Showing papers on "Matrix decomposition published in 1996"
TL;DR: This paper unify the formulation of these different approaches using transformation theory and an eigenvector analysis of the covariance or coherency matrix of the scattering matrix for target decomposition theory in radar polarimetry.
Abstract: In this paper, we provide a review of the different approaches used for target decomposition theory in radar polarimetry. We classify three main types of theorem; those based on the Mueller matrix and Stokes vector, those using an eigenvector analysis of the covariance or coherency matrix, and those employing coherent decomposition of the scattering matrix. We unify the formulation of these different approaches using transformation theory and an eigenvector analysis. We show how special forms of these decompositions apply for the important case of backscatter from terrain with generic symmetries.
2,369 citations
TL;DR: An approximate minimum degree (AMD) ordering algorithm for preordering a symmetric sparse matrix prior to numerical factorization is presented and produces results that are comparable in quality with the best orderings from other minimum degree algorithms.
Abstract: An approximate minimum degree (AMD) ordering algorithm for preordering a symmetric sparse matrix prior to numerical factorization is presented. We use techniques based on the quotient graph for matrix factorization that allow us to obtain computationally cheap bounds for the minimum degree. We show that these bounds are often equal to the actual degree. The resulting algorithm is typically much faster than previous minimum degree ordering algorithms and produces results that are comparable in quality with the best orderings from other minimum degree algorithms.
779 citations
TL;DR: The 2-D unitary ESPRIT is presented as an algorithm providing the same capabilities for a uniform rectangular array (URA) providing automatically paired source azimuth and elevation angle estimates.
Abstract: The UCA-ESPRIT is a closed-form algorithm developed for use in conjunction with a uniform circular array (UCA) that provides automatically paired source azimuth and elevation angle estimates. The 2-D unitary ESPRIT is presented as an algorithm providing the same capabilities for a uniform rectangular array (URA). In the final stage of the algorithm, the real and imaginary parts of the ith eigenvalue of a matrix are one-to-one related to the respective direction cosines of the ith source relative to the two major array axes. The 2-D unitary ESPRIT offers a number of advantages over other proposed ESPRIT based closed-form 2-D angle estimation techniques. First, except for the final eigenvalue decomposition of a dimension equal to the number of sources, it is efficiently formulated in terms of real-valued computation throughout. Second, it is amenable to efficient beamspace implementations that are presented. Third, it is applicable to array configurations that do not exhibit identical subarrays, e.g., two orthogonal linear arrays. Finally, the 2-D unitary ESPRIT easily handles sources having one member of the spatial frequency coordinate pair in common. Simulation results are presented verifying the efficacy of the method.
538 citations
TL;DR: A multilevel algorithm is presented for analyzing scattering from electrically large surfaces that accelerates the iterative solution of integral equations that arise in computational electromagnetics.
Abstract: A multilevel algorithm is presented for analyzing scattering from electrically large surfaces. The algorithm accelerates the iterative solution of integral equations that arise in computational electromagnetics. The algorithm permits a fast matrix-vector multiplication by decomposing the traditional method of moment matrix into a large number of blocks, with each describing the interaction between distant scatterers. The multiplication of each block by a trial solution vector is executed using a multilevel scheme that resembles a fast Fourier transform (FFT) and that only relies on well-known algebraic techniques. The computational complexity and the memory requirements of the proposed algorithm are O(N log/sup 2/ N).
364 citations
18 Jun 1996
TL;DR: This paper describes a family of factorization-based algorithms that recover 3D projective structure and motion from multiple uncalibrated perspective images of 3D points and lines that can be viewed as generalizations of the Tomasi-Kanade algorithm from affine to fully perspective cameras, and from points to lines.
Abstract: This paper describes a family of factorization-based algorithms that recover 3D projective structure and motion from multiple uncalibrated perspective images of 3D points and lines. They can be viewed as generalizations of the Tomasi-Kanade algorithm from affine to fully perspective cameras, and from points to lines. They make no restrictive assumptions about scene or camera geometry, and unlike most existing reconstruction methods they do not rely on 'privileged' points or images. All of the available image data is used, and each feature in each image is treated uniformly. The key to projective factorization is the recovery of a consistent set of projective depths (scale factors) for the image points: this is done using fundamental matrices and epipoles estimated from the image data. We compare the performance of the new techniques with several existing ones, and also describe an approximate factorization method that gives similar results to SVD-based factorization, but runs much more quickly for large problems.
