Perturbation Identities for Regularized Tikhonov Inverses and Weighted Pseudoinverses

We consider the perturbation analysis of two important problems for solving ill-conditioned or rank-deficient linear least squares problems. The Tikhonov regularized problem is a linear least squares problem with a regularization term balancing the size of the residual against the size of the weighted solution. The weight matrix can be a non-square matrix (usually with fewer rows than columns). The minimum-norm problem is the minimization of the size of the weighted solutions given by the set of solutions to the, possibly rank-deficient, linear least squares problem.It is well known that the solution of the Tikhonov problem tends to the minimum-norm solution as the regularization parameter of the Tikhonov problem tends to zero. Using this fact and the generalized singular value decomposition enable us to make a perturbation analysis of the minimum-norm problem with perturbation results for the Tikhonov problem. From the analysis we attain perturbation identities for Tikhonov inverses and weighted pseudoinverses.     

Randomized algorithms for total least squares problems

Motivated by the recently popular probabilistic methods for low‐rank approximations and randomized algorithms for the least squares problems, we develop randomized algorithms for the total least squares problem with a single right‐hand side. We present the Nyström method for the medium‐sized problems. For the large‐scale and ill‐conditioned cases, we introduce the randomized truncated total least squares with the known or estimated rank as the regularization parameter. We analyze the accuracy of the algorithm randomized truncated total least squares and perform numerical experiments to demonstrate the efficiency of our randomized algorithms. The randomized algorithms can greatly reduce the computational time and still maintain good accuracy with very high probability.     

加权总体最小二乘问题的解集和性质

本文讨论了加权总体最小二乘问题的等价解集，分析了加权总体最小二乘解与加权最小二乘问题的解之间的关系。推广了Ｇｏｌｕｂ和Ｖａｎ Ｌｏａｎ，Ｖａｎ Ｈｕｆｆｅｌ和Ｖａｎｄｅｗａｌｌｅ，及Ｗｅｉ的相应结果。     

Algorithms for the regularization of ill-conditioned least squares problems

Two regularization methods for ill-conditioned least squares problems are studied from the point of view of numerical efficiency. The regularization methods are formulated as quadratically constrained least squares problems, and it is shown that if they are transformed into a certain standard form, very efficient algorithms can be used for their solution. New algorithms are given, both for the transformation and for the regularization methods in standard form. A comparison to previous algorithms is made and it is shown that the overall efficiency (in terms of the number of arithmetic operations) of the new algorithms is better.     

Minimizing the sum of linear fractional functions over the cone of positive semidefinite matrices: Approximation and applications

The problem of maximizing the sum of two generalized Rayleigh quotients and the total least squares problem with nonsingular Tikhonov regularization are reformulated as a class of sum-of-linear-ratios minimizing over the cone of symmetric positive semidefinite matrices, which is shown to have a Fully Polynomial Time Approximation Scheme.     

Regularization tools and robust optimization for ill-conditioned least squares problem: A computational comparison

Least squares problems arise frequently in many disciplines such as image restorations. In these areas, for the given least squares problem, usually the coefficient matrix is ill-conditioned. Thus if the problem data are available with certain error, then after solving least squares problem with classical approaches we might end up with a meaningless solution. Tikhonov regularization, is one of the most widely used approaches to deal with such situations. In this paper, first we briefly describe these approaches, then the robust optimization framework which includes the errors in problem data is presented. Finally, our computational experiments on several ill-conditioned standard test problems using the regularization tools, a Matlab package for least squares problem, and the robust optimization framework, show that the latter approach may be the right choice.     

A weighted least squares method for scattered data fitting

In this paper, we present a weighted least squares method to fit scattered data with noise. Existence and uniqueness of a solution are proved and an error bound is derived. The numerical experiments illustrate that our weighted least squares method has better performance than the traditional least squares method in case of noisy data.     

Robust regression for mixed Poisson–Gaussian model

This paper focuses on efficient computational approaches to compute approximate solutions of a linear inverse problem that is contaminated with mixed Poisson–Gaussian noise, and when there are additional outliers in the measured data. The Poisson–Gaussian noise leads to a weighted minimization problem, with solution-dependent weights. To address outliers, the standard least squares fit-to-data metric is replaced by the Talwar robust regression function. Convexity, regularization parameter selection schemes, and incorporation of non-negative constraints are investigated. A projected Newton algorithm is used to solve the resulting constrained optimization problem, and a preconditioner is proposed to accelerate conjugate gradient Hessian solves. Numerical experiments on problems from image deblurring illustrate the effectiveness of the methods.     

Direct Implementation of Tikhonov Regularization for the First Kind Integral Equation

A common way to handle the Tikhonov regularization method for the first kind Fredholm integral equations, is first to discretize and then to work with the final linear system. This unavoidably inflicts discretization errors which may lead to disastrous results, especially when a quadrature rule is used. We propose to regularize directly the integral equation resulting in a continuous Tikhonov problem. The Tikhonov problem is reduced to a simple least squares problem by applying the Golub-Kahan bidiagonalization (GKB) directly to the integral operator. The regularization parameter and the iteration index are determined by the discrepancy principle approach. Moreover, we study the discrete version of the proposed method resulted from numerical evaluating the needed integrals. Focusing on the nodal values of the solution results in a weighted version of GKB-Tikhonov method for linear systems arisen from the Nyström discretization. Finally, we use numerical experiments on a few test problems to illustrate the performance of our algorithms.     

Exponential curve fitting with least squares

A comparison is made between standard least squares and a weighted least squares technique for exponential data.     

