Real-valued Case Research Articles

Synchronization of fractional complex-valued neural networks with pantograph delays and inhibitory factors

In this paper, the global synchronization of fractional complex-valued neural networks with pantograph delays and inhibitory factors is investigated, and some sufficient conditions are obtained. It deserves to mention that the attained sufficient conditions show that some factors including the pantograph coefficient, coupling connection weights, inhibitory factors, and the order of fractional derivative have an influence on network synchronization. Moreover, when complex-valued neural networks degenerate into real-valued cases and inhibitory factors are not taken into consideration, they are also discussed, and two corollaries are derived. Finally, to validate the feasibility of the theoretical results, a numerical example is presented. Meanwhile, some contrastive numerical results are given to illustrate the relationships among network synchronization, pantograph coefficient, and inhibitory factors.

Identification of Lossy Y-Type Two-Port Circuit Models under Measurement Uncertainties: Closed-Form Solution and Statistical–Perturbative Characterization

The present paper treats a black-box estimation of the three independent parameters of a reciprocal lossy two-port network whose terminals are supposed to be accessible to an impedance measurement device. The discussed estimation method is based on the availability of a number of data pairs made of external load admittances paired to equivalent external admittances affected by measurement errors. The proposed method is framed as a squared-estimation-error minimization problem that leads to a system of three nonlinear equations in the three unknown parameters. A key observation is, however, that a core subsystem of two equations may be turned exactly to a linear form and hence may be solved in closed form. The purely real-valued case is treated first since it serves to clarify the optimization problem at hand and the structure of its solution. In the purely real-valued case, a statistical analysis is carried out as well, which affords the evaluation of the effects of the measurement errors. The results of the statistical analysis afford quantifying the dependence of the estimation errors from the number of samples and from the variance of the measurement errors. Subsequently, the full complex-valued case is treated. Results of numerical simulations complement and illustrate the theoretical findings. The obtained numerical results confirm the statistical analysis and that the proposed external identification method is effective.

Stabilisation of stochastic complex-valued time-varying coupled systems with jump diffusion via aperiodically intermittent control

This paper considers the stabilisation of stochastic complex-valued time-varying coupled systems with jump diffusion by aperiodically intermittent control (AIC). In addition, as a special case of AIC, impulsive control is also used to achieve stabilisation. It is worth mentioning that the coupling strength of coupled systems is time-varying, which increases the difficulty of the research. In addition, the jump diffusion is taken into account in stochastic complex-valued coupled systems, which generalises the existing results about jump diffusion previously since the real-valued cases are mostly considered before. Based on the graph theory and the Lyapunov method, some stability criteria are given, and the sufficient conditions reflect that the realisation of stability is closely related to the strength of stochastic disturbance, coupling strength and uncontrolled rate. Then, our results are applied to stochastic complex-valued coupled oscillators models with time-varying coupling structure. Ultimately, two numerical examples are given to verify the effectiveness of the theoretical results.

New exploration on bifurcation for fractional-order quaternion-valued neural networks involving leakage delays.

During the past decades, many works on Hopf bifurcation of fractional-order neural networks are mainly concerned with real-valued and complex-valued cases. However, few publications involve the quaternion-valued neural networks which is a generalization of real-valued and complex-valued neural networks. In this present study, we explorate the Hopf bifurcation problem for fractional-order quaternion-valued neural networks involving leakage delays. Taking advantage of the Hamilton rule of quaternion algebra, we decompose the addressed fractional-order quaternion-valued delayed neural networks into the equivalent eight real valued networks. Then the delay-inspired bifurcation condition of the eight real valued networks are derived by making use of the stability criterion and bifurcation theory of fractional-order differential dynamical systems. The impact of leakage delay on the bifurcation behavior of the involved fractional-order quaternion-valued delayed neural networks has been revealed. Software simulations are implemented to support the effectiveness of the derived fruits of this study. The research supplements the work of Huang et al. (Neural Netw 117:67-93, 2019).

Bias Adjusted Sign Covariance Matrix

The spatial sign covariance matrix (SSCM), also known as the normalized sample covariance matrix (NSCM), has been widely used in signal processing as a robust alternative to the sample covariance matrix (SCM). It is well-known that the SSCM does not provide consistent estimates of the eigenvalues of the shape matrix (normalized scatter matrix). To alleviate this problem, we propose BASIC (Bias Adjusted SIgn Covariance), which performs an approximate bias correction to the eigenvalues of the SSCM under the assumption that the samples are generated from zero mean unspecified complex elliptically symmetric distributions (the real-valued case is also addressed). We then use the bias correction in order to develop a robust regularized SSCM based estimator, BASIC Shrinkage estimator (BASICS), which is suitable for high dimensional problems, where the dimension can be larger than the sample size. We assess the proposed estimator with several numerical examples as well as in a linear discriminant analysis (LDA) classification problem with real data sets. The simulations show that the proposed estimator compares well to competing robust covariance matrix estimators but has the advantage of being significantly faster to compute.

