Approximate Eigenvalues Research Articles

ASYMMETRY HELPS: EIGENVALUE AND EIGENVECTOR ANALYSES OF ASYMMETRICALLY PERTURBED LOW-RANK MATRICES.

This paper is concerned with the interplay between statistical asymmetry and spectral methods. Suppose we are interested in estimating a rank-1 and symmetric matrix , yet only a randomly perturbed version M is observed. The noise matrix M - M ⋆ is composed of independent (but not necessarily homoscedastic) entries and is, therefore, not symmetric in general. This might arise if, for example, we have two independent samples for each entry of M ⋆ and arrange them in an asymmetric fashion. The aim is to estimate the leading eigenvalue and the leading eigenvector of M ⋆. We demonstrate that the leading eigenvalue of the data matrix M can be times more accurate (up to some log factor) than its (unadjusted) leading singular value of M in eigenvalue estimation. Moreover, the eigen-decomposition approach is fully adaptive to heteroscedasticity of noise, without the need of any prior knowledge about the noise distributions. In a nutshell, this curious phenomenon arises since the statistical asymmetry automatically mitigates the bias of the eigenvalue approach, thus eliminating the need of careful bias correction. Additionally, we develop appealing non-asymptotic eigenvector perturbation bounds; in particular, we are able to bound the perterbation of any linear function of the leading eigenvector of M (e.g. entrywise eigenvector perturbation). We also provide partial theory for the more general rank-r case. The takeaway message is this: arranging the data samples in an asymmetric manner and performing eigen-decomposition could sometimes be quite beneficial.

Delay-coordinate maps, coherence, and approximate spectra of evolution operators

The problem of data-driven identification of coherent observables of measure-preserving, ergodic dynamical systems is studied using kernel integral operator techniques. An approach is proposed whereby complex-valued observables with approximately cyclical behavior are constructed from a pair of eigenfunctions of integral operators built from delay-coordinate mapped data. It is shown that these observables are \(\varepsilon \)-approximate eigenfunctions of the Koopman evolution operator of the system, with a bound \(\varepsilon \) controlled by the length of the delay-embedding window, the evolution time, and appropriate spectral gap parameters. In particular, \( \varepsilon \) can be made arbitrarily small as the embedding window increases so long as the corresponding eigenvalues remain sufficiently isolated in the spectrum of the integral operator. It is also shown that the time-autocorrelation functions of such observables are \(\varepsilon \)-approximate Koopman eigenvalues, exhibiting a well-defined characteristic oscillatory frequency (estimated using the Koopman generator) and a slowly decaying modulating envelope. The results hold for measure-preserving, ergodic dynamical systems of arbitrary spectral character, including mixing systems with continuous spectrum and no non-constant Koopman eigenfunctions in \(L^2\). Numerical examples reveal a coherent observable of the Lorenz 63 system whose autocorrelation function remains above 0.5 in modulus over approximately 10 Lyapunov timescales.

The Lichnerowicz–Obata theorem for the Kohn Laplacian in three dimensions

We prove rigidity for the Lichnerowicz-type eigenvalue estimate for the Kohn Laplacian on strictly pseudoconvex three-manifolds with nonnegative CR Paneitz operator and positive Webster curvature.

An efficient numerical method for estimating eigenvalues and eigenfunctions of fractional Sturm–Liouville problems

In this paper, we construct numerical schemes based on the Lagrange polynomial interpolation to solve Fractional Sturm–Liouville problems (FSLPs) in which the fractional derivatives are considered in the Caputo sense. First, we convert the differential equation with boundary conditions into integral form and discretize the fractional integral to generate a system of algebraic equations in the matrix form. Next, we calculate the set of approximate eigenvalues and corresponding eigenvectors. The eigenfunctions are approximated and some of their properties are investigated. The experimental rate of convergence of numerical calculations for the eigenvalues is reported and the order convergence of the numerical method is obtained. Finally, some examples are presented to illustrate the efficiency and accuracy of the numerical method.

A Note on Spacelike Submanifolds Through Light Cones in Lorentzian Space Forms

We analyse the intrinsic and extrinsic geometry of spacelike submanifolds in light cones $$\Lambda _{c}(p)$$ of de Sitter and anti-de Sitter spacetimes. Every light cone $$\Lambda _{c}(p)$$ contains the lightlike geodesics starting from p and, essentially, it coincides with the horisms $$E^{+}(p)\cup E^{-}(p)$$ . The analysis works by means of an explicit correspondence with the spacelike submanifolds through the light cone in the Lorentz–Minkowski spacetime. In particular, a characterization of totally umbilical compact surfaces through light cones in de Sitter and anti-de Sitter is shown and we obtain an estimation of the first eigenvalue of the Laplace operator on a compact spacelike surface in a light cone.