283 citations
TL;DR: This article proposes an alternative approach to RDA of discriminant analysis in the Gaussian framework, called EDDA, that is based on the reparameterization of the covariance matrix of a group Gk in terms of its eigenvalue decomposition.
Abstract: Friedman proposed a regularization technique (RDA) of discriminant analysis in the Gaussian framework. RDA uses two regularization parameters to design an intermediate classifier between the linear, the quadratic the nearest-means classifiers. In this article we propose an alternative approach, called EDDA, that is based on the reparameterization of the covariance matrix [Σ k ] of a group Gk in terms of its eigenvalue decomposition Σ k = λ k D k A k D k ′, where λk specifies the volume of density contours of Gk, the diagonal matrix of eigenvalues specifies its shape the eigenvectors specify its orientation. Variations on constraints concerning volumes, shapes orientations λ k , A k , and D k lead to 14 discrimination models of interest. For each model, we derived the normal theory maximum likelihood parameter estimates. Our approach consists of selecting a model by minimizing the sample-based estimate of future misclassification risk by cross-validation. Numerical experiments on simulated and rea...
210 citations
TL;DR: In this article, a new approach to solving the magnetic field integral equation (MFIE) for the current induced on a infinite perfectly conducting rough surface is presented, by splitting the propagator matrix into contributions from the left and from the right of the point of observation.
Abstract: A new approach to solving the magnetic field integral equation (MFIE) for the current induced on a infinite perfectly conducting rough surface is presented. By splitting the propagator matrix into contributions from the left and from the right of the point of observation, a second kind integral equation can be formed with a new Born term and a new kernel. Following discretization of this new integral equation, the unknown currents can be determined more rapidly and with significantly less storage requirements than conventional LU decomposition; where the time saving factor is roughly N/3 where N is the number of current samples on the surface and the usual storage requirements associated with matrix inversion are eliminated. While the new Born term is usually adequate for scattered field calculations, the new discretized integral equation can be iterated to amy desired accuracy with no apparent convergence problems. Results are presented for one-dimensional rough surfaces with rms heights exceeding one wavelength and rms slopes exceeding 40° which illustrate the robustness of the new Born term.
199 citations
11 Dec 1996
TL;DR: Using this semidefinite programming approach to FIR filter design with arbitrary upper and lower bounds on the frequency response magnitude, it is shown that the constraints can be expressed as linear matrix inequalities (LMIs), and hence they can be easily handled by interior-point methods.
Abstract: We present a semidefinite programming approach to FIR filter design with arbitrary upper and lower bounds on the frequency response magnitude. It is shown that the constraints can be expressed as linear matrix inequalities (LMIs), and hence they can be easily handled by interior-point methods. Using this LMI formulation, we can cast several interesting filter design problems as convex or quasi-convex optimization problems, e.g. minimizing the length of the FIR filter and computing the Chebychev approximation of a desired power spectrum or a desired frequency response magnitude on a logarithmic scale.
186 citations
TL;DR: Gill and Lewbel recently introduced a test for the rank of a matrix based on the LDU decomposition as mentioned in this paper, which is incorrect except in a very limited problem, and the test can be used to produce a valid test asymptotically equivalent to the minimum-X 2 test.
Abstract: Gill and Lewbel recently introduced a test for the rank of a matrix based on the LDU decomposition. Unfortunately, the asymptotic distribution suggested by them is incorrect except in a very limited problem. In general, the asymptotic distribution is that of a highly complicated nonlinear function of a normally distributed random vector that appears to defy useful characterization. The LDU decomposition can be used to produce a valid test asymptotically equivalent to the minimum-X 2 test.
101 citations
TL;DR: A unified framework for the construction of various synchronous and asynchronous parallel matrix multisplitting iterative methods, suitable to the SIMD and MIMD multiprocessor systems, is presented, and its convergence theory is established under rather weak conditions.
