Generalized Rayleigh Quotients Research Articles

On periodic and compactly supported least energy solutions to semilinear elliptic equations with non-Lipschitz nonlinearity

We discuss the existence and non-existence of periodic in one variable and compactly supported in the other variables least energy solutions for equations with non-Lipschitz nonlinearity of the form: − Δ u = λ u p − u q in R N + 1 , where 0 < q < p < 1 and λ ∈ R. The approach is based on the Nehari manifold method supplemented by a one-sided constraint given through the functional of the suitable Pohozaev identity. The limit value of the parameter λ, where the approach is applicable, corresponds to the existence of periodic in one variable and compactly supported in the other variables least energy solutions. This value is found through the extrem values of nonlinear generalized Rayleigh quotients and the so-called curve of the critical exponents of p, q. Important properties of the solutions are derived for suitable ranges of the parameters, such as that they are not trivial with respect to the periodic variable and do not coincide with compactly supported solutions on the entire space R N + 1 .

Existence of S-shaped type bifurcation curve with dual cusp catastrophe via variational methods

We discuss the existence of multiple positive solutions leading to the occurrence of an S-shaped bifurcation curve to the equations of the form −Δpu=f(u), with p>1. We deal with relatively unexplored cases when f is non-Lipschitz at 0, f(0)=0 and f(u)<0, u∈(0,r), for some r<+∞. Using the nonlinear generalized Rayleigh quotients method we find a range of parameters where the equation may have distinct branches of solutions. As a consequence, applying the variational methods, we prove that the equation has at least three positive solutions with two of them linearly unstable and one linearly stable. The results evidence that the bifurcation curve is S-shaped and exhibits the so-called dual cusp catastrophe. Our results are new even in the one-dimensional case and p=2.

Min-Max Principles with Nonlinear Generalized Rayleigh Quotients for Nonlinear Equations

Min-Max Principles with Nonlinear Generalized Rayleigh Quotients for Nonlinear Equations

Separating solutions of nonlinear problems using nonlinear generalized Rayleigh quotients

Separating solutions of nonlinear problems using nonlinear generalized Rayleigh quotients

Fundamental Frequency Solutions with Prescribed Action Value to Nonlinear Schrödinger Equations

We apply the nonlinear generalized Rayleigh quotients method to develop new tools that can be used to study ground states of nonlinear Schrodinger equations. We introduce a new type variational functional, the global minimizer of which corresponds to the so-called fundamental frequency solutions with a prescribed action value. We find the ground state of the problem and uniquely determine the corresponding values of the mass, frequency, and action level. Based on this approach, we obtain new results on the existence and absence of nonnegative solutions to the zero mass problem.

Rayleigh quotient minimization for absolutely one-homogeneous functionals

In this paper we examine the problem of minimizing generalized Rayleigh quotients of the form , where both J and H are absolutely one-homogeneous functionals. This can be viewed as minimizing J where the solution is constrained to be on a generalized sphere with , where H is any norm or semi-norm. The solution admits a nonlinear eigenvalue problem, based on the subgradients of J and H. We examine several flows which minimize the ratio. This is done both by time-continuous flow formulations and by discrete iterations. We focus on a certain flow, which is easier to analyze theoretically, following the theory of Brezis on flows with maximal monotone operators. A comprehensive theory is established, including convergence of the flow. We then turn into a more specific case of minimizing graph total variation on the L1 sphere, which approximates the Cheeger-cut problem. Experimental results show the applicability of such algorithms for clustering and classification of images.

L1 -Norm Heteroscedastic Discriminant Analysis Under Mixture of Gaussian Distributions.

