scispace - formally typeset
Search or ask a question

Showing papers on "Matrix (mathematics) published in 2001"


Proceedings ArticleDOI
25 Jun 2001
TL;DR: It is shown that the heuristic to replace the (nonconvex) rank objective with the sum of the singular values of the matrix, which is the dual of the spectral norm, can be reduced to a semidefinite program, hence efficiently solved.
Abstract: We describe a generalization of the trace heuristic that applies to general nonsymmetric, even non-square, matrices, and reduces to the trace heuristic when the matrix is positive semidefinite The heuristic is to replace the (nonconvex) rank objective with the sum of the singular values of the matrix, which is the dual of the spectral norm We show that this problem can be reduced to a semidefinite program, hence efficiently solved To motivate the heuristic, we, show that the dual spectral norm is the convex envelope of the rank on the set of matrices with norm less than one We demonstrate the method on the problem of minimum-order system approximation

1,111 citations


Journal ArticleDOI
TL;DR: A forward model of how data is affected by an inhomogeneous field at different object positions is suggested and a method to solve the inverse problem of estimating the field inhomogeneities and their derivatives with respect to object position is derived directly from the EPI data and estimated realignment parameters.

894 citations


Journal ArticleDOI
TL;DR: This work estimates both the number of sources and the mixing matrix by the maxima of a potential function along the circle of unit length, and obtains the minimal l1 norm representation of each data point by a linear combination of the pair of basis vectors that enclose it.

743 citations


Journal ArticleDOI
TL;DR: This book deals with regression analysis and suggests to shift the focus to problem solving, using existing (or developing new) mathematical-statistical and subject-matter theory rather than developing new theory to solve problems that could arise in the future.
Abstract: (2001). Introduction to Matrix Analytic Methods in Stochastic Modeling. Technometrics: Vol. 43, No. 3, pp. 379-380.

701 citations


Proceedings Article
03 Jan 2001
TL;DR: It is shown that a relaxed version of the trace maximization problem possesses global optimal solutions which can be obtained by Computing a partial eigendecomposition of the Gram matrix, and the cluster assignment for each data vectors can be found by computing a pivoted QR decomposition ofThe eigenvector matrix.
Abstract: The popular K-means clustering partitions a data set by minimizing a sum-of-squares cost function. A coordinate descend method is then used to find local minima. In this paper we show that the minimization can be reformulated as a trace maximization problem associated with the Gram matrix of the data vectors. Furthermore, we show that a relaxed version of the trace maximization problem possesses global optimal solutions which can be obtained by computing a partial eigendecomposition of the Gram matrix, and the cluster assignment for each data vectors can be found by computing a pivoted QR decomposition of the eigenvector matrix. As a by-product we also derive a lower bound for the minimum of the sum-of-squares cost function.

657 citations


Journal ArticleDOI
TL;DR: The Finite Integration Technique (FIT) as discussed by the authors is a consistent discretization scheme for Maxwell's equations in their integral form, which can be used for efficient numerical simulations on modern computers.
Abstract: The Finite Integration Technique (FIT) is a consistent discretization scheme for Maxwell's equations in their integral form. The resulting matrix equations of the discretized fields can be used for efficient numerical simulations on modern computers. In addition, the basic algebraic properties of this discrete electromagnetic field theory allow to analytically and algebraically prove conservation properties with respect to energy and charge of the discrete formulation and gives an explanation of the stability properties of numerical formulations in the time domain.

519 citations


Journal ArticleDOI
TL;DR: The singular value decomposition has been extensively used in engineering and statistical applications and certain properties of this decomposition are investigated as well as numerical algorithms.
Abstract: The singular value decomposition (SVD) has been extensively used in engineering and statistical applications. This method was originally discovered by Eckart and Young in [Psychometrika, 1 (1936), pp. 211--218], where they considered the problem of low-rank approximation to a matrix. A natural generalization of the SVD is the problem of low-rank approximation to high order tensors, which we call the multidimensional SVD. In this paper, we investigate certain properties of this decomposition as well as numerical algorithms.