Abstract: A unified framework for the construction of various synchronous and asynchronous parallel matrix multisplitting iterative methods, suitable to the SIMD and MIMD multiprocessor systems, respectively, is presented, and its convergence theory is established under rather weak conditions. These afford general method models and systematical convergence criterions for studying the parallel iterations in the sense of matrix multisplitting. In addition, how the known parallel matrix multisplitting iterative methods can be classified into this new framework, and what novel ones can be generated by it are shown in detail.
95 citations
TL;DR: This work studied the ‘column oriented’ or ‘left looking’ sparse variant of the Cholesky decomposition, which is a very popular method in large scale optimization and shows some techniques such as using supernodes and loop unrolling for improving the speed of computation.
Abstract: Every iteration of an interior point method of large scale linear programming requires computing at least one orthogonal projection. In practice, Cholesky decomposition seems to be the most efficient and sufficiently stable method. We studied the ‘column oriented’ or ‘left looking’ sparse variant of the Cholesky decomposition, which is a very popular method in large scale optimization. We show some techniques such as using supernodes and loop unrolling for improving the speed of computation. We show numerical results on a wide variety of large scale, real-life linear programming problems.
TL;DR: The problem of expressing a given nonlinear state-space system as the cascade connection of a lossless system and a stable, minimum-phase system (inner-outer factorization) is solved for the case of a stable system having state- space equations affine in the inputs.
Abstract: The problem of expressing a given nonlinear state-space system as the cascade connection of a lossless system and a stable, minimum-phase system (inner-outer factorization) is solved for the case of a stable system having state-space equations affine in the inputs. The solution is given in terms of the stabilizing solution of a certain Hamilton-Jacobi equation. The stable, minimum-phase factor is obtained as the solution of an associated nonlinear spectral factorization problem. As an application, one can arrive at the solution of the nonlinear H/sub /spl infin//-control problem for the disturbance feedforward case.
TL;DR: A fivefold speedup can be achieved over the fastest alternative approach over the basic SVD process when 3-D and 4-D extensions of the CORDIC algorithm for plane rotations are used.
Abstract: The singular value decomposition (SVD) of complex matrices is computed in a highly parallel fashion on a square array of processors using Kogbetliantz's analog of Jacobi's eigenvalue decomposition method. To gain further speed, new algorithms for the basic SVD operations are proposed and their implementation as specialized processors is presented. The algorithms are 3-D and 4-D extensions of the CORDIC algorithm for plane rotations. When these extensions are used in concert with an additive decomposition of 2/spl times/2 complex matrices, which enhances parallelism, and with low resolution rotations early on in the SVD process, which reduce operation count, a fivefold speedup can be achieved over the fastest alternative approach.
TL;DR: Based on structural VARs, a spectral decomposition which allows to infer the effects of changes in one variable on the other variables in the frequency domain has been proposed in this paper, where it is shown that there is a close relationship between this concept and conventional forecast error variance decomposition techniques for VAR.
Abstract: Based on structural VARs, this paper proposes a spectral decomposition which allows to infer the effects of changes in one variable on the other variables in the frequency domain. It is shown that there is a close relationship between this concept and conventional forecast error variance decomposition techniques for VARs. An empirical example demonstrates the usefulness of this additional tool in analyzing the relationships among time series.
24 Jun 1996
TL;DR: In this article, a higher-order-only technique for blind source separation is presented, where the identification problem is approached in a (linear and multilinear) algebraic framework: the solution can be obtained from the canonical decomposition (CANDECOMP) of a higher order cumulant tensor.
Abstract: Most conventional techniques for independent component analysis (or blind source separation) resort to second-order statistics to decorrelate the observed data. The prewhitening step makes these algorithms sensitive to the presence of additive Gaussian noise. A higher-order-only technique is presented. The identification problem is approached in a (linear and multilinear) algebraic framework: our derivation starts with the observation that the solution can be obtained from the canonical decomposition (CANDECOMP) of a higher-order cumulant tensor. Next, it is demonstrated that the CANDECOMP components follow from the simultaneous diagonalization, by congruence transformation, of a set of matrices. A reformulation in terms of orthogonal unknowns leads to a simultaneous Schur decomposition, which is solved by a Givens-type iteration. The technique can be considered as the higher-order-only equivalent of the popular JADE-algorithm.