Stabilization of complex-valued stochastic coupled systems with multiple time delays and regime-switching jump diffusion via periodically intermittent control

This paper is concerned with the stabilization of complex-valued stochastic coupled systems possessing multiple time delays and regime-switching jump diffusion (RSJD) via periodically intermittent control. Moreover, most of the existing results about RSJD are considered in real-valued cases, which is extended to complex-valued cases in this paper. Combining the Lyapunov method with the graph theory, some sufficient conditions can be obtained, which reflect that the realization of stability is related to stochastic disturbance strength, coupling strength and control rate. Additionally, the theoretical results are applied to complex-valued stochastic coupled oscillators with multiple time delays and RSJD. In the end, some simulation results are presented to demonstrate the validity and availability of the theoretical results.

Exponential passive filter design for switched neural networks with time-delay and reaction-diffusion terms

This paper investigates the problem of exponential passive filter design for switched neural networks with time-delay and reaction-diffusion terms. With the aid of a suitable Lyapunov–Krasovskii functional and some inequalities, a linear matrix inequality-based design method is developed that not only makes the filtering error system exponentially stable but also forces it to be passive from external interference to output error. Then, the filter design is extended to the complex-valued case via separating the system into real-valued and complex-valued parts. Finally, a numerical example is utilized to illustrate the effectiveness of the filter design methods for the real-valued and complex-valued cases, respectively.

Phase retrieval of complex-valued objects via a randomized Kaczmarz method

Abstract This paper investigates the convergence of the randomized Kaczmarz algorithm for the problem of phase retrieval of complex-valued objects. Although this algorithm has been studied for the real-valued case in [ 28], its generalization to the complex-valued case is nontrivial and has been left as a conjecture. This paper applies a different approach by establishing the connection between the convergence of the algorithm and the convexity of an objective function. Based on the connection, it demonstrates that when the sensing vectors are sampled uniformly from a unit sphere in ${\mathcal{C}}^n$ and the number of sensing vectors $m$ satisfies $m>O(n\log n)$ as $n, m\rightarrow \infty $, then this algorithm with a good initialization achieves linear convergence to the solution with high probability. The method can be applied to other statistical models of sensing vectors as well. A similar convergence result is established for the unitary model, where the sensing vectors are from the columns of random orthogonal matrices. 2000 Math Subject Classification: 68W20, 68W27, 92D25.

Nano-Crossbar Weighted Memristor-Based Convolution Neural Network Architecture for High-Performance Artificial Intelligence Applications.

Nano memristor crossbar arrays, which can represent analog signals with smaller silicon areas, are popularly used to describe the node weights of the neural networks. The crossbar arrays provide high computational efficiency, as they can perform additions and multiplications at the same time at a cross-point. In this study, we propose a new approach for the memristor crossbar array architecture consisting of multi-weight nano memristors on each cross-point. As the proposed architecture can represent multiple integer-valued weights, it can enhance the precision of the weight coefficients in comparison with the existing memristor-based neural networks. This study presents a Radix-11 nano memristor crossbar array with weighted memristors; it validates the operations of the circuits, which use the arrays through circuit-level simulation. With the proposed Radix-11 approach, it is possible to represent eleven integer-valued weights. In addition, this study presents a neural network designed using the proposed Radix-11 weights, as an example of high-performance AI applications. The neural network implements a speech-keyword detection algorithm, and it was designed on a TensorFlow platform. The implemented keyword detection algorithm can recognize 35 Korean words with an inferencing accuracy of 95.45%, reducing the inferencing accuracy only by 2% when compared to the 97.53% accuracy of the real-valued weight case.

Constant-time energy-normalization for the Phong specular BRDFs

The Phong and Modified Phong specular BRDFs, although of limited physical basis, are nevertheless some of the simplest BRDFs exhibiting glossy and specular qualities to understand and to implement, making them useful for validation and teaching. Unfortunately, although it is well-known how to make these BRDFs conserve energy (that is, never gain energy), making them energy-normalized (that is, never lose nor gain energy) is far more difficult. Lesser-known algorithms exist, but require the specular exponent n to be integer-valued, and have O(n) runtime cost. We express these algorithms as mathematical formulae and generalize to the real-valued specular exponent case. We then simplify and optimize to finally attain an algorithm that is O(1). Energy normalization makes the Phong BRDFs more physically plausible and therefore both more practically and theoretically useful—and our improvements allow for this energy normalization to be done efficiently and without arbitrary limitations.

