Xiao-Wen Chang

Solving Box-Constrained Integer Least Squares Problems

A box-constrained integer least squares problem (BILS) arises from several wireless communications applications. Solving a BILS problem usually has two stages: reduction (or preprocessing) and search. This paper presents a reduction algorithm and a search algorithm. Unlike the typical reduction algorithms, which use only the information of the lattice generator matrix, the new reduction algorithm also uses the information of the given input vector and the box constraint and is very effective for search. The new search algorithm overcomes some shortcomings of the existing search algorithms and gives some other improvement. Simulation results indicate the combination of the new reduction algorithm and the new search algorithm can be much more efficient than the existing algorithms, in particular when the least squares residual is large.

Wavelet estimation of partially linear models

A wavelet approachis presented for estimating a partially linear model (PLM). We /nd an estimator of the PLM by minimizing the square of the l2 norm of the residual vector while penalizing the l1 norm of the wavelet coe2cients of the nonparametric component. This approach, an extension of the wavelet approach for nonparametric regression problems, avoids the restrictive smoothness requirements for the nonparametric function of the traditional smoothing approaches for PLM, such as smoothing spline, kernel and piecewise polynomial methods. To solve the optimization problem, an e2cient descent algorith m withan exact line searchis presented. Simulation results are given to demonstrate e6ectiveness of our method. c 2003 Elsevier B.V. All rights reserved.

Approximate Newton Methods for Nonsmooth Equations

We develop general approximate Newton methods for solving Lipschitz continuous equations by replacing the iteration matrix with a consistently approximated Jacobian, thereby reducing the computation in the generalized Newton method. Locally superlinear convergence results are presented under moderate assumptions. To construct a consistently approximated Jacobian, we introduce two main methods: the classic difference approximation method and the ε-generalized Jacobian method. The former can be applied to problems with specific structures, while the latter is expected to work well for general problems. Numerical tests show that the two methods are efficient. Finally, a norm-reducing technique for the global convergence of the generalized Newton method is briefly discussed.

Effects of the LLL Reduction on the Success Probability of the Babai Point and on the Complexity of Sphere Decoding

A common method to estimate an unknown integer parameter vector in a linear model is to solve an integer least squares (ILS) problem. A typical approach to solving an ILS problem is sphere decoding. To make a sphere decoder faster, the well-known LLL reduction is often used as preprocessing. The Babai point produced by the Babai nearest plane algorithm is a suboptimal solution of the ILS problem. First, we prove that the success probability of the Babai point as a lower bound on the success probability of the ILS estimator is sharper than the lower bound given by Hassibi and Boyd [1]. Then, we show rigorously that applying the LLL reduction algorithm will increase the success probability of the Babai point and give some theoretical and numerical test results. We give examples to show that unlike LLLs column permutation strategy, two often used column permutation strategies SQRD and V-BLAST may decrease the success probability of the Babai point. Finally, we show rigorously that applying the LLL reduction algorithm will also reduce the computational complexity of sphere decoders, which is measured approximately by the number of nodes in the search tree in the literature.

On the sensitivity of the LU factorization

This paper gives sensitivity analyses by two approaches forL andU in the factorizationA=LU for general perturbations inA which are sufficiently small in norm. By the matrix-vector equation approach, we derive the condition numbers for theL andU factors. By the matrix equation approach we derive corresponding condition estimates. We show how partial pivoting and complete pivoting affect the sensitivity of the LU factorization.

Rigorous Perturbation Bounds of Some Matrix Factorizations

This article presents rigorous normwise perturbation bounds for the Cholesky, LU, and QR factorizations with normwise or componentwise perturbations in the given matrix. The considered componentwise perturbations have the form of backward rounding errors for the standard factorization algorithms. The used approach is a combination of the classic and refined matrix equation approaches. Each of the new rigorous perturbation bounds is a small constant multiple of the corresponding first-order perturbation bound obtained by the refined matrix equation approach in the literature and can be estimated efficiently. These new bounds can be much tighter than the existing rigorous bounds obtained by the classic matrix equation approach, while the conditions for the former to hold are almost as moderate as the conditions for the latter to hold.

Solving Ellipsoid-Constrained Integer Least Squares Problems

A new method is proposed to solve an ellipsoid-constrained integer least squares (EILS) problem arising in communications. In this method, the LLL reduction, which is cast as a QRZ factorization of a matrix, is used to transform the original EILS problem to a reduced EILS problem, and then a search algorithm is proposed to solve the reduced EILS problem. Simulation results indicate the new method can be much more computationally efficient than the existing method. The method is extended to solve a more general EILS problem.

Componentwise perturbation analyses for the QR factorization

Summary. This paper gives componentwise perturbation analyses for Q and R in the QR factorization A=QR,

PERTURBATION ANALYSIS OF THE QR FACTOR R IN THE CONTEXT OF LLL LATTICE BASIS REDUCTION

Q^\mathrm{T}Q=I

An Orthogonal Transformation Algorithm for GPS Positioning

, R upper triangular, for a given real