Matrix Approximation Problem Research Articles

A tensor product matrix approximation problem in quantum physics

We consider a matrix approximation problem arising in the study of entanglement in quantum physics. This notion represents a certain type of correlations between subsystems in a composite quantum system. The states of a system are described by a density matrix, which is a positive semidefinite matrix with trace one. The goal is to approximate such a given density matrix by a so-called separable density matrix, and the distance between these matrices gives information about the degree of entanglement in the system. Separability here is expressed in terms of tensor products. We discuss this approximation problem for a composite system with two subsystems and show that it can be written as a convex optimization problem with special structure. We investigate related convex sets, and suggest an algorithm for this approximation problem which exploits the tensor product structure in certain subproblems. Finally some computational results and experiences are presented.

Approximation of Matrices and a Family of Gander Methods for Polar Decomposition

Two matrix approximation problems are considered: approximation of a rectangular complex matrix by subunitary matrices with respect to unitarily invariant norms and a minimal rank approximation with respect to the spectral norm. A characterization of a subunitary approximant of a square matrix with respect to the Schatten norms, given by Maher, is extended to the case of rectangular matrices and arbitrary unitarily invariant norms. Iterative methods, based on the family of Gander methods and on Higham’s scaled method for polar decomposition of a matrix, are proposed for computing subunitary and minimal rank approximants. Properties of Gander methods are investigated in details.

Solving Hankel matrix approximation problem using semidefinite programming

Positive semidefinite Hankel matrices arise in many important applications. Some of their properties may be lost due to rounding or truncation errors incurred during evaluation. The problem is to find the nearest matrix to a given matrix to retrieve these properties. The problem is converted into a semidefinite programming problem as well as a problem comprising a semidefined program and second-order cone problem. The duality and optimality conditions are obtained and the primal–dual algorithm is outlined. Explicit expressions for a diagonal preconditioned and crossover criteria have been presented. Computational results are presented. A possibility for further improvement is indicated.

Semidefinite and second-order cone optimization approach for the Toeplitz matrix approximation problem

The nearest positive semidefinite symmetric Toeplitz matrix to an arbitrary data covariance matrix is useful in many areas of engineering, including stochastic filtering and digital signal processing applications. In this paper, the interior point primal-dual path-following method will be used to solve our problem after reformulating it into different forms, first as a semidefinite programming problem, then into the form of a mixed semidefinite and second-order cone optimization problem. Numerical results, comparing the performance of these methods against the modified alternating projection method will be reported.

Semidefinite and second-order cone optimization approach for the Toeplitz matrix approximation problem

The nearest positive semidefinite symmetric Toeplitz matrix to an arbitrary data covariance matrix is useful in many areas of engineering, including stochastic filtering and digital signal processing applications. In this paper, the interior point primal-dual path-following method will be used to solve our problem after reformulating it into different forms, first as a semidefinite programming problem, then into the form of a mixed semidefinite and second-order cone optimization problem. Numerical results, comparing the performance of these methods against the modified alternating projection method will be reported.

On the Low-Rank Approximation of Data on the Unit Sphere

In various applications, data in multidimensional space are normalized to unit length. This paper considers the problem of best fitting given points on the m-dimensional unit sphere Sm-1 by k-dimensional great circles with k much less than m. The task is cast as an algebraically constrained low-rank matrix approximation problem. Using the fidelity of the low-rank approximation to the original data as the cost function, this paper offers an analytic expression of the projected gradient which, on one hand, furnishes the first order optimality condition and, on the other hand, can be used as a numerical means for solving this problem.

SQP algorithms for solving Toeplitz matrix approximation problem

AbstractGiven an n × n matrix F, we find the nearest symmetric positive semi‐definite Toeplitz matrix T to F. The problem is formulated as a non‐linear minimization problem with positive semi‐definite Toeplitz matrix as constraints. Then a computational framework is given. An algorithm with rapid convergence is obtained by l1 Sequential Quadratic Programming (SQP) method. Copyright © 2002 John Wiley & Sons, Ltd.

The Solution to a Structured Matrix Approximation Problem Using GrassmanCoordinates

A method for finding the best approximation of a matrix A by a full rank Hankel matrix is given. The initial problem of best approximation of one matrix by another is transformed to a problem involving best approximation of a given vector by a second vector whose elements are constrained so that its inverse image is a Hankel matrix. The map from a matrix to a vector is the invertible map between a subspace represented as the row space of the matrix A and the Grassman vector representing that subspace. The relation between the principle angles associated with a pair of subspaces and the angle between the Grassman vectors associated with the subspaces is established.