Fisher's criterion is one of the most popular discriminant criteria for feature extraction. It is defined as the generalized Rayleigh quotient of the between-class scatter distance to the within-class scatter distance. Consequently, Fisher's criterion does not take advantage of the discriminant information in the class covariance differences, and hence, its discriminant ability largely depends on the class mean differences. If the class mean distances are relatively large compared with the within-class scatter distance, Fisher's criterion-based discriminant analysis methods may achieve a good discriminant performance. Otherwise, it may not deliver good results. Moreover, we observe that the between-class distance of Fisher's criterion is based on the l2 -norm, which would be disadvantageous to separate the classes with smaller class mean distances. To overcome the drawback of Fisher's criterion, in this paper, we first derive a new discriminant criterion, expressed as a mixture of absolute generalized Rayleigh quotients, based on a Bayes error upper bound estimation, where mixture of Gaussians is adopted to approximate the real distribution of data samples. Then, the criterion is further modified by replacing l2 -norm with l1 one to better describe the between-class scatter distance, such that it would be more effective to separate the different classes. Moreover, we propose a novel l1 -norm heteroscedastic discriminant analysis method based on the new discriminant analysis (L1-HDA/GM) for heteroscedastic feature extraction, in which the optimization problem of L1-HDA/GM can be efficiently solved by using the eigenvalue decomposition approach. Finally, we conduct extensive experiments on four real data sets and demonstrate that the proposed method achieves much competitive results compared with the state-of-the-art methods.

Efficient algorithms for physical layer security in one-way relay systems

In this paper, we solve two problems aiming at improving the secure communications in a one-way relay system. The system under consideration consists of one source, one destination, N cooperative relays that use amplify and forward, and one eavesdropper. First, we minimize the relays power under information rate constraints for both the legitimate destination and the eavesdropper. When the source power is given, the problem is formulated as a non-convex quadratically constraint quadratic programming, which can be solved using a semidefinite programming (SDP). However, since SDP suffers from high complexity, we propose a novel approach, using generalized eigenvalue, that provides a closed form of the optimal solution in most cases. Then we prove that the relays power can be further decreased as the source power increases. Second, we solve the problem of maximizing the secrecy rate by looking for the optimal relays beamforming vector. We show that the optimization problem of the beamforming vector is a product of two correlated generalized Rayleigh quotients. For this problem, we show that the optimal beamforming vector can be obtained using a series of SDPs, then we significantly simplify the problem by solving it using a series of eigenvalue problem. Numerical results show that the proposed approaches achieve the optimal solution of the relay beamforming vector and enhance the physical layer security with power allocation.

An efficient global optimization algorithm for maximizing the sum of two generalized Rayleigh quotients

Maximizing the sum of two generalized Rayleigh quotients (SRQ) can be reformulated as a one-dimensional optimization problem, where the function value evaluations are reduced to solving semi-definite programming (SDP) subproblems. In this paper, we first use the dual SDP subproblem to construct an explicit overestimation and then propose a branch-and-bound algorithm to globally solve (SRQ). Numerical results demonstrate that it is even more efficient than the recent SDP-based heuristic algorithm.

Minimizing the sum of linear fractional functions over the cone of positive semidefinite matrices: Approximation and applications

The problem of maximizing the sum of two generalized Rayleigh quotients and the total least squares problem with nonsingular Tikhonov regularization are reformulated as a class of sum-of-linear-ratios minimizing over the cone of symmetric positive semidefinite matrices, which is shown to have a Fully Polynomial Time Approximation Scheme.

Matrix positivity preservers in fixed dimension. I

A classical theorem proved in 1942 by I.J. Schoenberg describes all real-valued functions that preserve positivity when applied entrywise to positive semidefinite matrices of arbitrary size; such functions are necessarily analytic with non-negative Taylor coefficients. Despite the great deal of interest generated by this theorem, a characterization of functions preserving positivity for matrices of fixed dimension is not known.In this paper, we provide a complete description of polynomials of degree N that preserve positivity when applied entrywise to matrices of dimension N. This is the key step for us then to obtain negative lower bounds on the coefficients of analytic functions so that these functions preserve positivity in a prescribed dimension. The proof of the main technical inequality is representation theoretic, and employs the theory of Schur polynomials. Interpreted in the context of linear pencils of matrices, our main results provide a closed-form expression for the lowest critical value, revealing at the same time an unexpected spectral discontinuity phenomenon.Tight linear matrix inequalities for Hadamard powers of matrices and a sharp asymptotic bound for the matrix-cube problem involving Hadamard powers are obtained as applications. Positivity preservers are also naturally interpreted as solutions of a variational inequality involving generalized Rayleigh quotients. This optimization approach leads to a novel description of the simultaneous kernels of Hadamard powers, and a family of stratifications of the cone of positive semidefinite matrices.

