A Gauss--Newton iteration for Total Least Squares problems

TL;DR

This paper introduces a globally convergent Gauss-Newton iteration tailored for Total Least Squares problems, which involves solving a perturbed least squares problem at each step to find the solution minimizing backward error.

Contribution

It develops a novel Gauss-Newton based method with global convergence guarantees specifically for Total Least Squares problems, involving rank-one perturbations.

Findings

The method converges globally to the TLS solution.

Each iteration involves solving a rank-one perturbed least squares problem.

The approach effectively minimizes the backward error in TLS problems.

Abstract

The Total Least Squares solution of an overdetermined, approximate linear equation $Ax \approx b$ minimizes a nonlinear function which characterizes the backward error. We show that a globally convergent variant of the Gauss--Newton iteration can be tailored to compute that solution. At each iteration, the proposed method requires the solution of an ordinary least squares problem where the matrix $A$ is perturbed by a rank-one term.