461 citations



Book
Victor Y. Pan1
01 Jan 2001
TL;DR: This book covers most fundamental numerical and algebraic computations with Toeplitz, Hankel, Vandermonde, Cauchy, and other popular structured matrices, enabling both a unified treatment of various matrix structures and dramatic saving of computer time and memory.
Abstract: Structure matrices serve as a natural bridge between the areas of algebraic computations with polynomials and numerical matrix computations, allowing cross-fertilization of both fields. This book covers most fundamental numerical and algebraic computations with Toeplitz, Hankel, Vandermonde, Cauchy, and other popular structured matrices. Throughout the computations, the matrices are represented by their compressed images, called displacements, enabling both a unified treatment of various matrix structures and dramatic saving of computer time and memory. The resulting superfast algorithms allow further dramatic parallel acceleration using FFT and fast sine and cosine transforms.

394 citations


Journal ArticleDOI
TL;DR: Algebraic meshQuality metrics for structured and unstructured mesh generation are placed within an algebraic framework to form a mathematical theory of mesh quality metrics and equivalence of the element condition number and mean ratio metrics is proved.
Abstract: Quality metrics for structured and unstructured mesh generation are placed within an algebraic framework to form a mathematical theory of mesh quality metrics. The theory, based on the Jacobian and related matrices, provides a means of constructing, classifying, and evaluating mesh quality metrics. The Jacobian matrix is factored into geometrically meaningful parts. A nodally invariant Jacobian matrix can be defined for simplicial elements using a weight matrix derived from the Jacobian matrix of an ideal reference element. Scale and orientation-invariant algebraic mesh quality metrics are defined. The singular value decomposition is used to study relationships between metrics. Equivalence of the element condition number and mean ratio metrics is proved. The condition number is shown to measure the distance of an element to the set of degenerate elements. Algebraic measures for skew, length ratio, shape, volume, and orientation are defined abstractly, with specific examples given. Two combined metrics, shape-volume and shape-volume orientation, are algebraically defined and examples of such metrics are given. Algebraic mesh quality metrics are extended to nonsimplicial elements. A series of numerical tests verifies the theoretical properties of the metrics defined.

350 citations


Journal ArticleDOI
TL;DR: In this article, the concept of a Leonard system was introduced, and it was shown that for any Leonard pair A,A* on V, there exists a sequence of scalars β,γ,γ*,ϱ,ϱ* taken from K such that both

Journal ArticleDOI
TL;DR: It is shown that a truncated matrix linking just a few modes is a good approximation of the full time translation matrix, which suggests that the number of essential connections among the genes is small.
Abstract: We describe the time evolution of gene expression levels by using a time translational matrix to predict future expression levels of genes based on their expression levels at some initial time. We deduce the time translational matrix for previously published DNA microarray gene expression data sets by modeling them within a linear framework by using the characteristic modes obtained by singular value decomposition. The resulting time translation matrix provides a measure of the relationships among the modes and governs their time evolution. We show that a truncated matrix linking just a few modes is a good approximation of the full time translation matrix. This finding suggests that the number of essential connections among the genes is small.

Journal ArticleDOI
TL;DR: In this article, ideas from elementary graph theory and linear matrix inequalities are combined with logic-based switching to shed light on the various control strategies which are feasible in the leader-following framework for the formation flying of multiple spacecraft.
Abstract: Ideas from elementary graph theory and linear matrix inequalities are combined with logic-based switching to shed light on the various control strategies which are feasible in the leader-following framework for the formation flying of multiple spacecraft.

