control. First, the state-based switched inertial complex-valued neural networks with multiple delays are modeled. Second, by separating the real and imaginary parts of complex values, the state-based switched inertial complex-valued neural networks are transformed into two state-based switched inertial real-valued neural networks. Through the variable substitution method, the model of the second-order inertial neural networks is transformed into a model of the first-order neural networks. Third, an event-triggered controller with the transmission sequence is designed to study the exponential stabilization issue of neural networks constructed above. Then, by constructing the Lyapunov functions and based on some inequalities, we obtain sufficient conditions for exponential stabilization of the proposed neural networks. Furthermore, it is proved that the Zeno phenomenon cannot happen under the designed event-triggered controller. Finally, a simulation example is given to illustrate the correctness of the results.

Quaternion Projection Rule for Rotor Hopfield Neural Networks

A rotor Hopfield neural network (RHNN) is an extension of a complex-valued Hopfield neural network (CHNN) and has excellent noise tolerance. The RHNN decomposition theorem says that an RHNN decomposes into a CHNN and a symmetric CHNN. For a large number of training patterns, the projection rule for RHNNs generates large self-feedbacks, which deteriorates the noise tolerance. To remove self-feedbacks, we propose a projection rule using quaternions based on the decomposition theorem. Using computer simulations, we show that the quaternion projection rule improves noise tolerance.

Quaternion-Valued Twin-Multistate Hopfield Neural Networks With Dual Connections

Dual connections (DCs) utilize the noncommutativity of quaternions and improve the noise tolerance of quaternion Hopfield neural networks (QHNNs). In this article, we introduce DCs to twin-multistate QHNNs. We conduct computer simulations to investigate the noise tolerance. The QHNNs with DCs were weak against an increase in the number of training patterns, but they were robust against increased resolution factor. The simulation results can be explained from the standpoints of storage capacities and rotational invariance.

Event-Triggered Exponential Synchronization for Complex-Valued Memristive Neural Networks With Time-Varying Delays

This article solves the event-triggered exponential synchronization problem for a class of complex-valued memristive neural networks with time-varying delays. The drive-response complex-valued memristive neural networks are translated into two real-valued memristive neural networks through the method of separating the complex-valued memristive neural networks into real and imaginary parts. In order to reduce the information exchange frequency between the sensor and the controller, a novel event-triggered mechanism with the event-triggering functions is introduced in wireless communication networks. Some sufficient conditions are established to achieve the event-triggered exponential synchronization for drive-response complex-valued memristive neural networks with time-varying delays. In addition, to guarantee that the Zeno behavior cannot occur, a positive lower bound for the interevent times is explicitly derived. Finally, numerical simulations are provided to illustrate the effectiveness and superiority of the obtained theoretical results.

Stability Analysis for Memristor-Based Complex-Valued Neural Networks with Time Delays

In this paper, the problem of stability analysis for memristor-based complex-valued neural networks (MCVNNs) with time-varying delays is investigated extensively. This paper focuses on the exponential stability of the MCVNNs with time-varying delays. By means of the Brouwer's fixed-point theorem and M-matrix, the existence, uniqueness, and exponential stability of the equilibrium point for MCVNNs are studied, and several sufficient conditions are obtained. In particular, these results can be applied to general MCVNNs whether the activation functions could be explicitly described by dividing into two parts of the real parts and imaginary parts or not. Two numerical simulation examples are provided to illustrate the effectiveness of the theoretical results.

