Singular Value Decomposition Problem Research Articles

On singular value decomposition for split quaternion matrices and applications in split quaternionic mechanics

In the theoretical studies and numerical computations of split quaternionic quantum mechanics, the singular value decomposition theorem of split quaternion matrices plays an essential role. In general the problem of singular value decomposition over split quaternion algebra has hitherto remained tangential for split quaternion matrices. In this paper, by means of a real representation matrix of a split quaternion matrix, the singular value decomposition of split quaternion matrices is studied, the singular value decomposition theorem and the corresponding algorithm for split quaternion matrices are given. Finally, three applications for solving the split quaternion least squares problem and the wave separation problem are given based on the singular value decomposition over split quaternion algebra.

Facilitated low-rank multi-view subspace clustering

Low-rank multi-view subspace clustering has recently attracted increasing attention in the multi-view learning research. Despite significant progress, most existing approaches still suffer from two issues. First, they mostly focus on exploiting the low-rank consistency across multiple views, but often ignore the low-rank structure within each view. Second, they often encounter the expensive time overhead, typically owing to the costs of matrix inversion and singular value decomposition (SVD) at each iteration. In light of this, we propose an efficient and effective approach termed Facilitated Low-rank Multi-view Subspace Clustering (FLMSC) in this paper. In terms of efficiency, our approach factorizes the view-specific representation matrix into two small factor matrices, i.e., an orthogonal dictionary and a latent representation, which mitigates the computation cost of solving SVD problems. In terms of effectiveness, our approach preserves the latent low-rank structure within each view, while at the same time encourages the structural consistency across different views by imposing an agreement term. Based on this, our approach can fully explore the underlying subspace structure of multiple views, so as to better serve for the following spectral clustering. Besides, a facilitated iterative algorithm is developed for the resultant optimization problem, upon which the matrix inversion operation is substantially accelerated during iterations. Experimental results on a variety of multi-view data sets demonstrate the effectiveness and efficiency of our approach. The source code and data sets are available at: https://www.researchgate.net/publication/365349930_FLMSC_v1.

The Impact of Intelligent Accounting Information Management on Corporate Governance Information Transparency

In order to improve the transparency of corporate governance information, this paper combines intelligent accounting information technology to analyze the information transparency in the process of corporate governance, analyzes the fast adaptive neighbor clustering algorithm, and realizes the joint spectral embedding of data by creating block diagonal anchor graphs. Moreover, this paper re-selects anchor points from the spectral embedding data and constructs a new block diagonal anchor graph. In addition, using the principle of adaptive nearest neighbor clustering, this paper performs structured graph learning on the block diagonal anchor graph matrix. After derivation, the Laplace eigenvalue decomposition problem of the whole graph in adaptive nearest neighbor clustering can be transformed into a block matrix singular value decomposition problem. The regression analysis shows that there is a linear correlation between the advanced degree of intelligent accounting information management technology and the transparency of corporate governance information.

Norm-based user selection algorithm for single-RF space modulation techniques with geometric mean decomposition based precoding scheme

Joint user selection algorithm and fully digital geometric mean decomposition (GMD)-based precoding scheme is considered in this paper for single radio frequency (RF) space modulation techniques (SMTs), namely, spatial modulation (SM) and space shift keying (SSK) schemes. The objective is to jointly perform the Frobenius norm-based user selection algorithm and design GMD-based precoded SMTs with single-RF chain in order to reduce the cost and the power consumption in multiple input multiple output (MIMO) systems, and to avoid the complicated bit-allocation problem of singular value decomposition (SVD)-based precoding technique. Based on these schemes, the GMD-based precoding transmission carried out in the context of a single-user SMTs can readily be extended to the multi-user (MU) case. Simulation results demonstrate that single-RF SMTs with GMD-based precoding scheme is capable of outperforming SMTs with SVD-based precoding technique. Meanwhile, MU-SMTs with GMD-based precoding scheme provide significant performance gains over the conventional SM- and SSK-MIMO counterparts and single-user SMTs with GMD-based precoding algorithm, which increase the energy efficiency and the reachability using these schemes. Furthermore, better error performance in MU-SMTs with fully digital GMD-based precoding technique is obtained by selecting any number of users. Therefore, MU-SMTs with GMD-based precoding scheme can be effectively used in various 5G wireless networks.