Proceedings ArticleDOI
01 Dec 2001
TL;DR: A novel solution for flow-based tracking and 3D reconstruction of deforming objects in monocular image sequences using a linear combination of 3D basis shapes and the rank constraint is used to achieve robust and precise low-level optical flow estimation.
Abstract: This paper presents a novel solution for flow-based tracking and 3D reconstruction of deforming objects in monocular image sequences. A non-rigid 3D object undergoing rotation and deformation can be effectively approximated using a linear combination of 3D basis shapes. This puts a bound on the rank of the tracking matrix. The rank constraint is used to achieve robust and precise low-level optical flow estimation without prior knowledge of the 3D shape of the object. The bound on the rank is also exploited to handle occlusion at the tracking level leading to the possibility of recovering the complete trajectories of occluded/disoccluded points. Following the same low-rank principle, the resulting flow matrix can be factored to get the 3D pose, configuration coefficients, and 3D basis shapes. The flow matrix is factored in an iterative manner, looping between solving for pose, configuration, and basis shapes. The flow-based tracking is applied to several video sequences and provides the input to the 3D non-rigid reconstruction task. Additional results on synthetic data and comparisons to ground truth complete the experiments.

Posted Content
TL;DR: In this article, a new representation of the diagonal Vech model is given using the Hadamard product, and sufficient conditions on parameter matrices are provided to ensure the positive definiteness of covariance matrices from the new representation.
Abstract: A new representation of the diagonal Vech model is given using the Hadamard product. Sufficient conditions on parameter matrices are provided to ensure the positive definiteness of covariance matrices from the new representation. Based on this, some new and simple models are discussed. A set of diagnostic tests for multivariate ARCH models is proposed. The tests are able to detect various model misspecifications by examing the orthogonality of the squared normalized residuals. A small Monte-Carlo study is carried out to check the small sample performance of the test. An empirical example is also given as guidance for model estimation and selection in the multivariate framework. For the specific data set considered, it is found that the simple one and two parameter models and the constant conditional correlation model perform fairly well.

Journal ArticleDOI
TL;DR: The main result is that a minimal confidence ellipsoid for the state, consistent with the measured output and the uncertainty description, may be recursively computed in polynomial time, using interior-point methods for convex optimization.
Abstract: This note presents a new approach to finite-horizon guaranteed state prediction for discrete-time systems affected by bounded noise and unknown-but-bounded parameter uncertainty. Our framework handles possibly nonlinear dependence of the state-space matrices on the uncertain parameters. The main result is that a minimal confidence ellipsoid for the state, consistent with the measured output and the uncertainty description, may be recursively computed in polynomial time, using interior-point methods for convex optimization. With n states, l uncertain parameters appearing linearly in the state-space matrices, with rank-one matrix coefficients, the worst-case complexity grows as O(l(n + l)/sup 3.5/) With unstructured uncertainty in all system matrices, the worst-case complexity reduces to O(n/sup 3.5/).

Journal ArticleDOI
TL;DR: It is shown, with a variety of modeling techniques, that matrix quality can be extremely important in determining metapopulation dynamics, and that increasing matrix quality will lower the probability of global extinction of a population.
Abstract: In both strictly theoretical and more applied contexts it has been historically assumed that metapopulations exist within a featureless, uninhabitable matrix and that dynamics within the matrix are unimportant. In this article, we explore the range of theoretical consequences that result from relaxing this assumption. We show, with a variety of modeling techniques, that matrix quality can be extremely important in determining metapopulation dynamics. A higher‐quality matrix generally buffers against extinction. However, in some situations, an increase in matrix quality can generate chaotic subpopulation dynamics, where stability had been the rule in a lower‐quality matrix. Furthermore, subpopulations acting as source populations in a low‐quality matrix may develop metapopulation dynamics as the quality of the matrix increases. By forcing metapopulation dynamics on a formerly heterogeneous (but stable within subpopulations) population, the probability of simultaneous extinction of all subpopulati...

Posted Content
TL;DR: In this paper, a new approach via Hurwitz numbers to Kontsevich's combinatorial/matrix model for the intersection theory of the moduli space of curves is presented.
Abstract: The main goal of the paper is to present a new approach via Hurwitz numbers to Kontsevich's combinatorial/matrix model for the intersection theory of the moduli space of curves. A secondary goal is to present an exposition of the circle of ideas involved: Hurwitz numbers, Gromov-Witten theory of the projective line, matrix integrals, and the theory of random trees. Further topics will be treated in a sequel.