Generalized Rayleigh quotients and generating vectors

This article is concerned with the following problem: given , of which either A or B is Hermitian, and given , determine a vector such that with , referred to as a generalized Rayleigh quotient. Such a vector w is called a generating vector for z. Our approach builds on the fact that the field of values of a linear pencil can be accurately approximated under the compression into bidimensional linear pencils, in which case the problem has an exact solution. The efficiency of the presented algorithms is discussed and numerical examples are included.

Generalized Rayleigh quotient based innovation covariance testing applied to sensor/actuator fault detection

A new approach based on the generalized Rayleigh quotient for testing the innovation covariance of the Kalman filter is proposed. The optimization process of testing quality is reduced to the classical problem of maximization of the generalized Rayleigh quotient. In the simulations, the longitudinal and lateral dynamics of the F-16 aircraft model are considered, and the detection procedure of sensor/actuator faults, which affect the innovation covariance, is examined. Comparison of the proposed generalized Rayleigh quotients based algorithms for testing the innovation covariance is performed in the sense of the fastest detection of a fault and the detected minimum fault rate. Some recommendations for the fastest detection of the fault are given.

Achievable rate region with secrecy constraints for secure communication in two‐way relay networks

SUMMARYIn this paper, we consider a two‐way relay network consisted of two sources and multiple relays in the presence of an eavesdropper, where the cooperative beamforming strategy is applied to exploit the cooperative diversity to support the secure communication as illustrated in Figure 1. Naturally, we are interested in the beamforming strategy and power allocation to maximize the achievable sum secrecy rate. However, the corresponding problem is equivalent to solve a product of three correlated generalized Rayleigh quotients problem and difficult to solve in general. Because of the openness of wireless medium, the information rate leakage to the eavesdropper cannot be canceled perfectly. To some extent, ‘almost perfect secrecy’, where the rate leakage to the eavesdropper is limited, is more interesting from the practical point of view. In this case, we concern ourself mainly the achievable rate region for general case where the rate at the eavesdropper is regarded as the measurement of secrecy level. Two beamforming approaches, optimal beamforming and null space beamforming, are applied to investigate the achievable rate region with total power constraint and the rate constraint at the eavesdropper, which can be obtained by solving a sequence of the weighted sum inverse‐signal‐to‐noise‐ratio minimization (WSISM) problem. Because of the non‐convexity of WSISM problem, an alternating iteration algorithm is proposed to optimize the relay beamforming vector and two sources' transmit power, where two subproblems need to be solved in each iteration. Meanwhile, we provide the convergence analysis of proposed algorithm. Through the numerical simulations, we verify the effectiveness of proposed algorithm. Copyright © 2013 John Wiley &amp; Sons, Ltd.

Generalized Rayleigh quotient and finite element two-grid discretization schemes

This study discusses generalized Rayleigh quotient and high efficiency finite element discretization schemes. Some results are as follows: 1) Rayleigh quotient accelerate technique is extended to nonselfadjoint problems. Generalized Rayleigh quotients of operator form and weak form are defined and the basic relationship between approximate eigenfunction and its generalized Rayleigh quotient is established. 2) New error estimates are obtained by replacing the ascent of exact eigenvalue with the ascent of finite element approximate eigenvalue. 3) Based on the work of Xu Jinchao and Zhou Aihui, finite element two-grid discretization schemes are established to solve nonselfadjoint elliptic differential operator eigenvalue problems and these schemes are used in both conforming finite element and nonconforming finite element. Besides, the efficiency of the schemes is proved by both theoretical analysis and numerical experiments. 4) Iterated Galerkin method, interpolated correction method and gradient recovery for selfadjoint elliptic differential operator eigenvalue problems are extended to nonselfadjoint elliptic differential operator eigenvalue problems.