Complex Gaussian Processes for Regression

In this paper, we propose a novel Bayesian solution for nonlinear regression in complex fields. Previous solutions for kernels methods usually assume a complexification approach, where the real-valued kernel is replaced by a complex-valued one. This approach is limited. Based on the results in complex-valued linear theory and Gaussian random processes, we show that a pseudo-kernel must be included. This is the starting point to develop the new complex-valued formulation for Gaussian process for regression (CGPR). We face the design of the covariance and pseudo-covariance based on a convolution approach and for several scenarios. Just in the particular case where the outputs are proper, the pseudo-kernel cancels. Also, the hyperparameters of the covariance can be learned maximizing the marginal likelihood using Wirtinger's calculus and patterned complex-valued matrix derivatives. In the experiments included, we show how CGPR successfully solves systems where the real and imaginary parts are correlated. Besides, we successfully solve the nonlinear channel equalization problem by developing a recursive solution with basis removal. We report remarkable improvements compared to previous solutions: a 2-4-dB reduction of the mean squared error with just a quarter of the training samples used by previous approaches.

Multiobjective Evolution of Biped Robot Gaits Using Advanced Continuous Ant-Colony Optimized Recurrent Neural Networks

This paper proposes the optimization of a fully connected recurrent neural network (FCRNN) using advanced multiobjective continuous ant colony optimization (AMO-CACO) for the multiobjective gait generation of a biped robot (the NAO). The FCRNN functions as a central pattern generator and is optimized to generate angles of the hip roll and pitch, the knee pitch, and the ankle pitch and roll. The performance of the FCRNN-generated gait is evaluated according to the walking speed, trajectory straightness, oscillations of the body in the pitch and yaw directions, and walking posture, subject to the basic constraints that the robot cannot fall down and must walk forward. This paper formulates this gait generation task as a constrained multiobjective optimization problem and solves this problem through an AMO-CACO-based evolutionary learning approach. The AMO-CACO finds Pareto optimal solutions through ant-path selection and sampling operations by introducing an accumulated rank for the solutions in each single-objective function into solution sorting to improve learning performance. Simulations are conducted to verify the AMO-CACO-based FCRNN gait generation performance through comparisons with different multiobjective optimization algorithms. Selected software-designed Pareto optimal FCRNNs are then applied to control the gait of a real NAO robot.

Singularities of Three-Layered Complex-Valued Neural Networks With Split Activation Function

There are three important concepts related to learning processes in neural networks: reducibility, nonminimality, and singularity. Although the definitions of these three concepts differ, they are equivalent in real-valued neural networks. This is also true of complex-valued neural networks (CVNNs) with hidden neurons not employing biases. The situation of CVNNs with hidden neurons employing biases, however, is very complicated. Exceptional reducibility was found, and it was shown that reducibility and nonminimality are not the same. Irreducibility consists of minimality and exceptional reducibility. The relationship between minimality and singularity has not yet been established. In this paper, we describe our surprising finding that minimality and singularity are independent. We also provide several examples based on exceptional reducibility.

Quantum-Inspired Multidirectional Associative Memory With a Self-Convergent Iterative Learning

Quantum-inspired computing is an emerging research area, which has significantly improved the capabilities of conventional algorithms. In general, quantum-inspired hopfield associative memory (QHAM) has demonstrated quantum information processing in neural structures. This has resulted in an exponential increase in storage capacity while explaining the extensive memory, and it has the potential to illustrate the dynamics of neurons in the human brain when viewed from quantum mechanics perspective although the application of QHAM is limited as an autoassociation. We introduce a quantum-inspired multidirectional associative memory (QMAM) with a one-shot learning model, and QMAM with a self-convergent iterative learning model (IQMAM) based on QHAM in this paper. The self-convergent iterative learning enables the network to progressively develop a resonance state, from inputs to outputs. The simulation experiments demonstrate the advantages of QMAM and IQMAM, especially the stability to recall reliability.

Content-Driven Associative Memories for Color Image Patterns

This paper presents a novel content-driven associative memory (CDAM) to associate large-scale color images based on the subjects that represent the images' content. Compared to traditional associative memories, CDAM inherits their tolerance to random noise in images and possesses greater robustness against correlated noise that distorts an image's spatial contextual structure. A three-layer recurrent neural tensor network (RNTN) is designed as the network model of CDAM. Multiple salient objects detection algorithm and partial radial basis function (PRBF) kernel are proposed for subject determination and content-driven association, respectively. Convergence of the RNTN is analyzed based on the properties of PRBF kernels. Extensive comparative experiment results are provided to verify the CDAM's efficiency, robustness, and accuracy.

