Eigenvalue Decomposition Research Articles

Spectral clustering eigenvector selection of hyperspectral image based on the coincidence degree of data distribution

ABSTRACT Spectral clustering is a well-regarded subspace clustering algorithm that exhibits outstanding performance in hyperspectral image classification through eigenvalue decomposition of the Laplacian matrix. However, its classification accuracy is severely limited by the selected eigenvectors, and the commonly used eigenvectors not only fail to guarantee the inclusion of detailed discriminative information, but also have high computational complexity. To address these challenges, we proposed an intuitive eigenvector selection method based on the coincidence degree of data distribution (CDES). First, the clustering result of improved k-means, which can well reflect the spatial distribution of various types was used as the reference map. Then, the adjusted Rand index and adjusted mutual information were calculated to assess the data distribution consistency between each eigenvector and the reference map. Finally, the eigenvectors with high coincidence degrees were selected for clustering. A case study on hyperspectral mineral mapping demonstrated that the mapping accuracies of CDES are approximately 56.3%, 15.5%, and 10.5% higher than those of the commonly used top, high entropy, and high relevance eigenvectors, and CDES can save more than 99% of the eigenvector selection time. Especially, due to the unsupervised nature of k-means, CDES provides a novel solution for autonomous feature selection of hyperspectral images.

Canonical systems whose Weyl coefficients have dominating real part

For a two-dimensional canonical system y′ (t) = zJH(t)y(t) on the half-line (0, ∞) whose Hamiltonian H is a.e. positive semi-definite, denote by qH its Weyl coefficient. De Branges’ inverse spectral theorem states that the assignment H ↦ qH is a bijection between Hamiltonians (suitably normalised) and Nevanlinna functions.The main result of the paper is a criterion when the singular integral of the spectral measure, i.e. Re qH(iy), dominates its Poisson integral Im qH(iy) for y → +∞. Two equivalent conditions characterising this situation are provided. The first one is analytic in nature, very simple, and explicit in terms of the primitive M of H. It merely depends on the relative size of the off-diagonal entries of M compared with the diagonal entries. The second condition is of geometric nature and technically more complicated. It involves the relative size of the off-diagonal entries of H, a measurement for oscillations of the diagonal of H, and a condition on the speed and smoothness of the rotation of H.

Quantum canonical correlation analysis algorithm

Canonical correlation analysis (CCA) is a fundamental technique used to analyze data correlation in various fields, including video and medical data analysis. In this paper, we propose a quantum canonical correlation analysis (QCCA) algorithm. First, we introduce a combined density matrix representation method that transforms CCA into generalized eigenvalue decomposition. Moreover, to address the challenge of performing generalized eigenvalue decomposition in high-dimensional scenarios, we propose a quantum method for extracting the canonical principal axes. In this method, two sets of variables are transformed into a reduced density matrix, so that the product of variable matrices can be accelerated by phase estimation and controlled rotation. Complexity analysis shows that the QCCA algorithm achieves exponential acceleration in variable dimensions n, p and variable size m compared to classical algorithms. The QCCA algorithm serves as a foundation for the subsequent development of quantum algorithms for classification, regression, and other machine learning tasks.

A hybrid method for an inverse scattering problem related to cylindrical bodies buried in a half-space

A hybrid method which is suitable to find numerical solutions for two-dimensional inverse scattering problems connected with cylindrical bodies buried in a half-space with the known orientation is established. The mentioned hybrid method is based on the classical moment method in connection with vertical coordinate while it involves the horizontal coordinate through a spectral expansion. The structure of the spectral expansion which is imposed by the Green's function contributes to improvement of the ill-posedness of the problem along with Tikhonov regularization. By taking this particular structure into consideration one determines the basis functions such that the effect of the ill-posedness is reduced to a minimum. Some illustrative examples show the applicability and accuracy of the method.