Implementation Complexity and Communication Performance Tradeoffs in Discrete Multitone Modulation Equalizers

Several high-speed communication standards modulate encoded data on multiple-carrier frequencies using the fast Fourier transform (FFT). The real part of the quantized inverse FFT samples form a symbol. The symbol is periodically extended by prepending a copy of its last few samples, also known as a cyclic prefix. When the cyclic prefix is longer than the channel order, amplitude and phase distortion can be equalized entirely in the frequency domain. In the receiver, prior to the FFT, a time-domain equalizer, in the form of a finite-impulse response filter, shortens the effective channel impulse response. Alternately, a bank of equalizers tuned to each carrier frequency can be used. In earlier literature, we unified optimal multicarrier equalizer design algorithms as a product of generalized Rayleigh quotients. In this paper, we convert the unified theoretical framework into a framework for fast design algorithms. The relevant literature is reviewed and classified according to this framework. We analyze the achieved bit rate versus implementation complexity (in terms of multiply-and-accumulate operations) tradeoffs in the original and fast design algorithms. The comparison includes multiple implementations of each of 16 different equalizer structures and design algorithms using synthetic and measured discrete multitone modulated data

Unification and evaluation of equalization structures and design algorithms for discrete multitone modulation systems

To ease equalization in a multicarrier system, a cyclic prefix (CP) is typically inserted between successive symbols. When the channel order exceeds the CP length, equalization can be accomplished via a time-domain equalizer (TEQ), which is a finite impulse response (FIR) filter. The TEQ is placed in cascade with the channel to produce an effective shortened impulse response. Alternatively, a bank of equalizers can remove the interference tone-by-tone. This paper presents a unified treatment of equalizer designs for multicarrier receivers, with an emphasis on discrete multitone systems. It is shown that almost all equalizer designs share a common mathematical framework based on the maximization of a product of generalized Rayleigh quotients. This framework is used to give an overview of existing designs (including an extensive literature survey), to apply a unified notation, and to present various common strategies to obtain a solution. Moreover, the unification emphasizes the differences between the methods, enabling a comparison of their advantages and disadvantages. In addition, 16 different equalizer structures and design procedures are compared in terms of computational complexity and achievable bit rate using synthetic and measured data.

Generalized Rayleigh Quotients for Eigenvalue Estimates of Neutron Balance Equations

Generalized Rayleigh quotients are developed to provide estimates of the eigenvalues of the continuous-energy transport equation and its diffusion approximation. The new variational principles extend the applicability of the quotient to perturbations of the boundary as well as the boundary condition of the system. As a result, all three (operator, boundary condition, and external boundary) perturbation types can now be treated simultaneously, and the standard Rayleigh quotient appears as a special case of the variational principles given in this paper. The correctness of the principles are verified by reproducing the first-order perturbation results and considering some numerical examples for the case of boundary perturbation.

Spectral inequalities for generalized Rayleigh quotients

We show that the s-spaces of Carlson and Sa´, with a few natural axioms added, are rich enough to develop unified abstract versions of theorems leading to very general families of inequalities involving eigenvalues of sums of Hermitian matrices, singular values of products of complex matrices, and invariant factors of products of matrices over principal-ideal domains.

Global analysis of Oja's flow for neural networks

A detailed study of Oja's learning equation in neural networks is undertaken in this paper. Not only are such fundamental issues as existence, uniqueness, and representation of solutions completely resolved, but also the convergence issue is resolved. It is shown that the solution of Oja's equation is exponentially convergent to an equilibrium from any initial value. Moreover, the necessary and sufficient conditions are given on the initial value for the solution to converge to a dominant eigenspace of the associated autocorrelation matrix. As a by-product, this result confirms one of Oja's conjectures that the solution converges to the principal eigenspace from almost all initial values. Some other characteristics of the limiting solution are also revealed. These facilitate the determination of the limiting solution in advance using only the initial information. Two examples are analyzed demonstrating the explicit dependence of the limiting solution on the initial value. In another respect, it is found that Oja's equation is the gradient flow of generalized Rayleigh quotients on a Stiefel manifold.