Location-Free Robust Scale Estimates for Fuzzy Data

In analyzing fuzzy-valued imprecise data statistically, scale measures/estimates play an important role. Scale measures/estimates of data sets are often considered, among others, to descriptively summarize them, to compare the dispersion or the spread of different data sets, standardize data, state rules for detecting outliers, formulate regression objective functions, etc. To be robust, an estimate of scale should have a finite breakdown point close to 50% (i.e., around half data should be replaced by “outliers” to make the estimate break down, either in the sense of exploding to infinity or imploding to zero). In this respect, the median distance deviation about the median (MDD) for fuzzy data sets has already been introduced and its robust behavior has been proved. In contrast to the real-valued case, computation of the MDD for fuzzy data is much more complex and cannot be exactly but approximately performed in general. These computational inconveniences are mainly associated with the fact that, in general, the “median of the fuzzy data set” cannot be exactly calculated, but simply approximated through some levels, and it does not preserve the shape of the fuzzy data. The same happens with the distances between data and the approximate median. Consequently, the use of location-free scale measures would be especially appropriate-to-use in this fuzzy-valued environment. This article aims to extend some robust global scale estimates, and to prove that the extension remains robust. Furthermore, it will be shown that these estimates can be easily and exactly computed for fuzzy trapezoidal data, the assumption of considering trapezoidal data not implying an important loss of generality in the setting of scale estimation.

Long range scattering for the complex-valued Klein-Gordon equation with quadratic nonlinearity in two dimensions

In this paper, we study large time behavior of complex-valued solutions to nonlinear Klein-Gordon equation with a gauge invariant quadratic nonlinearity in two spatial dimensions. To find a possible asymptotic behavior, we consider the final value problem. It turns out that one possible behavior is a linear solution with a logarithmic phase correction as in the real-valued case. However, the shape of the logarithmic correction term has one more parameter which is also given by the final data. In the real case the parameter is constant so one cannot see its effect. However, in the complex case it varies in general. The one dimensional case is also discussed.

Random Phaseless Sampling for Causal Signals in Shift-Invariant Spaces: A Zero Distribution Perspective

We proved that the phaseless sampling (PLS) in the linear-phase modulated shift-invariant space (SIS) $V(e^{\textbf{i}\alpha \cdot}\varphi), \alpha\neq0,$ is impossible even though the real-valued function $\varphi$ enjoys the full spark property (so does $e^{\textbf{i}\alpha \cdot}\varphi$). Stated another way, the PLS in the complex-generated SISs is essentially different from that in the real-generated ones. Motivated by this, we first establish the condition on the complex-valued generator $\phi$ such that the PLS of nonseparable causal (NC) signals in $V(\phi)$ can be achieved by random sampling. The condition is established from the generalized Haar condition (GHC) perspective. Based on the proposed reconstruction approach, it is proved that if the GHC holds then with probability $1$, the random sampling density (SD) $=3$ is sufficient for the PLS of NC signals in the complex-generated SISs. For the real-valued case we also prove that, if the GHC holds then with probability $1$, the random SD $=2$ is sufficient for the PLS of real-valued NC signals in the real-generated SISs. For the local reconstruction of highly oscillatory signals such as chirps, a great number of deterministic samples are required. Compared with deterministic sampling, the proposed random approach enjoys not only the greater sampling flexibility but the much smaller number of samples. To verify our results, numerical simulations were conducted to reconstruct highly oscillatory NC signals in the chirp-modulated SISs.

Performance evaluation of the maximum complex correntropy criterion with adaptive kernel width update

The complex correntropy is a recently defined similarity measure that extends the advantages of conventional correntropy to complex-valued data. As in the real-valued case, the maximum complex correntropy criterion (MCCC) employs a free parameter called kernel width, which affects the convergence rate, robustness, and steady-state performance of the method. However, determining the optimal value for such parameter is not always a trivial task. Within this context, several works have introduced adaptive kernel width algorithms to deal with this free parameter, but such solutions must be updated to manipulate complex-valued data. This work reviews and updates the most recent adaptive kernel width algorithms so that they become capable of dealing with complex-valued data using the complex correntropy. Besides that, a novel gradient-based solution is introduced to the Gaussian kernel and its respective convergence analysis. Simulations compare the performance of adaptive kernel width algorithms with different fixed kernel sizes in an impulsive noise environment. The results show that the iterative kernel adjustment improves the performance of the gradient solution for complex-valued data.