Nonlinear eigenvector algorithms for local optimization in multivariate data analysis

Many multivariate statistical procedures, such as principal component, canonical correlation, correspondence, and discriminant analysis, are based on the solution of a certain matrix approximation problem, one of the most important being the reduced-rank approximation. Its solution hinges on the singular value or the eigendecomposition of a certain matrix. Therefore, numerical techniques to determine eigenvalues and eigenvectors play an important role in these statistical applications. Furthermore, various modifications, generalizations, or refinements of the classical methods lead to some sort of nonlinear eigenproblem with no immediate unique solution. However, the nonlinear eigenvector approach has proved to be quite effective in solving these types of constrained optimization problems in diverse fields of multivariate data analysis. Since a general analysis of the properties and performance of the nonlinear eigenvector algorithm seems to be long overdue, this paper collects the common features in the applications mentioned above and gives an overview from a superior perspective. It unifies the treatment by contrasting the nonlinear eigenvector algorithm with the well-known inverse iteration, iteratively reweighted least squares, and majorization algorithms; extracts the mathematical tools—basically from convex analysis and matrix algebra—needed for the convergence proofs; and discusses convergence properties.

Order reduction of 2-D FIR filters with applications

The author develops two algorithms for the order reduction of two-dimensional (2-D) FIR digital filters, which lead to two indirect design methods. Specifically, it is shown how an optimal L/sub 2/ design can be derived as a matrix approximation problem in the Frohenius norm, and how a suboptimal L/sub /spl infin// design can he obtained by solving a matrix approximation problem in the 2-norm using the Davis-Kahan-Weinberger theorem on norm-preserving dilations. A case study is included to demonstrate the usefulness of the proposed methods.

On approximation problems with zero-trace matrices

We consider some approximation problems in the linear space of complex matrices with respect to unitarily invariant norms. We deal with special cases of approximation of a matrix by zero-trace matrices. Moreover, some characterizations of zero-trace matrices are given by means of matrix approximation problems.

GMRES/CR and Arnoldi/Lanczos as Matrix Approximation Problems

The and algorithms, which reduce to the CR and Lanczos algorithms in the symmetric case, both minimize ||p(A)b|| over polynomials p of degree n. The difference is that p is normalized at z=0 for and at z=infinity for Arnoldi. Analogous GMRES and Arnoldi problems are obtained if one removes b from the discussion and minimizes ||p(A)|| instead. Investigation of these true and ideal approximation problems gives insight into how fast converges and how the iteration locates eigenvalues.

On matrix approximation problems with Ky Fank norms

LetA be a realm xn matrix whose elements depend onl free parameters forming the vectorx. Then a class of approximation problems can be defined by the requirement thatx be chosen to minimize ∥A(x)∥, for a given matrix normon m ×n matrices. For example, it may be required to approximate a given matrix by a particular type of matrix, or by a linear combination of matrices. In the derivation of effective algorithms for such problems, a prerequisite is the provision of appropriate conditions satisfied by a solution, and the subdifferential of the matrix norm plays a crucial role in this. Therefore a characterization of the subdifferential is important, and this is considered for a class of orthogonally invariant norms known as Ky Fank norms, which include as special cases the spectral norm and the trace norm. The results lead to a consideration of efficient algorithms.

Superresolution by structured matrix approximation

The bearing estimation problem is formulated as a matrix-approximation problem. The columns of a matrix X are formed by the snapshot vectors from an N-element array. The matrix X is then approximated by a matrix in the least-square sense. The rank as well as the partial structure of the space spanned by the columns of the approximated X matrix are prespecified. After the approximated X matrix is computed, the bearings of the sources and, consequently, the spatial correlation of the source signals are estimated. The performance of the proposed technique is compared with two existing methods using simulation. The comparison is made in terms of bias, mean-squared error, failure rates, and confidence intervals for the mean and the variance estimates for all three methods at different signal-to-noise ratios. When the sources are moving slowly and the number of snapshot vectors available for processing is large, a simple online adaptive algorithm is suggested. >

State reduction in hidden Markov chains used for speech recognition

This paper examines the problem of reducing the number of states in the Markov chain models used in speech recognition algorithms by statistical methods. It is shown that the equivalence requirements between the original and reduced model lead to entirely different reduction equations depending on whether the reduction occurs inside a word model or between the models of different words. In general, the exact solution of these reduction equations is not possible and one shows that a least-squares solution can be reformulated as a matrix approximation problem in Euclidean norm.

Relationship between the Karhunen- Loéve transform and the Courant - Fischer theorem (Corresp.)

It is shown that the Karhunen-Loeve transform problem can be formulated as a matrix approximation problem with Hilbert-Schmidt error norm. On the other hand, the Courant-Fischer minimax theorem provides a characterization for the best matrix approximation when the spectral norm is used. It appears that the optimality conditions of the Karhunen-Loeve problem lead to the selection of a particular solution among the set of solutions to the Courant-Fischer problem.

Maximum likelihood estimation and factor analysis

Fisher's method of maximum likelihood is applied to the problem of estimation in factor analysis, as initiated by Lawley, and found to lead to a generalization of the Eckart matrix approximation problem. The solution of this in a special case is applied to show how test fallability enters into factor determination, it being noted that the method of communalities underestimates the number of factors.