Detection of outliers in weighted least squares regression

In multiple linear regression model, we have presupposed assumptions (independence, normality, variance homogeneity and so on) on error term. When case weights are given because of variance heterogeneity, we can estimate efficiently regression parameter using weighted least squares estimator. Unfortunately, this estimator is sensitive to outliers like ordinary least squares estimator. Thus, in this paper, we proposed some statistics for detection of outliers in weighted least squares regression.     

Error estimation for a weighted minimum-norm least squares solution with positive definite weights

The weighted least squares problem is considered. Given a generally inconsistent system of linear algebraic equations, error estimates are obtained for its weighted minimum-norm least squares solution under perturbations of the matrix and the right-hand side, including the case of rank modifications of the perturbed matrix.     

Regularization of differential-algebraic equations

Linear systems of ordinary differential equations with an identically singular or rectangular matrix multiplying the derivative of the unknown vector function are numerically solved by applying the least squares method and Tikhonov regularization. The deviation of the solution of the regularized problem from the solution set of the original problem is estimated depending on the regularization parameter.     

A weighted pseudoinverse,generalized singular values,and constrained least squares problems

The weighted pseudoinverse providing the minimum semi-norm solution of the weighted linear least squares problem is studied. It is shown that it has properties analogous to those of the Moore-Penrose pseudoinverse. The relation between the weighted pseudoinverse and generalized singular values is explained. The weighted pseudoinverse theory is used to analyse least squares problems with linear and quadratic constraints. A numerical algorithm for the computation of the weighted pseudoinverse is briefly described.This work was supported in part by the Swedish Institute for Applied Mathematics.     

Total variation-penalized Poisson likelihood estimation for ill-posed problems

The noise contained in data measured by imaging instruments is often primarily of Poisson type. This motivates, in many cases, the use of the Poisson negative-log likelihood function in place of the ubiquitous least squares data fidelity when solving image deblurring problems. We assume that the underlying blurring operator is compact, so that, as in the least squares case, the resulting minimization problem is ill-posed and must be regularized. In this paper, we focus on total variation regularization and show that the problem of computing the minimizer of the resulting total variation-penalized Poisson likelihood functional is well-posed. We then prove that, as the errors in the data and in the blurring operator tend to zero, the resulting minimizers converge to the minimizer of the exact likelihood function. Finally, the practical effectiveness of the approach is demonstrated on synthetically generated data, and a nonnegatively constrained, projected quasi-Newton method is introduced.     

TLS和LS问题的比较

There are a number of articles discussing the total least squares(TLS) and the least squares(LS) problems.M.Wei(M.Wei, Mathematica Numerica Sinica 20(3)(1998),267-278) proposed a new orthogonal projection method to improve existing perturbation bounds of the TLS and LS problems.In this paper,wecontinue to improve existing bounds of differences between the squared residuals,the weighted squared residuals and the minimum norm correction matrices of the TLS and LS problems.     

Latent scale linear models for multivariate ordinal responses and analysis by the method of weighted least squares

Summary A multivariate latent scale linear model is defined for multivariate ordered categorical responses and inference procedures based on the weighted least squares method are developed. Several applications of the model are suggested and illustrated through an analysis of real data. Asymptotic properties of the weighted least squares method are examined and some consequences of misspecification of the model are also discussed.     

Weighted least squares solutions to general coupled Sylvester matrix equations

This paper is concerned with weighted least squares solutions to general coupled Sylvester matrix equations. Gradient based iterative algorithms are proposed to solve this problem. This type of iterative algorithm includes a wide class of iterative algorithms, and two special cases of them are studied in detail in this paper. Necessary and sufficient conditions guaranteeing the convergence of the proposed algorithms are presented. Sufficient conditions that are easy to compute are also given. The optimal step sizes such that the convergence rates of the algorithms, which are properly defined in this paper, are maximized and established. Several special cases of the weighted least squares problem, such as a least squares solution to the coupled Sylvester matrix equations problem, solutions to the general coupled Sylvester matrix equations problem, and a weighted least squares solution to the linear matrix equation problem are simultaneously solved. Several numerical examples are given to illustrate the effectiveness of the proposed algorithms.     

ON STABLE PERTURBATIONS OF THE STIFFLY WEIGHTED PSEUDOINVERSE AND WEIGHTED LEAST SQUARES PROBLEM

In this paper we study perturbations of the stiffly weighted pseudoinverse （W^1/2 A）^＋W^1/2 and the related stiffly weighted least squares problem, where both the matrices A and W are given with W positive diagonal and severely stiff. We show that the perturbations to the stiffly weighted pseudoinverse and the related stiffly weighted least squares problem are stable, if and only if the perturbed matrices A = A ＋ δA satisfy several row rank preserving conditions.     

Local Weighted Composite Quantile Estimating for Varying Coefficient Models

A generalization of classical linear models is varying coefficient models, which offer a flexible approach to modeling nonlinearity between covariates. A method of local weighted composite quantile regression is suggested to estimate the coefficient functions. The local Bahadur representation of the local estimator is derived and the asymptotic normality of the resulting estimator is established. Comparing to the local least squares estimator, the asymptotic relative efficiency is examined for the local weighted composite quantile estimator. Both theoretical analysis and numerical simulations reveal that the local weighted composite quantile estimator can obtain more efficient than the local least squares estimator for various non-normal errors. In the normal error case, the local weighted composite quantile estimator is almost as efficient as the local least squares estimator. Monte Carlo results are consistent with our theoretical findings. An empirical application demonstrates the potential of the proposed method.