Multichannel spectral factorization algorithm using polynomial matrix eigenvalue decomposition

In this paper, we present a new multichannel spectral factorization algorithm which can be utilized to calculate the approximate spectral factor of any para-Hermitian polynomial matrix. The proposed algorithm is based on an iterative method for polynomial matrix eigenvalue decomposition (PEVD). By using the PEVD algorithm, the multichannel spectral factorization problem is simply broken down to a set of single channel problems which can be solved by means of existing one-dimensional spectral factorization algorithms. In effect, it transforms the multichannel spectral factorization problem into one which is much easier to solve

The Matrix Ridge Approximation: Algorithms and Applications

We are concerned with an approximation problem for a symmetric positive semidefinite matrix due to motivation from a class of nonlinear machine learning methods. We discuss an approximation approach that we call {matrix ridge approximation}. In particular, we define the matrix ridge approximation as an incomplete matrix factorization plus a ridge term. Moreover, we present probabilistic interpretations using a normal latent variable model and a Wishart model for this approximation approach. The idea behind the latent variable model in turn leads us to an efficient EM iterative method for handling the matrix ridge approximation problem. Finally, we illustrate the applications of the approximation approach in multivariate data analysis. Empirical studies in spectral clustering and Gaussian process regression show that the matrix ridge approximation with the EM iteration is potentially useful

A new multichannel spectral factorisation algorithm for parahermitian polynomial matrices

A novel multichannel spectral factorization algorithm is illustrated in this paper. This new algorithm is based on an iterative method for polynomial eigenvalue decomposition (PEVD) called the second order sequential best rotation (SBR2) algorithm [1]. By using the SBR2 algorithm, multichannel spectral factorization problems are simply broken down to a set of single channel problems which can be solved by means of existing one dimensional spectral factorization algorithms. In effect, it transforms the multichannel spectral factorization problem into one which is much easier to solve. The proposed algorithm can be used to calculate the approximate spectral factor of any parahermitian polynomial matrix. Two worked examples are presented in order to demonstrate its ability to find a valid spectral factor, and indicate the level of accuracy which can be achieved

A state-space approach to blind estimation of MIMO wireless channels

This dissertation focuses on blind channel estimation in wireless communications such that the estimated channel admits the minimum phase property. It assumes only the second order statistics of the transmitted signal at the receive side. Our proposed approach is based on the generalized spectral factorization because of the deficient normal rank for the power spectral density (PSD) function of the received signal. We will show the relationship between the generalized spectral factorization and inner-outer factorizations where the inner is square with smaller size. The inner-outer factorization is in turn related to the generalized Kalman filtering in which the dimension of the input noise processes is greater than the dimension of the output measurement and thus the covariance matrix is always singular. A dual problem is the generalized LQR control in which the dimension of the control input is smaller than the dimension of the controlled output and thus the weighting matrix on control signal is always singular. Iterative algorithms are proposed to obtain stabilizing solutions to algebraic Riccati equations (ARE) associated with the generalized Kalman filtering, LQR control, and spectral factorization. We will show the convergence of the proposed iterative algorithm that provides an effective algorithm for blind channel estimation. Examples are worked out to illustrate our proposed spectral factorization approach to blind channel estimation with comparisons to the existing method in the literature

Factorizing the Stochastic Galerkin System

Recent work has explored solver strategies for the linear system of equations arising from a spectral Galerkin approximation of the solution of PDEs with parameterized (or stochastic) inputs. We consider the related problem of a matrix equation whose matrix and right hand side depend on a set of parameters (e.g. a PDE with stochastic inputs semidiscretized in space) and examine the linear system arising from a similar Galerkin approximation of the solution. We derive a useful factorization of this system of equations, which yields bounds on the eigenvalues, clues to preconditioning, and a flexible implementation method for a wide array of problems. We complement this analysis with (i) a numerical study of preconditioners on a standard elliptic PDE test problem and (ii) a fluids application using existing CFD codes; the MATLAB codes used in the numerical studies are available online.Comment: 13 pages, 4 figures, 2 table

Fast model-fitting of Bayesian variable selection regression using the iterative complex factorization algorithm

Bayesian variable selection regression (BVSR) is able to jointly analyze genome-wide genetic datasets, but the slow computation via Markov chain Monte Carlo (MCMC) hampered its wide-spread usage. Here we present a novel iterative method to solve a special class of linear systems, which can increase the speed of the BVSR model-fitting tenfold. The iterative method hinges on the complex factorization of the sum of two matrices and the solution path resides in the complex domain (instead of the real domain). Compared to the Gauss-Seidel method, the complex factorization converges almost instantaneously and its error is several magnitude smaller than that of the Gauss-Seidel method. More importantly, the error is always within the pre-specified precision while the Gauss-Seidel method is not. For large problems with thousands of covariates, the complex factorization is 10 -- 100 times faster than either the Gauss-Seidel method or the direct method via the Cholesky decomposition. In BVSR, one needs to repetitively solve large penalized regression systems whose design matrices only change slightly between adjacent MCMC steps. This slight change in design matrix enables the adaptation of the iterative complex factorization method. The computational innovation will facilitate the wide-spread use of BVSR in reanalyzing genome-wide association datasets.Comment: Accepted versio