Independent Subspace Analysis of the Sea Surface Temperature Variability: Non-Gaussian Sources and Sensitivity to Sampling and Dimensionality

We propose an expansion of multivariate time-series data into maximally independent source subspaces. The search is made among rotations of prewhitened data which maximize non-Gaussianity of candidate sources. We use a tensorial invariant approximation of the multivariate negentropy in terms of a linear combination of squared coskewness and cokurtosis. By solving a high-order singular value decomposition problem, we extract the axes associated with most non-Gaussianity. Moreover, an estimate of the Gaussian subspace is provided by the trailing singular vectors. The independent subspaces are obtained through the search of “quasi-independent” components within the estimated non-Gaussian subspace, followed by the identification of groups with significant joint negentropies. Sources result essentially from the coherency of extremes of the data components. The method is then applied to the global sea surface temperature anomalies, equatorward of 65°, after being tested with non-Gaussian surrogates consistent with the data anomalies. The main emerging independent components and subspaces, supposedly generated by independent forcing, include different variability modes, namely, The East-Pacific, the Central Pacific, and the Atlantic Niños, the Atlantic Multidecadal Oscillation, along with the subtropical dipoles in the Indian, South Pacific, and South-Atlantic oceans. Benefits and usefulness of independent subspaces are then discussed.

Polarimetric–Anisotropic Decomposition and Anisotropic Entropies of High-Resolution SAR Images

In the booming era of high-resolution synthetic aperture radar (SAR) technology, SAR advanced information retrieval is critical for effective utilization of huge-volume SAR data. One important aspect of high-resolution SAR interpretation is to explore the anisotropic and dispersive information embedded among subaperture and subband SAR images. This paper formulates the polarimetric subaperture analysis as a singular-value decomposition problem, where polarimetric and anisotropic features can be simultaneously decomposed. The decomposed singular values and left singular vectors are equivalent to eigenanalysis-based polarimetric target decomposition, whereas the right singular vectors give the corresponding anisotropic feature vectors. A physics-based parameterization is proposed for anisotropic patterns, where two anisotropic entropy parameters, namely, compactness and directivity, are proposed. Both simulation results and real SAR image analyses demonstrate that these proposed anisotropic entropies can effectively identify specific types of scatterers depending on their geometric scale, curvature, and form of spatial distribution. The proposed anisotropic entropies could be applied to single- and dual-polarization high-resolution SAR data as well.

Nonlinear tensor product approximation of functions

We are interested in approximation of a multivariate function f(x1,…,xd) by linear combinations of products u1(x1)⋯ud(xd) of univariate functions ui(xi), i=1,…,d. In the case d=2 it is the classical problem of bilinear approximation. In the case of approximation in the L2 space the bilinear approximation problem is closely related to the problem of singular value decomposition (also called Schmidt expansion) of the corresponding integral operator with the kernel f(x1,x2). There are known results on the rate of decay of errors of best bilinear approximation in Lp under different smoothness assumptions on f. The problem of multilinear approximation (nonlinear tensor product approximation) in the case d≥3 is more difficult and much less studied than the bilinear approximation problem. We will present results on best multilinear approximation in Lp under mixed smoothness assumption on f.

IFESIS: Instantaneous frequencies estimation via subspace invariance properties of wavelet structures

According to the proposed method, a set of wavelet transforms of the signal is first obtained, using a structure of Complex Shifted Morlet Wavelets. No specific constraints are imposed on the center frequencies and the bandwidths of the individual wavelets, as well as on the number of wavelets used. In this way, a set of complex signals result in the time domain, equal to the number of the wavelets used. Then, the instantaneous frequencies of the signals are estimated by applying an appropriate subspace algorithm (as for e.g. ESPRIT), to the entire set of the resulting complex wavelet transforms, exploiting the corresponding subspace rotational invariance property of this set of complex signals. Since the method proposes the application of the subspace algorithm after the signal has been appropriately transformed by an appropriate wavelet structure, contrary to the classical subspace methods which are applied to the signal itself, the desired time–frequency features of the signal are enhanced, while simultaneously, the undesired frequency components, as well as the noise, are suppressed. In this way, the method combines the advantages of the Complex Shifted Morlet Wavelets with the advantages of subspace based approaches. Moreover, the method provides a means for estimating the number of the resulting harmonic components, using as a relevant indicator the number of the non-zero singular values of the corresponding singular value decomposition problem. It should be noted that the resulting singular values are also time dependent, providing valuable information on the possibly time variable dynamic structure of the signal.

Tensor Approximation Approach to Calculation of Singular Values and Vectors for SVD Problem

A new method of calculation of singular values and left and right singular vectors of arbitrary nonsquare matrices is proposed. The method allows one to avoid solutions of high rank systems of linear equations of singular value decomposition problems, which makes it not sensitive to ill-conditioness of the decomposed matrix. On the base of the Eckart–Young theorem, it was shown that each second order r-rank tensor can be represented as a sum of the first rank r-order “coordinate” tensors. A new system of equations for “coordinate” tensor generator vectors was obtained. An iterative method of solution of the system was elaborated. Results of the method were compared with classical methods of solutions of singular value decomposition problems.