An Improved Two-Stage Spherical Harmonic ESPRIT-Type Algorithm

Sensor arrays are gradually becoming a current research hotspot due to their flexible beam control, high signal gain, robustness against extreme interference, and high spatial resolution. Among them, spherical microphone arrays with complex rotational symmetry can capture more sound field information than planar arrays and can convert the collected multiple speech signals into the spherical harmonic domain for processing through spherical modal decomposition. The subspace class direction of arrival (DOA) estimation algorithm is sensitive to noise and reverberation, and its performance can be improved by introducing relative sound pressure and frequency-smoothing techniques. The introduction of the relative sound pressure can increase the difference between the eigenvalues corresponding to the signal subspace and the noise subspace, which is helpful to estimate the number of active sound sources. The eigenbeam estimation of signal parameters via the rotational invariance technique (EB-ESPRIT) is a well-known subspace-based algorithm for a spherical microphone array. The EB-ESPRIT cannot estimate the DOA when the elevation angle approaches 90°. Huang et al. proposed a two-step ESPRIT (TS-ESPRIT) algorithm to solve this problem. The TS-ESPRIT algorithm estimates the elevation and azimuth angles of the signal independently, so there is a problem with DOA parameter pairing. In this paper, the DOA parameter pairing problem of the TS-ESPRIT algorithm is solved by introducing generalized eigenvalue decomposition without increasing the computation of the algorithm. At the same time, the estimation of the elevation angle is given by the arctan function, which increases the estimation accuracy of the elevation angle of the algorithm. The robustness of the algorithm in a noisy environment is also enhanced by introducing the relative sound pressure into the algorithm. Finally, the simulation and field-testing results show that the proposed method not only solves the problem of DOA parameter pairing, but also outperforms the traditional methods in DOA estimation accuracy.

Totally (m, n)-paranormal operators

In this paper, we investigate the class of totally (m, n)-paranormal operators on Hilbert space and prove some additional properties of totally (m, n)-paranormal operators on Hilbert space. Also, we show that Weyl's theorem holds for operators in this class. Moreover, we also show that for totally (m, n)-paranormal operators, the Weyl spectrum satisfies the spectral mapping theorem.

Spectral theorems for generalized Weyl nodes with impurities in a magnetic field

Spectral theorems for generalized Weyl nodes with impurities in a magnetic field

Regularity and stability of wave equations with variable coefficients and Wentzell type boundary conditions

We investigate regularity and stability of wave equations with variable coefficients and the frictional damping, where the damping effect is only on Wentzell boundary and there is no interior damping at all. Based on the spectral approach and the Reimannian geometry method, we present a new way of studying the challenging Wentzell-type problem concerning “the lack of interior damping” and the variable coefficients adiv(A(x)∇u). First, we establish a high-order regularity estimate for the solution of variable coefficient Wentzell problem and a spectral theorem for the operator derived from this problem. And then, after a series of specialized analysis and investigation on the special structure and basic properties of some auxiliary systems, the lower-order term and the resolvent of the operator associated to the variable coefficient Wentzell system by virtue of the Riemannian geometry method and other ideas, we prove eventually that the energies of the system decay with the rate 1/t. As can be seen, even for the constant coefficient case, the decay rate 1/t obtained in this paper is essentially better than the decay rate t−4/5 given in the previous work.

Sampling Gaussian stationary random fields: A stochastic realization approach

Generating large-scale samples of stationary random fields is of great importance in the fields such as geomaterial modeling and uncertainty quantification. Traditional methodologies based on covariance matrix decomposition have the difficulty of being computationally expensive, which is even more serious when the dimension of the random field is large. This paper proposes an efficient stochastic realization approach for sampling Gaussian stationary random fields from a systems and control point of view. Specifically, we take the exponential and squared exponential covariance functions as examples and make a decoupling assumption when there are multiple dimensions. Then a rational spectral density is constructed in each dimension using techniques from covariance extension, and the corresponding autoregressive moving-average (ARMA) model is obtained via spectral factorization. As a result, samples of the random field with a specific covariance function can be generated very efficiently in the space domain by implementing the ARMA recursion using a Gaussian white noise input. Such a procedure is computationally cheap due to the fact that the constructed ARMA model has a low order. Furthermore, the same method is integrated to multiscale simulations where interpolations of the generated samples are achieved when one zooms into finer scales. Both theoretical analysis and simulation results show that our approach performs favorably compared with covariance matrix decomposition methods.

Positive bidiagonal factorization of tetradiagonal Hessenberg matrices

Recently, a spectral Favard theorem was presented for bounded banded lower Hessenberg matrices that possess a positive bidiagonal factorization. The paper establishes conditions, expressed in terms of continued fractions, under which an oscillatory tetradiagonal Hessenberg matrix can have such a positive bidiagonal factorization. Oscillatory tetradiagonal Toeplitz matrices are examined as a case study of matrices that admit a positive bidiagonal factorization. Furthermore, the paper proves that oscillatory banded Hessenberg matrices are organized in rays, where the origin of the ray does not have a positive bidiagonal factorization, but all the interior points of the ray do have such a positive bidiagonal factorization.