Journal ArticleDOI
TL;DR: A finite element formulation for the numerical solution of the stationary incompressible Navier–Stokes equations including Coriolis forces and the permeability of the medium using the algebraic version of the sub-grid scale approach.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the local spacing distribution of the distance between nearest neighbor eigenvalues in some part of the spectrum is, in the limit as N→∞, the same as that of hermitian random matrices from GUE.
Abstract: Consider an N×N hermitian random matrix with independent entries, not necessarily Gaussian, a so-called Wigner matrix. It has been conjectured that the local spacing distribution, i.e. the distribution of the distance between nearest neighbour eigenvalues in some part of the spectrum is, in the limit as N→∞, the same as that of hermitian random matrices from GUE. We prove this conjecture for a certain subclass of hermitian Wigner matrices.

Journal ArticleDOI
TL;DR: This work takes a different approach in which the CRB is derived as the solution to an unconstrained quadratic maximization problem, which enables it to handle the singular case in a simple yet rigorous manner.
Abstract: The case of a singular Fisher information matrix (FIM) represents a significant complication for the theory of the Cramer-Rao lower bound (CRB) that is usually handled by resorting to the pseudoinverse of the Fisher matrix. We take a different approach in which the CRB is derived as the solution to an unconstrained quadratic maximization problem, which enables us to handle the singular case in a simple yet rigorous manner. When the Fisher matrix is singular, except under unusual circumstances, any estimator having the specified bias derivatives that figure in the CRB must have infinite variance.

Journal ArticleDOI
TL;DR: This article describes some techniques, outlines a few proofs, and discusses some exceptional results of linear preserver problems, an active research area in matrix and operator theory.
Abstract: Linear preserver problems is an active research area in matrix and operator theory. These problems involve certain linear operators on spaces of matrices or operators. We give a general introduction to the subject in this article. In the first three sections, we discuss motivation, results, and problems. In the last three sections, we describe some techniques, outline a few proofs, and discuss some exceptional results. 1. EXAMPLES AND TYPICAL PROBLEMS. Let Mm,n be the set of m × n complex matrices, and let Mn = Mn,n. Suppose that M, N ∈ Mn satisfy det( MN ) = 1. Then the mapping φ : Mn → Mn given by

Journal ArticleDOI
TL;DR: In this article, an operational matrix of integration P based on Legendre wavelets is presented, and a general procedure for forming this matrix is given. Illustrative examples are included to demonstrate the validity and applicability of the matrix P.
Abstract: An operational matrix of integration P based on Legendre wavelets is presented. A general procedure for forming this matrix is given. Illustrative examples are included to demonstrate the validity and applicability of the matrix P.

Journal ArticleDOI
01 May 2001
TL;DR: In this article, Component averaging (CAV) is introduced as a new iterative parallel technique suitable for large and sparse unstructured systems of linear equations, which simultaneously projects the current iterate onto all the system's hyperplanes, and is thus inherently parallel.
Abstract: Component averaging (CAV) is introduced as a new iterative parallel technique suitable for large and sparse unstructured systems of linear equations. It simultaneously projects the current iterate onto all the system's hyperplanes, and is thus inherently parallel. However, instead of orthogonal projections and scalar weights (as used, for example, in Cimmino's method), it uses oblique projections and diagonal weighting matrices, with weights related to the sparsity of the system matrix. These features provide for a practical convergence rate which approaches that of algebraic reconstruction technique (ART) (Kaczmarz's row-action algorithm) – even on a single processor. Furthermore, the new algorithm also converges in the inconsistent case. A proof of convergence is provided for unit relaxation, and the fast convergence is demonstrated on image reconstruction problems of the Herman head phantom obtained within the SNARK93 image reconstruction software package. Both reconstructed images and convergence plots are presented. The practical consequences of the new technique are far reaching for real-world problems in which iterative algorithms are used for solving large, sparse, unstructured and often inconsistent systems of linear equations.