A note on Lyapunov-type inequalities for fractional boundary value problems with Sturm-Liouville boundary conditions

In this note, we consider fractional Strum-Liouville boundary value problem containing Caputo derivative of order $\alpha$, $ 1<\alpha\leq 2$ with mixed boundary conditions. We establish Cauchy-Schwarz-type inequality to determine a lower bound for the smallest eigenvalues. We give comparison between the smallest eigenvalues and its lower bounds obtained from the Lyapunov-type and Cauchy-Schwarz-type inequalities. Result shows that the Lyapunov-type inequality gives the worse and Cauchy-Schwarz-type inequality gives better lower bound estimates for the smallest eigenvalues. We then use these inequalities to obtain an interval where a linear combination of certain Mittag-Leffler functions have no real zeros.

Fourier Method in Initial Boundary Value Problems for Regions with Curvilinear Boundaries

The algorithm of the generalized Fourier method associated with the use of orthogonal splines is presented on the example of an initial boundary value problem for a region with a curvilinear boundary. It is shown that the sequence of finite Fourier series formed by the method algorithm converges at each moment to the exact solution of the problem – an infinite Fourier series. The structure of these finite Fourier series is similar to that of partial sums of an infinite Fourier series. As the number of grid nodes increases in the area under consideration with a curvilinear boundary, the approximate eigenvalues and eigenfunctions of the boundary value problem converge to the exact eigenvalues and eigenfunctions, and the finite Fourier series approach the exact solution of the initial boundary value problem. The method provides arbitrarily accurate approximate analytical solutions to the problem, similar in structure to the exact solution, and therefore belongs to the group of analytical methods for constructing solutions in the form of orthogonal series. The obtained theoretical results are confirmed by the results of solving a test problem for which both the exact solution and analytical solutions of discrete problems for any number of grid nodes are known. The solution of test problem confirm the findings of the theoretical study of the convergence of the proposed method and the proposed algorithm of the method of separation of variables associated with orthogonal splines, yields the approximate analytical solutions of initial boundary value problem in the form of a finite Fourier series with any desired accuracy. For any number of grid nodes, the method leads to a generalized finite Fourier series which corresponds with high accuracy to the partial sum of the Fourier series of the exact solution of the problem.

Eigenvalue Decomposition of a Parahermitian Matrix: Extraction of Analytic Eigenvalues

An analytic parahermitian matrix admits an eigenvalue decomposition (EVD) with analytic eigenvalues and eigenvectors except in the case of multiplexed data. In this paper, we propose an iterative algorithm for the estimation of the analytic eigenvalues. Since these are generally transcendental, we find a polynomial approximation with a defined error. Our approach operates in the discrete Fourier transform (DFT) domain and for every DFT length generates a maximally smooth association through EVDs evaluated in DFT bins; an outer loop iteratively grows the DFT order and is shown, in general, to converge to the analytic eigenvalues. In simulations, we compare our results to existing approaches.

Ricci curvature and eigenvalue estimates for the magnetic Laplacian on manifolds

In this paper, we present a Lichnerowicz type estimate and (higher order) Buser type estimates for the magnetic Laplacian on a closed Riemannian manifold with a magnetic potential. These results relate eigenvalues, magnetic fields, Ricci curvature, and Cheeger type constants.

Asymptotic approximations for eigenvalues and eigenfunctions of a spectral problem in a thin graph-like junction with a concentrated mass in the node

A spectral problem is considered in a thin 3D graph-like junction that consists of three thin curvilinear cylinders that are joined through a domain (node) of the diameter [Formula: see text] where [Formula: see text] is a small parameter. A concentrated mass with the density [Formula: see text] [Formula: see text] is located in the node. The asymptotic behavior of the eigenvalues and eigenfunctions is studied as [Formula: see text] i.e. when the thin junction is shrunk into a graph. There are five qualitatively different cases in the asymptotic behavior [Formula: see text] of the eigenelements depending on the value of the parameter [Formula: see text] In this paper three cases are considered, namely, [Formula: see text] [Formula: see text] and [Formula: see text] Using multiscale analysis, asymptotic approximations for eigenvalues and eigenfunctions are constructed and justified with a predetermined accuracy with respect to the degree of [Formula: see text] For irrational [Formula: see text] a new kind of asymptotic expansions is introduced. These approximations show how to account the influence of local geometric and physical inhomogeneity of the node in the corresponding limit spectral problem on the graph for different values of the parameter [Formula: see text]

AN EFFICIENT NUMERICAL METHOD BASED ON LEGENDRE-GALERKIN APPROXIMATION FOR THE STEKLOV EIGENVALUE PROBLEM IN SPHERICAL DOMAIN