The Perron solution for vector-valued equations

Given a continuous function on the boundary of a bounded open set in $$\mathbb {R}^d$$ there exists a unique bounded harmonic function, called the Perron solution, taking the prescribed boundary values at least at all regular points (in the sense of Wiener) of the boundary. We extend this result to vector-valued functions and consider several methods of constructing the Perron solution which are classical in the real-valued case. Special emphasis is on the case where the codomain is a Banach lattice. In this case we investigate Perron’s classical construction via sub- and supersolutions. We also apply our results to solve elliptic and parabolic boundary value problems of vector-valued functions.

Efficient retrieval of phase information from real-valued electromagnetic field data

While analytic calculations may give access to complex-valued electromagnetic field data which allow trivial access to envelope and phase information, the majority of numeric codes uses a real-valued representation. This typically increases the performance and reduces the memory footprint, albeit at a price: In the real-valued case it is much more difficult to extract envelope and phase information, even more so if counter propagating waves are spatially superposed. A novel method for the analysis of real-valued electromagnetic field data is presented in this paper. We show that, by combining the real-valued electric and magnetic field at a single point in time, we can directly reconstruct the full information of the electromagnetic fields in the form of complex-valued spectral coefficients (k→-space) at a low computational cost of only three Fourier transforms. The method allows for counter propagating plane waves to be accurately distinguished as well as their complex spectral coefficients, i.e. spectral amplitudes and spectral phase to be calculated. From these amplitudes, the complex-valued electromagnetic fields and also the complex-valued vector potential can be calculated from which information about spatiotemporal phase and amplitude is readily available. Additionally, the complex fields allow for efficient vacuum propagation allowing to calculate far field data or boundary input data from near field data. An implementation of the new method is available as part of PostPic1, a data analysis toolkit written in the Python programming language.

Ostrowski and Ostrowski-Gruss type inequalities for vector-valued functions with k points via a parameter

In this paper, we provide an extension of the Ostrowski's inequality for vector-valued functions for $k$ points via a parameter. We also provide a sharp Ostrowski-Gruss type inequality for vector valued functions for $k$ points. Our results generalize some of the results for the real-valued case in the literature.

On Multiple Covariance Equality Testing with Application to SAR Change Detection

This paper deals with the problem of testing the equality of $M$ covariance matrices. We first identify a suitable group of transformations leaving the problem invariant and obtain the corresponding maximal invariant statistic. Then, the Generalized Likelihood Ratio Test (GLRT) is recalled and explicit expressions for Rao, Wald, Gradient, and Durbin tests are provided. Also, equivalences among them and with other well-known tests proposed in open literature (mostly for the real-valued case) are analyzed and compared. Finally, the application of the proposed framework to the relevant signal processing application of multipass Coherent Change Detection (CCD) in polarimetric synthetic aperture radar is demonstrated both on simulated and on live data.

Widely Linear Complex-Valued Kernel Methods for Regression

Usually, complex-valued RKHS are presented as an straightforward application of the real-valued case. In this paper we prove that this procedure yields a limited solution for regression. We show that another kernel, here denoted as pseudo kernel, is needed to learn any function in complex-valued fields. Accordingly, we derive a novel RKHS to include it, the widely RKHS (WRKHS). When the pseudo-kernel cancels, WRKHS reduces to complex-valued RKHS of previous approaches. We address the kernel and pseudo-kernel design, paying attention to the kernel and the pseudo-kernel being complex-valued. In the experiments included we report remarkable improvements in simple scenarios where real a imaginary parts have different similitude relations for given inputs or cases where real and imaginary parts are correlated. In the context of these novel results we revisit the problem of non-linear channel equalization, to show that the WRKHS helps to design more efficient solutions.

A Stochastic Control Verification Theorem for the Dequantized Schrödinger Equation Not Requiring a Duration Restriction

A stochastic control representation for solution of the Schrodinger equation is obtained, utilizing complex-valued diffusion processes. The Maslov dequantization is employed, where the domain is complex-valued in the space variable. The notion of stationarity is utilized to relate the Hamilton–Jacobi form of the dequantized Schrodinger equation to its stochastic control representation. Convexity is not required, and consequently, there is no restriction on the duration of the problem. Additionally, existence is reduced to a real-valued domain case.