Proceedings ArticleDOI
01 Dec 2001
TL;DR: It is found that for regression the tensor-rank coding, as a dimensionality reduction technique, significantly outperforms other techniques like PCA.
Abstract: Given a collection of images (matrices) representing a "class" of objects we present a method for extracting the commonalities of the image space directly from the matrix representations (rather than from the vectorized representation which one would normally do in a PCA approach, for example). The general idea is to consider the collection of matrices as a tensor and to look for an approximation of its tensor-rank. The tensor-rank approximation is designed such that the SVD decomposition emerges in the special case where all the input matrices are the repeatition of a single matrix. We evaluate the coding technique both in terms of regression, i.e., the efficiency of the technique for functional approximation, and classification. We find that for regression the tensor-rank coding, as a dimensionality reduction technique, significantly outperforms other techniques like PCA. As for classification, the tensor-rank coding is at is best when the number of training examples is very small.

Journal ArticleDOI
TL;DR: The spherical harmonics for fuzzy spheres of even and odd dimensions are constructed, generalizing the correspondence between finite Matrix algebras and fuzzy two-spheres and ensuring the correct classical limit.

Journal ArticleDOI
TL;DR: Sanderson et al.'s Knight's Tour generates large variances and does not sample matrices equiprobably, and the Sequential Swap generates results that are very similar to those of an unbiased Random Knight'sTour, and is not overly prone to Type I or Type II errors.
Abstract: Community assembly rules are often inferred from patterns in presence-absence matrices. A challenging problem in the analysis of presence-absence matrices has been to devise a null model algorithm to produce random matrices with fixed row and column sums. Previous studies by Roberts and Stone [(1990) Oecologia 83:560–567] and Manly [(1995) Ecology 76:1109–1115] used a "Sequential Swap" algorithm in which submatrices are repeatedly swapped to produce null matrices. Sanderson et al. [(1998) Oecologia 116:275–283] introduced a "Knight's Tour" algorithm that fills an empty matrix one cell at a time. In an analysis of the presence-absence matrix for birds of the Vanuatu islands, Sanderson et al. obtained different results from Roberts and Stone and concluded that "results from previous studies are generally flawed". However, Sanderson et al. did not investigate the statistical properties of their algorithm. Using simple probability calculations, we demonstrate that their Knight's Tour is biased and does not sample all unique matrices with equal frequency. The bias in the Knight's Tour arises because the algorithm samples exhaustively at each step before retreating in sequence. We introduce an unbiased Random Knight's Tour that tests only a small number of cells and retreats by removing a filled cell from anywhere in the matrix. This algorithm appears to sample unique matrices with equal frequency. The Random Knight's Tour and Sequential Swap algorithms generate very similar results for the large Vanuatu matrix, and for other presence-absence matrices we tested. As a further test of the Sequential Swap, we constructed a set of 100 random matrices derived from the Vanuatu matrix, analyzed them with the Sequential Swap, and found no evidence that the algorithm is prone to Type I errors (rejecting the null hypothesis too frequently). These results support the original conclusions of Roberts and Stone and are consistent with Gotelli's [(2000) Ecology 81:2606–2621] Type I and Type II error tests for the Sequential Swap. In summary, Sanderson et al.'s Knight's Tourgenerates large variances and does not sample matrices equiprobably. In contrast, the Sequential Swap generates results that are very similar to those of an unbiased Random Knight's Tour, and is not overly prone to Type I or Type II errors. We suggest that the statistical properties of proposed null model algorithms be examined carefully, and that their performance judged by comparisons with artificial data sets of known structure. In this way, Type I and Type II error frequencies can be quantified, and different algorithms and indices can be compared meaningfully.