<abstract> We present in this paper an efficient numerical method based on Legendre-Galerkin approximation for the Steklov eigenvalue problem in spherical domain. Firstly, by means of spherical coordinate transformation and spherical harmonic expansion, the original problem is reduced to a sequence of equivalent one-dimensional eigenvalue problems that can be solved individually in parallel. Through the introduction of the appropriate weighted Sobolev spaces, the weak form and corresponding discrete scheme are established for each one-dimensional eigenvalue problem. Then from the approximate property of orthogonal polynomials in the weighted Sobolev spaces, we prove the error estimates of approximate eigenvalues for each one dimensional eigenvalue problem. Finally, some numerical examples are provided to illustrate the validity of our algorithms. </abstract>

Spurious relationships in high-dimensional systems with strong or mild persistence

This paper is concerned with the interactions of persistence and dimensionality in the context of the eigenvalue estimation problem of large covariance matrices arising in cointegration and principal component analysis. Following a review of the early and more recent developments in this area, we investigate the behaviour of these eigenvalues in a vector autoregression setting that blends pure unit root, local to unit root and mildly integrated components. Our results highlight the seriousness of spurious relationships that may arise in such big data environments even when the degree of persistence of the variables involved is mild and affects only a small proportion of a large data matrix, with important implications for forecasts based on principal component regressions and related methods. We argue that, prior to principal component analysis, first-differencing may be suitable even in stationary or nearly stationary environments.

Efficient fault diagnosis for CORDIC Algorithm by using Hamming Codes

The attractiveness of CORDIC Algorithm lies in the fact that by simple shift-add operations, it can perform several computing tasks such as the calculation of logarithmic, hyperbolic and trigonometric functions, real and complex multiplications, square-root, Eigen value estimation and many others. As a consequence, CORDIC has been used diversely for applications in diverse areas such as communication systems, 3- D graphics, signal and image processing apart from general scientific and technical computation. The digital computation is very much prone to errors; hence the error detection and correction became important for this algorithm. In this work it is achieved by integrating the Hamming codes to the CORDIC algorithm and hence achieving the error-free CORDIC algorithm. The proposed design are benchmarked through Xilinx FPGA and hence the reliability constraints be achieved.

On extremal eigenvalues of the graph Laplacian * *A preprint appears in the PhD thesis of the author [

Upper and lower estimates of eigenvalues of the Laplacian on a metric graph have been established in 2017 by Berkolaiko, Kennedy, Kurasov, and Mugnolo. Both these estimates can be achieved at the same time only by highly degenerate eigenvalues which we call maximally degenerate. By comparison with the maximal eigenvalue multiplicity proved by Kac and Pivovarchik in 2011, we characterize the graphs exhibiting maximally degenerate eigenvalues which are the figure-of-eight graph, the 3-watermelon graph, and the lasso trees—namely, trees decorated with lasso graphs.

Linear stability of transversely modulated finite‐amplitude capillary waves on deep water

AbstractWe investigate the three‐dimensional linear stability of the periodic motion of pure capillary waves progressing in permanent form on water of infinite depth for the whole range of wave amplitudes. After introducing a coordinate transformation based on a conformal map for two‐dimensional steady capillary waves, we perform linear stability analysis of finite‐amplitude capillary waves in the transformed space. To solve the linearized equations for small amplitude disturbances, it is assumed that the wavelengths of the disturbances in the transverse direction are much longer than those in the propagation direction and, therefore, the disturbances are weakly three‐dimensional. This assumption along with the periodicity of solutions allows us to write the linearized equations as an eigenvalue problem in matrix form. Following a perturbation theory for matrices, we expand the solutions of this eigenvalue problem in terms of a small parameter measuring the weak three‐dimensionality, and numerically obtain approximate eigenvalues. For weakly three‐dimensional superharmonic disturbances, the numerical results demonstrate that the pure capillary waves are two‐dimensionally stable, but three‐dimensionally unstable for almost all wave amplitudes. On the other hand, for subharmonic disturbances that are known to be two‐dimensionally unstable, it is found that the long‐wavelength disturbances in the transverse direction reduce the two‐dimensional growth rate near the critical amplitude beyond which the pure capillary waves are unstable.