TL;DR: The proposed fast steepest descent path algorithm (FASDPA) complements previously developed fast solvers, notably the fast multipole method and the matrix decomposition algorithm, and permits a matrix-vector multiplication in O(N4/3) operations.
Abstract: A novel algorithm for accelerating the iterative solution of integral equations governing scattering from surfaces is presented. The proposed fast steepest descent path algorithm (FASDPA) complements previously developed fast solvers, notably the fast multipole method (FMM) and the matrix decomposition algorithm (MDA). Whereas the computational complexity per iteration of a two-level FMM and MDA is O(N3/2), the FASDPA permits a matrix-vector multiplication in O(N4/3) operations.
TL;DR: The conditions for achieving the desired rank of the block Hankel matrix are provided and it is shown that the rank should be equal to the number of 2-D frequencies inherent in the data in the absence of noise.
Abstract: For detection and estimation of 2-D frequencies from a 2-D array of data using a subspace decomposition method, one needs to construct a block Hankel matrix. For reliable detection and estimation, the rank of the block Hankel matrix should be made equal to the number of 2-D frequencies inherent in the data in the absence of noise. In this work, we provide the conditions for achieving the desired rank.
TL;DR: A pseudo- Jacobian matrix is introduced, which leaves the stationary points untouched and can be calculated analytically and by slightly changing the pseudo-Jacobian matrix, a better approximation of the Hessian can be obtained resulting in faster convergence.
Abstract: The Gauss-Newton algorithm is often used to minimize a nonlinear least-squares loss function instead of the original Newton-Raphson algorithm. The main reason is the fact that only first-order derivatives are needed to construct the Jacobian matrix. Some applications as, for instance multivariable system identification, give rise to "weighted" nonlinear least-squares problems for which it can become quite hard to obtain an analytical expression of the Jacobian matrix. To overcome that struggle, a pseudo-Jacobian matrix is introduced, which leaves the stationary points untouched and can be calculated analytically. Moreover, by slightly changing the pseudo-Jacobian matrix, a better approximation of the Hessian can be obtained resulting in faster convergence.
TL;DR: The spectral and inner-outer factorization problems are solved for discrete-time systems with transfer matrices of arbitrary rank by derived simple and explicit state-space formulas expressed in terms of the stabilizing solution to the discrete- time algebraic Riccati system.
Abstract: The spectral and inner-outer factorization problems are solved for discrete-time systems with transfer matrices of arbitrary rank. Simple and explicit state-space formulas expressed in terms of the stabilizing solution to the discrete-time algebraic Riccati system are derived. The results are numerically effective since a reliable algorithm for computing such a stabilizing solution is available.
04 Sep 1996
TL;DR: Three algorithms which permute and partition a sparse matrix are presented as tools for the improved solution of Markov chain problems and if the partition obtained from the ordering algorithm is used as an aggregation scheme, an iterative aggregation method performs better with this partition than with others found in the literature.
Abstract: Three algorithms which permute and partition a sparse matrix are presented as tools for the improved solution of Markov chain problems. One is the algorithm PABLO [SIAM J. Sci. Stat. Computing, vol.11, pp.811-823, 1990] while the other two are modifications of it. In the new algorithms, in addition to the location of the nonzeros, the values of the entries are taken into account. The permuted matrices are well suited for block iterative methods that find the corresponding probability distribution, as well as for block diagonal preconditioners of Krylov-based methods. Also, if the partition obtained from the ordering algorithm is used as an aggregation scheme, an iterative aggregation method performs better with this partition than with others found in the literature. Numerical experiments illustrate the performance of the iterative methods with the new orderings.
TL;DR: A new decomposition algorithm to route a rearrangeable three-stage Clos network in O(nr/sup 2/) time, which is faster than all existing decomposition algorithms, by performing a row-wise matrix decomposition.
Abstract: We give a new decomposition algorithm to route a rearrangeable three-stage Clos network in O(nr/sup 2/) time, which is faster than all existing decomposition algorithms. By performing a row-wise matrix decomposition, this algorithm routes all possible permutations, thus overcoming the limitation on realizable permutations exhibited by many other routing algorithms. This algorithm is extended to the fault tolerant Clos network which has extra switches in each stage, where it provides fault tolerance under faulty conditions and reduces routing time under submaximal fault conditions.