Journal ArticleDOI
TL;DR: The Finite Integration Technique (FIT) is a consistent discretization scheme for Maxwell's equations in their integral form as mentioned in this paper, which can be used for efficient numerical simulations on modern computers.
Abstract: The Finite Integration Technique (FIT) is a consistent discretization scheme for Maxwell's equations in their integral form. The resulting matrix equations of the discretized fields can be used for efficient numerical simulations on modern computers. In addition, the basic algebraic properties of this discrete electromagnetic field theory allow to analytically and algebraically prove conservation properties with respect to energy and charge of the discrete formulation and gives an explanation of the stability properties of numerical time domain formulations.

BookDOI
01 Jan 2001
TL;DR: In this paper, the authors present a generalization of the Stochastic Canonical Equation (SCE) for symmetric random matrices with infinitely small entries, which is a regularized version of the SCE.
Abstract: List of basic notations and assumptions. How the stochastic canonical equation was found. 1. Canonical equation K1. 2. Canonical equation K2* Necessary and sufficient modified Lindeberg's condition. The Wigner and Cubic laws. 3. Regularized stochastic canonical equation K3 for symmetric random matrices with infinitely small entries. 4. Stochastic canonical equation K4 for symmetric random matrices with infinitely small entries. Necessary and sufficient conditions for the convergence of normalized spectral functions. 5. Canonical equation K5 for symmetric random matrices with infinitely small entries. 6. Canonical equation K6 for symmetric random matrices with identically distributed entries. 7. Canonical equation K7 for Gram random matrices. 8. Canonical equation K8. 9. Canonical equation K9 for random matrices whose entries have identical variances. 10. Canonical equation K10* Necessary and sufficient modified Lindeberg condition. 11. Canonical equation K11* Limit theorem for normalized spectral functions of empirical covariance matrices under the modified Lindeberg condition. 12. Canonical Equation K12 for random Gram matrices with infinitely small entries. 13. Canonical Equation K13 for random Gram matrices with infinitely small entries. 14. The method of random determinants for estimating the permanents of matrices and the canonical equation K14 for random Gram matrices. 15. Canonical EquationK15 for random Gram matrices with identically distributed entries. 16. Canonical Equation K16 for sample covariance matrices. 17. Canonical Equation K17 for identically distributed independent vector observations and the G2-estimators of the real Stieltjes transforms of the normalized spectral functions of the covariance matrices. 18. Canonical equation K18 for the special structure of vector observations. 19. Canonical equation K19. 20. Canonical equation K20* Strong law for normalized spectral functions of nonselfadjoint random matrices with independent row vectors. Simple rigorous proof of the strong Circular law. 21. Canonical equation K21 for random matrices with independent pairs of entries with zero expectations. Circular and Elliptic laws. 22. Canonical equation K22 for random matrices with independent pairs of entries. 23. Canonical equation K23 for random matrices with independent pairs of entries with different variances and equal covariances. 24. Canonical equation K24 for random G-matrices with infinitesimally small random entries. 25. Canonical equation K25 for random G-matrices. Strong V-law. 26. Class of canonical V-equation K26 for a single matrix and a product of two matrices. The V-density of eigenvalues of random matrices such that the variances of their entries form a doubly stochastic matrix. 27. Canonical equation K27 for normalized spectral functions of random symmetric block matrices.

Journal ArticleDOI
TL;DR: It is proved that basic perturbation theorems for absolute errors for sine and cosine of principal angles with improved constants are proved.
Abstract: Computation of principal angles between subspaces is important in many applications, e.g., in statistics and information retrieval. In statistics, the angles are closely related to measures of dependency and covariance of random variables. When applied to column-spaces of matrices, the principal angles describe canonical correlations of a matrix pair. We highlight that all popular software codes for canonical correlations compute only cosine of principal angles, thus making impossible, because of round-off errors, finding small angles accurately. We review a combination of sine and cosine based algorithms that provide accurate results for all angles. We generalize the method to the computation of principal angles in an A-based scalar product for a symmetric and positive definite matrix A. We provide a comprehensive overview of interesting properties of principal angles. We prove basic perturbation theorems for absolute errors for sine and cosine of principal angles with improved constants. Numerical examples and a detailed description of our code are given.