On the Smoothness of a Function at the Convergence Point of Its Spectral Expansion Associated with B-Elliptic Operators

On the Smoothness of a Function at the Convergence Point of Its Spectral Expansion Associated with B-Elliptic Operators

Riesz idempotent, spectral mapping theorem and Weyl's theorem for (m,n)*-paranormal operators

Abstract In this paper, we show that the spectral mapping theorem holds for ( m , n ) * {(m,n)^{*}} -paranormal operators. We also exhibit the self-adjointness of the Riesz idempotent E λ {E_{\lambda}} of ( m , n ) * {(m,n)^{*}} -paranormal operators concerning for each isolated point λ of σ ⁢ ( T ) {\sigma(T)} . Moreover, we show Weyl’s theorem for ( m , n ) * {(m,n)^{*}} -paranormal operators and f ⁢ ( T ) {f(T)} for every f ∈ ℋ ⁢ ( σ ⁢ ( T ) ) {f\in\mathcal{H}(\sigma(T))} . Furthermore, we investigate the class of totally ( m , n ) * {(m,n)^{*}} -paranormal operators and show that Weyl’s theorem holds for operators in this class.

Multi-view kernel PCA for time series forecasting

In this paper, we propose a kernel principal component analysis model for multi-variate time series forecasting, where the training and prediction schemes are derived from the multi-view formulation of Restricted Kernel Machines. The training problem is simply an eigenvalue decomposition of the summation of two kernel matrices corresponding to the views of the input and output data. When a linear kernel is used for the output view, it is shown that the forecasting equation takes the form of kernel ridge regression. When that kernel is non-linear, a pre-image problem has to be solved to forecast a point in the input space. We evaluate the model on several standard time series datasets, perform ablation studies, benchmark with closely related models and discuss its results.

Hypergraph partitioning using tensor eigenvalue decomposition.

Hypergraphs have gained increasing attention in the machine learning community lately due to their superiority over graphs in capturing super-dyadic interactions among entities. In this work, we propose a novel approach for the partitioning of k-uniform hypergraphs. Most of the existing methods work by reducing the hypergraph to a graph followed by applying standard graph partitioning algorithms. The reduction step restricts the algorithms to capturing only some weighted pairwise interactions and hence loses essential information about the original hypergraph. We overcome this issue by utilizing tensor-based representation of hypergraphs, which enables us to capture actual super-dyadic interactions. We extend the notion of minimum ratio-cut and normalized-cut from graphs to hypergraphs and show that the relaxed optimization problem can be solved using eigenvalue decomposition of the Laplacian tensor. This novel formulation also enables us to remove a hyperedge completely by using the "hyperedge score" metric proposed by us, unlike the existing reduction approaches. We propose a hypergraph partitioning algorithm inspired from spectral graph theory and also derive a tighter upper bound on the minimum positive eigenvalue of even-order hypergraph Laplacian tensor in terms of its conductance, which is utilized in the partitioning algorithm to approximate the normalized cut. The efficacy of the proposed method is demonstrated numerically on synthetic hypergraphs generated by stochastic block model. We also show improvement for the min-cut solution on 2-uniform hypergraphs (graphs) over the standard spectral partitioning algorithm.

Subspace-based higher-order compressive sensing algorithm for raypath separation in a shallow-water waveguide

In underwater acoustics, compressive sensing was applied to the separation of multiple raypaths based on direction of arrival (DOA) estimation. While it is impossible to separate raypaths directly without eliminating the interference between raypaths. This paper proposes a subspace-based higher-order compressive sensing algorithm, which enables separating raypaths interrupted by colored noise. In the algorithm, a trispectrum matrix is first calculated from the signal received on the reception array. To improve the robustness against noise, the trispectrum matrix is decomposed into signal and noise subspaces using an eigenvalue decomposition. Then, a subspace-based higher-order convex optimization problem is designed based on the signal subspace. The separation of raypaths is cast as the optimization problem solved in a greedy strategy approximation. The results obtained in both simulations and experiments show that the present method not only achieves the more precise separation of raypaths, but it also provides better suppression of noise as compared to the other algorithms.