Theory and Algorithms for Weighted Total Least-Squares Fitting of Lines, Planes, and Parallel Planes to Support Tolerancing Standards

We present the theory and algorithms for fitting a line, a plane, two parallel planes (corresponding to a slot or a slab), or many parallel planes in a total (orthogonal) least-squares sense to coordinate data that is weighted. Each of these problems is reduced to a simple 3 × 3 matrix eigenvalue/eigenvector problem or an equivalent singular value decomposition problem, which can be solved using reliable and readily available commercial software. These methods were numerically verified by comparing them with brute-force minimization searches. We demonstrate the need for such weighted total least-squares fitting in coordinate metrology to support new and emerging tolerancing standards, for instance, ISO 14405-1:2010. The widespread practice of unweighted fitting works well enough when point sampling is controlled and can be made uniform (e.g., using a discrete point contact coordinate measuring machine). However, we show by example that nonuniformly sampled points (arising from many new measurement technologies) coupled with unweighted least-squares fitting can lead to erroneous results. When needed, the algorithms presented also solve the unweighted cases simply by assigning the value one to each weight. We additionally prove convergence from the discrete to continuous cases of least-squares fitting as the point sampling becomes dense.

A DFT-based approximate eigenvalue and singular value decomposition of polynomial matrices

Abstract In this article, we address the problem of singular value decomposition of polynomial matrices and eigenvalue decomposition of para-Hermitian matrices. Discrete Fourier transform enables us to propose a new algorithm based on uniform sampling of polynomial matrices in frequency domain. This formulation of polynomial matrix decomposition allows for controlling spectral properties of the decomposition. We set up a nonlinear quadratic minimization for phase alignment of decomposition at each frequency sample, which leads to a compact order approximation of decomposed matrices. Compact order approximation of decomposed matrices makes it suitable in filterbank and multiple-input multiple-output (MIMO) precoding applications or any application dealing with realization of polynomial matrices as transfer function of MIMO systems. Numerical examples demonstrate the versatility of the proposed algorithm provided by relaxation of paraunitary constraint, and its configurability to select different properties.

BetaDock: Shape-Priority Docking Method Based on Beta-Complex

This paper presents an approach and a software, BetaDock, to the docking problem by putting the priority on shape complementarity between a receptor and a ligand. The approach is based on the theory of the β-complex. Given the Voronoi diagram of the receptor whose topology is stored in the quasi-triangulation, the β-complex corresponding to water molecule is computed. Then, the boundary of the β-complex defines the β-shape which has the complete proximity information among all atoms on the receptor boundary. From the β-shape, we first compute pockets where the ligand may bind. Then, we quickly place the ligand within each pocket by solving the singular value decomposition problem and the assignment problem. Using the conformations of the ligands within the pockets as the initial solutions, we run the genetic algorithm to find the optimal solution for the docking problem. The performance of the proposed algorithm was verified through a benchmark test and showed that BetaDock is superior to a popular docking software AutoDock 4.

SVD for imaging systems with discrete rotational symmetry

The singular value decomposition (SVD) of an imaging system is a computationally intensive calculation for tomographic imaging systems due to the large dimensionality of the system matrix. The computation often involves memory and storage requirements beyond those available to most end users. We have developed a method that reduces the dimension of the SVD problem towards the goal of making the calculation tractable for a standard desktop computer. In the presence of discrete rotational symmetry we show that the dimension of the SVD computation can be reduced by a factor equal to the number of collection angles for the tomographic system. In this paper we present the mathematical theory for our method, validate that our method produces the same results as standard SVD analysis, and finally apply our technique to the sensitivity matrix for a clinical CT system. The ability to compute the full singular value spectra and singular vectors will augment future work in system characterization, image-quality assessment and reconstruction techniques for tomographic imaging systems.

Some properties of LSQR for large sparse linear least squares problems

It is well-known that many Krylov solvers for linear systems, eigenvalue problems, and singular value decomposition problems have very simple and elegant formulas for residual norms. These formulas not only allow us to further understand the methods theoretically but also can be used as cheap stopping criteria without forming approximate solutions and residuals at each step before convergence takes place. LSQR for large sparse linear least squares problems is based on the Lanczos bidiagonalization process and is a Krylov solver. However, there has not yet been an analogously elegant formula for residual norms. This paper derives such kind of formula. In addition, the author gets some other properties of LSQR and its mathematically equivalent CGLS.