A New Local Bipolar Autoassociative Memory Based on External Inputs of Discrete Recurrent Neural Networks With Time Delay

In this paper, local bipolar auto-associative memories are presented based on discrete recurrent neural networks with a class of gain type activation function. The weight parameters of neural networks are acquired by a set of inequalities without the learning procedure. The global exponential stability criteria are established to ensure the accuracy of the restored patterns by considering time delays and external inputs. The proposed methodology is capable of effectively overcoming spurious memory patterns and achieving memory capacity. The effectiveness, robustness, and fault-tolerant capability are validated by simulated experiments.In this paper, local bipolar auto-associative memories are presented based on discrete recurrent neural networks with a class of gain type activation function. The weight parameters of neural networks are acquired by a set of inequalities without the learning procedure. The global exponential stability criteria are established to ensure the accuracy of the restored patterns by considering time delays and external inputs. The proposed methodology is capable of effectively overcoming spurious memory patterns and achieving memory capacity. The effectiveness, robustness, and fault-tolerant capability are validated by simulated experiments.

Symmetric Complex-Valued Hopfield Neural Networks

Complex-valued neural networks, which are extensions of ordinary neural networks, have been studied as interesting models by many researchers. Especially, complex-valued Hopfield neural networks (CHNNs) have been used to process multilevel data, such as gray-scale images. CHNNs with Hermitian connection weights always converge using asynchronous update. The noise tolerance of CHNNs deteriorates extremely as the resolution increases. Noise tolerance is one of the most controversial problems for CHNNs. It is known that rotational invariance reduces noise tolerance. In this brief, we propose symmetric CHNNs (SCHNNs), which have symmetric connection weights. We define their energy function and prove that the SCHNNs always converge. In addition, we show that the SCHNNs improve noise tolerance through computer simulations and explain this improvement from the standpoint of rotational invariance.

Exponential Stability of Complex-Valued Memristive Recurrent Neural Networks

In this brief, we establish a novel complex-valued memristive recurrent neural network (CVMRNN) to study its stability. As a generalization of real-valued memristive neural networks, CVMRNN can be separated into real and imaginary parts. By means of M -matrix and Lyapunov function, the existence, uniqueness, and exponential stability of the equilibrium point for CVMRNNs are investigated, and sufficient conditions are presented. Finally, the effectiveness of obtained results is illustrated by two numerical examples.

Two Fast Complex-Valued Algorithms for Solving Complex Quadratic Programming Problems

In this paper, we propose two fast complex-valued optimization algorithms for solving complex quadratic programming problems: 1) with linear equality constraints and 2) with both an l₁ -norm constraint and linear equality constraints. By using Brandwood's analytic theory, we prove the convergence of the two proposed algorithms under mild assumptions. The two proposed algorithms significantly generalize the existing complex-valued optimization algorithms for solving complex quadratic programming problems with an l₁ -norm constraint only and unconstrained complex quadratic programming problems, respectively. Numerical simulations are presented to show that the two proposed algorithms have a faster speed than conventional real-valued optimization algorithms.

A Complex-Valued Projection Neural Network for Constrained Optimization of Real Functions in Complex Variables

In this paper, we present a complex-valued projection neural network for solving constrained convex optimization problems of real functions with complex variables, as an extension of real-valued projection neural networks. Theoretically, by developing results on complex-valued optimization techniques, we prove that the complex-valued projection neural network is globally stable and convergent to the optimal solution. Obtained results are completely established in the complex domain and thus significantly generalize existing results of the real-valued projection neural networks. Numerical simulations are presented to confirm the obtained results and effectiveness of the proposed complex-valued projection neural network.