Exponential integrator for stochastic strongly damped wave equation based on the Wong–Zakai approximation

We consider strong convergence of numerical approximations for a stochastic strongly damped wave equation driven by a class of additive space–time noises characterized by a parameter β∈(−1,2]. With the help of the Wong–Zakai (WZ) approximation to the noise, i.e., replacing the driven noise with its finite spectral expansion truncation, we obtain an approximate equation. Based on the consistency and high regularity of the approximate equation, we develop two exponential integrators for time-stepping discretization and use the spectral Galerkin method in space to develop full-discrete schemes. Performing error estimates in the strong sense, we show that the optimal strong order in time of the proposed WZ-approximation-based exponential Euler scheme is min{1,1+β/2−ϵ}, which is |β|/2 order higher than the order of min{1,1+β} in the existing works when β∈(−1,0). Moreover, we prove that the proposed WZ-approximation-based exponential trapezoidal scheme is of min{3/2,1+β/2−ϵ}-order strong convergence in time when β∈[−1/4,2], so it can break the first order barrier and obtain a higher accurate numerical solution. Numerical examples are performed to verify and develop the theoretical findings.

Properties of Fredholm, Weyl and Jeribi essential S-spectra in a right quaternionic Hilbert space

The paper aims to extend the concept of Fredholm, Weyl and Jeribi essential spectra in the quaternionic setting. Furthermore, some properties and stability of the corresponding spectra of Fredholm and Weyl operators have been investigated in this setting. To achieve the goal, a characterization of the sum of two invariant bounded linear operators has been obtained in order to explore various properties of the Fredholm operator and Weyl operator under some assumptions in quaternionic setting. Also, various sequential properties of the pseudo-resolvent operator, right quaternionic linear operator, Weyl operator, Weyl S-spectrum, Jeribi essential S-spectrum and some properties of [Formula: see text] block operator matrices have been discussed. The spectral mapping theorem of essential S-spectrum, Weyl S-spectrum and Jeribi essential S-spectrum for self-adjoint operators has been established. A characterization of the essential S-spectrum and Weyl S-spectrum of the sum of two bounded linear operators concludes this investigation.

Spectral multiplier theorems for abstract harmonic oscillators on UMD lattices

Spectral multiplier theorems for abstract harmonic oscillators on UMD lattices

Monitoring Surface Subsidence Using Distributed Scatterer InSAR with an Improved Statistically Homogeneous Pixel Selection Method in Coalfield Fire Zones

Statistically homogeneous pixel (SHP) selection is an important process in the distributed scatterer interferometric synthetic aperture radar (DS-InSAR) approach. However, prevalent methods struggle to appropriately balance the efficiency and accuracy of selection. To solve this problem, the authors of this study improved the Hypothesis Test of Confidence Interval (HTCI) to propose an adaptive method to select the level of saliency and confidence interval for the HTCI, called Adp-HTCI. The proposed method can accurately select homogeneous pixels while inheriting the high efficiency of the HTCI. Once homogeneous pixels have been chosen, the eigenvalue decomposition of the covariance matrix is used to optimize their phase and perform temporal processing. We used the proposed method along with data on 31 scenes from the Sentinel-1 satellite from 2 June 2021 to 28 May 2022 to monitor the deformation of the surface of the fire zone in the Sikeshu coalfield. The selection results of homogeneous pixels indicate that the proposed Adp-HTCI algorithm can increase the number of selected homogeneous pixels while ensuring the accuracy of the selection results, thereby enhancing the estimation accuracy and reliability of subsequent parameter solving. The DS-InSAR results showed that the cumulative maximum subsidence in the study area within a year reached—138 mm and the point density used by the DS-InSAR approach was 17.28 times higher than that used by the StaMPS approach. The results of cross-analysis with the results of StaMPS verified the accuracy of the DS-InSAR-based approach.

Analysis of punitive damages in the context of economic law based on the singular value analysis method

Abstract Based on some problems in the punitive damages system that are not conducive to economic development, we take the perspective of economic law to play a positive role in promoting the further improvement of punitive damages. In this paper, the rank of punitive damages is first defined by the singular value analysis algorithm, and the function is set up as a diagonal matrix composed of different vectors, and the square root of the decomposition eigenvalue is the singular value of the matrix. Then the matrix nodes are calculated, and each node is divided into two column blocks and predicted, and the number of nodes is input for the eigenvalue decomposition. The matrix regression coefficients of the independent variables are examined after the decomposition is performed, and the fit of the singular value analysis algorithm in dealing with the compensation problem is derived from this coefficient. The experimental results from a region of the United States show that among 100 large punitive damages cases exceeding $100 million between January 2009 and December 2018, the number of cases of punitive damages in 2010, 2011, and 2013 was only one, and the highest number reached 12 in 2018 illustrating the accuracy of singular value analysis in locating punitive damages, which helps to promote the construction and development of the punitive compensation system and contribute to the stable development of the social economy.