TL;DR: Experiments on the hypercube confirm that the proposed method for parallel load-flow calculations based on an effective decomposition of the network is indeed effective, particularly for problems where a good initial approximation is available (such as outage assessment).
Abstract: The objective of this paper is to present a new method for parallel load-flow calculations based on an effective decomposition of the network. In the solution process, the authors utilize the block-parallel Newton method which involves only diagonal blocks of the Jacobian. The underlying structure is obtained by applying the epsilon decomposition algorithm which eliminates weak coupling elements from the matrix. They demonstrate that the iterative process can be significantly accelerated by making certain modifications in mismatch evaluation for buses connecting different blocks. Experiments on the hypercube confirm that the proposed method is indeed effective, particularly for problems where a good initial approximation is available (such as outage assessment).
TL;DR: In this paper, the authors presented a new matrix decomposition for a matrix pair (A, B) with A Hermitian, where the hyperbolic SVD comes as a special case of the decomposition with A set to be the signature matrix.
Patent•
06 Dec 1996
TL;DR: In this article, a method for establishing key agreement between two communicating parties using a general linear protocol in finite and infinite dimensional spaces is presented, in particular Euclidean spaces and a non-trivial degenerate linear operator.
Abstract: A method for establishing key agreement between two communicating parties using a general linear protocol in finite and infinite dimensional spaces. Two topological linear spaces, in particular Euclidean spaces, and a non-trivial degenerate linear operator are selected. Each party respectively selects a secret element, and exchanges with the other party an image under the transformation of a matrix. Key agreement is therefore mutually established between the two communicating parties having the same cryptographic key. Various illustrative embodiments of the general linear operator are disclosed, including a rectangular matrix, a square matrix, a symmetric matrix, a skew symmetric matrix, an upper triangular square matrix, a lower triangular square matrix, a special type of skew symmetric matrix to generate a modified cross product protocol, a series of matrices to generate a sequential key protocol, and a combination of circulant matrices.
TL;DR: A numerical algorithm for computing a few extreme generalized singular values and corresponding vectors of a sparse or structured matrix pair based on the CS decomposition and the Lanczos bidiagonalization process is presented.
Abstract: We present a numerical algorithm for computing a few extreme generalized singular values and corresponding vectors of a sparse or structured matrix pair \(\{A,B\}\). The algorithm is based on the CS decomposition and the Lanczos bidiagonalization process. At each iteration step of the Lanczos process, the solution to a linear least squares problem with\((A^{\rm T},B^{\rm T})^{\rm T}\) as the coefficient matrix is approximately computed, and this consists the only interface of the algorithm with the matrix pair \(\{A,B\}\). Numerical results are also given to demonstrate the feasibility and efficiency of the algorithm.
TL;DR: An algorithm for the computation of the maximum likelihood and the maximum a posteriori estimates of the parameters of PMD models is presented and is a special case of a more general algorithm that can be used for the whole class of LRMs.
Abstract: In this paper, we consider a class of models for two-way matrices with binary entries of 0 and 1. First, we considerBoolean matrix decomposition, conceptualize it as alatent response model (LRM) and, by making use of this conceptualization, generalize it to a larger class of matrix decomposition models. Second,probability matrix decomposition (PMD) models are introduced as a probabilistic version of this larger class of deterministic matrix decomposition models. Third, an algorithm for the computation of the maximum likelihood (ML) and the maximum a posteriori (MAP) estimates of the parameters of PMD models is presented. This algorithm is an EM-algorithm, and is a special case of a more general algorithm that can be used for the whole class of LRMs. And fourth, as an example, a PMD model is applied to data on decision making in psychiatric diagnosis.
22 Apr 1996
TL;DR: The kinematics and dynamics of kinematically redundant manipulators are to be decomposed in consistent way with the joint space decomposition induced by the weighted pseudo-inverse operation.