Accelerating the reduction to upper Hessenberg, tridiagonal, and bidiagonal forms through hybrid GPU-based computing

We present a Hessenberg reduction (HR) algorithm for hybrid systems of homogeneous multicore with GPU accelerators that can exceed 25× the performance of the corresponding LAPACK algorithm running on current homogeneous multicores. This enormous acceleration is due to proper matching of algorithmic requirements to architectural strengths of the system’s hybrid components. The results described in this paper are significant because the HR has not been properly accelerated before on homogeneous multicore architectures, and it plays a significant role in solving non-symmetric eigenvalue problems. Moreover, the ideas from the hybrid HR are used to develop a hybrid tridiagonal reduction algorithm (for symmetric eigenvalue problems) and a bidiagonal reduction algorithm (for singular value decomposition problems). Our approach demonstrates a methodology that streamlines the development of a large and important class of algorithms on modern computer architectures of multicore and GPUs. The new algorithms can be directly used in the software stack that relies on LAPACK.

A singular value decomposition algorithm based on solving hyperplane constrained nonlinear systems

A new algorithm for singular value decomposition (SVD) is presented through relating SVD problem to nonlinear systems whose solutions are constrained on hyperplanes. The hyperplane constrained nonlinear systems are solved with the help of Newton’s iterative method. It is proved that our SVD algorithm has the quadratic convergence substantially and all singular pairs are computable. These facts are also confirmed by some numerical examples.

Hierarchical taxonomy preparation for text categorization using consistent bipartite spectral graph copartitioning

Multiclass classification has been investigated for many years in the literature. Recently, the scales of real-world multiclass classification applications have become larger and larger. For example, there are hundreds of thousands of categories employed in the Open Directory Project (ODP) and the Yahoo! directory. In such cases, the scalability of classification methods turns out to be a major concern. To tackle this problem, hierarchical classification is proposed and widely adopted to get better trade-off between effectiveness and efficiency. Unfortunately, many data sets are not explicitly organized in hierarchical forms and, therefore, hierarchical classification cannot be used directly. In this paper, we propose a novel algorithm to automatically mine a hierarchical structure from the flat taxonomy of a data corpus as a preparation for the adoption of hierarchical classification. In particular, we first compute matrices to represent the relations among categories, documents, and terms. And, then, we cocluster the three substances at different scales through consistent bipartite spectral graph copartitioning, which is formulated as a generalized singular value decomposition problem. At last, a hierarchical taxonomy is constructed from the category clusters. Our experiments showed that the proposed algorithm could discover very reasonable taxonomy hierarchy and help improve the classification accuracy.

Lifetime Prediction for Fatigue Damage in Bonded Joints

A procedure is outlined and programmed in software in order to predict the fatigue threshold in composite adhesively bonded joints. Two different joint configurations are considered in this study, namely, Double Lap (DL) and Lap Strap (LS). The substrates are made of either uni-directional (UD) or multi-directional (MD) composite laminates and bonded together using an epoxy film adhesive. The theory of continuum damage mechanics has been used to develop damage evolution laws for UD/DL joints at different temperatures. The number of cycles to failure was then expressed in terms of the stresses in the adhesive layer and material constants (damage parameters). The stresses were calculated from non-linear finite element analyses. In order to calculate the damage parameters, a system of equations, in which the number of unknowns is less than the number of equations, has been derived using a combination of experimental data, finite element results and the damage law. The system of equations has been solved using the concept of pseudo-inverse by solving a singular value decomposition problem. Software has been developed to automatically calculate the damage parameters and produce the required load-number of cycle to failure curves. From those curves, the fatigue crack initiation thresholds were predicted for several joints, e.g. MD/DL, UD/LS and MD/LS joints. Good predictions were found.

Missing values in principal component analysis

Calculation schemes for principal component analysis are considered for the case when some matrix elements are missing. Iterative solutions are proposed—either a set of multilinear regression problems or as singular-value decomposition problems with iterative imputation of missing values. If mean values are subtracted from the data matrix, they should also be included in the iteration scheme. Test calculations using Matlab show that the regression approach is somewhat faster than the imputation approach. The results with a substantial amount of missing data are different and superior to those obtained with the naive NIPALS algorithm in common use in chemometrics.

Matrix Methods for Estimating the Coherence Functions from Estimates of the Cross-Spectral Density Matrix

It is shown that the usual method for estimating the coherence functions (ordinary, partial, and multiple) for a general multiple-input! multiple-output problem can be expressed as a modified form of Cholesky decomposition of the cross-spectral density matrix of the input and output records. The results can be equivalently obtained using singular value decomposition (SVD) of the cross-spectral density matrix. Using SVD suggests a new form of fractional coherence. The formulation as a SVD problem also suggests a way to order the inputs when a natural physical order of the inputs is absent.