Estimating Number of Factors by Adjusted Eigenvalues Thresholding

Determining the number of common factors is an important and practical topic in high-dimensional factor models. The existing literature is mainly based on the eigenvalues of the covariance matrix. Owing to the incomparability of the eigenvalues of the covariance matrix caused by the heterogeneous scales of the observed variables, it is not easy to find an accurate relationship between these eigenvalues and the number of common factors. To overcome this limitation, we appeal to the correlation matrix and demonstrate, surprisingly, that the number of eigenvalues greater than 1 of the population correlation matrix is the same as the number of common factors under certain mild conditions. To use such a relationship, we study random matrix theory based on the sample correlation matrix to correct biases in estimating the top eigenvalues and to take into account of estimation errors in eigenvalue estimation. Thus, we propose a tuning-free scale-invariant adjusted correlation thresholding (ACT) method for determining the number of common factors in high-dimensional factor models, taking into account the sampling variabilities and biases of top sample eigenvalues. We also establish the optimality of the proposed ACT method in terms of minimal signal strength and the optimal threshold. Simulation studies lend further support to our proposed method and show that our estimator outperforms competing methods in most test cases. Supplementary materials for this article are available online.

An orthogonally equivariant estimator of the covariance matrix in high dimensions and for small sample sizes

We introduce an estimation method of covariance matrices in a high-dimensional setting, i.e., when the dimension of the matrix, p, is larger than the sample size n. Specifically, we propose an orthogonally equivariant estimator. The eigenvectors of such estimator are the same as those of the sample covariance matrix. The eigenvalue estimates are obtained from an adjusted profile likelihood function derived by approximating the integral of the density function of the sample covariance matrix over its eigenvectors, which is a challenging problem in its own right. Exact solutions to the approximate likelihood equations are obtained and employed to construct estimates that involve a tuning parameter. Bootstrap and cross-validation based algorithms are proposed to choose this tuning parameter under various loss functions. Finally, comparisons with two well-known orthogonally equivariant estimators of the covariance matrix are given, which are based on Monte-Carlo risk estimates for simulated data and misclassification errors in real data analyses. In addition, Monte-Carlo risk estimates are also provided to compare our estimates of eigenvalues to those of a consistent estimator of population eigenvalues.

Non-conforming Crouzeix-Raviart element approximation for Stekloff eigenvalues in inverse scattering

In this paper, we use the non-conforming Crouzeix-Raviart element method to solve a Stekloff eigenvalue problem arising in inverse scattering. The weak formulation corresponding to this problem is non-self-adjoint and indefinite, and its Crouzeix-Raviart element discretization does not meet the condition of the Strang lemma. We use the standard duality technique to prove an extension of the Strang lemma. And we prove the convergence and error estimate of discrete eigenvalues and eigenfunctions using the spectral perturbation theory for compact operators. Finally, we present some numerical examples not only on uniform meshes but also on adaptive refined meshes to show that the Crouzeix-Raviart method is efficient for computing real and complex eigenvalues as expected.

Some limitations on: Analytic approximate eigenvalues by a new technique. Application to sextic anharmonic potentials, D.D. Almeida and P. Martin, Results. in. Phys. 8, 140–145 (2018)

New technique suggested by Almeida and Martin, can be only used to calculate eigenvalues of single well oscillators. As such the method will fail drastically even to compute eigenvalues of single properly non-linear oscillator of the type H=p2+x2+<x2>x4. Further new technique can hardly ever be used to calculate eigenvalues of any double-well oscillator. Without the calculation of wave characteristics mod square |Ψ|2 or oblique way by way of expectation value <A> the stated paper loses its intention as widely usual method in eigenvalue relation.

NESZ Estimation and Calibration for Gaofen-3 Polarimetric Products by the Minimum Noise Envelope Estimator

The Chinese Gaofen-3 satellite currently provides us with an open way to access fully polarimetric data in the C-band frequency. The noise equivalent sigma zero (NESZ) is a crucial factor when calibrating the additive noise in radar imagery. For most radar sensors, NESZ coefficients are stored in header files, but these are not provided for the Gaofen-3 products. The minimum eigenvalue estimator (MEE) and maximum likelihood estimator (MLE) are the two most common techniques used to derive the NESZ from polarimetric imagery. Nevertheless, the bias has been found to be higher than 5 dB compared with the noise measurement circuit (NMC) of the hardware. In this article, we propose a minimum noise envelope estimator (MNEE) for the robust estimation of the Gaofen-3 NESZ. In this article, we carried out an in-depth investigation to analyze the error sources of the MEE and MLE techniques. Based on our analysis, the MNEE framework requires the use of the ocean surface as a reference, and MNEE is combined with the minimum operation to suppress overestimation. In the experimental section, we describe how we validated the proposed algorithm with Radarsat-2 images, and the MNEE is treated as a tool to estimate the NESZ of Gaofen-3 polarimetric products. We found that the Gaofen-3 NESZ is generally less than -20 dB, which satisfies the design specification. The range-dependent NESZ coefficients are provided here to allow convenient noise correction for Gaofen-3 data users.