Abstract: The kinematics and dynamics of kinematically redundant manipulators are to be decomposed in consistent way with the joint space decomposition induced by the weighted pseudo-inverse operation. The weighted kinematically decoupled joint space decomposition results to a family of decomposed kinematics and dynamics, parametrized by the symmetric positive-definite weight matrix, in the task motion and weighted null motion space. The inertially-decoupled weighted decomposition is shown to be only possible with the inertia as the weight. Also, the feedback motion controller which is consistent with a specific joint space decomposition is developed. The controller realizes the decoupled and linearized reference acceleration tracking controller in the task motion and weighted null motion space.
Proceedings Article•
01 Sep 1996TL;DR: A new Jacobi-type method to calculate a simultaneous Schur decomposition (SSD) of several real-valued, non-symmetric matrices by minimizing an appropriate cost function, which achieves automatic pairing of the estimated R-dimensional modes via a closed-form procedure that neither requires any search nor any other heuristic pairing strategy.
Abstract: This paper presents a new Jacobi-type method to calculate a simultaneous Schur decomposition (SSD) of several real-valued, non-symmetric matrices by minimizing an appropriate cost function. Thereby, the SSD reveals the “average eigenstructure” of these non-symmetric matrices. This enables an R-dimensional extension of Unitary ESPRIT to estimate several undamped R-dimensional modes or frequencies along with their correct pairing in multidimensional harmonic retrieval problems. Unitary ESPRIT is an ESPRIT-type high-resolution frequency estimation technique that is formulated in terms of real-valued computations throughout. For each of the R dimensions, the corresponding frequency estimates are obtained from the real eigenvalues of a real-valued matrix. The SSD jointly estimates the eigenvalues of all R matrices and, thereby, achieves automatic pairing of the estimated R-dimensional modes via a closed-form procedure, that neither requires any search nor any other heuristic pairing strategy. Finally, we show how R-dimensional harmonic retrieval problems (with R ≥ 3) occur in array signal processing and model-based object recognition applications.
TL;DR: The authors formulate a problem of maximizing the response as an optimization problem for an applied current intensity distribution on the surface, and find the optimal intensity distribution of the current, which maximizes the response from the inclusion.
Abstract: The first problem which needs to be solved when planning any geoelectrical survey is a choice of a particular electrode configuration that can give the maximal response from a target inhomogeneity. The authors formulate a problem of maximizing the response as an optimization problem for an applied current intensity distribution on the surface. The solution of this problem is the optimal intensity distribution of the current, which maximizes the response from the inclusion. This problem is solved numerically with singular value decomposition of an impedance matrix. The optimal current array is modeled as a current of varying optimal intensity injected at different electrodes. The problem does not need any information about the inclusion but its measured impedance matrix. Thus an optimal current array can be designed for every particular resistivity distribution. The optimal current patterns are found for a number of models of a conductive inclusion, and responses due to the optimal current are compared with responses due to conventional arrays. This method can be applied to any background and inclusion resistivity distribution.
TL;DR: In this paper, an iterative solution scheme for calculating the eigenvector sensitivity in which only the lowest eigencharacteristics are required was developed, which is applicable to situations where only a partial eigensolution of the lowest Eigenvectors and eigenvalues is available.
Abstract: There are a number of applications where the sensitivity of eigenvectors with respect to physical parameters is desired. We develop an iterative solution scheme for calculating the eigenvector sensitivity in which only the lowest eigencharacteristics are required. It uses a least-squares formulation for the eigenvector sensitivity including the relation from the basic eigenvalue problem and the orthogonality and normality conditions with respect to the mass matrix. The iterative scheme uses the band structure of the stiffness matrix and an efficient use of the Householder transformation to reduce the number of calculations. Since only the lowest eigensolutions are used in the formulation, it is applicable to situations where only a partial eigensolution of the lowest eigenvectors and eigenvalues is available. The eigenvalues are assumed to be distinct and only the first-order variation is calculated. The stiffness matrix is assumed to be nonsingular and a Choleski decomposition of the stiffness matrix is required, but this is the only large matrix that needs to be decomposed. The least-squares solution to the eigenvector sensitivity is shown to reduce to the modal expansion method when appropriate weights are incorporated. From this expression, we show why the modal expansion is not always adequate for eigenvector sensitivity and give a criterion for evaluating this method in a given application.