A Gauss--Newton iteration for Total Least Squares problems
For practitioners needing robust solutions to overdetermined linear systems with errors in both A and b, this offers a computationally efficient and globally convergent alternative.
The paper presents a globally convergent Gauss-Newton iteration for solving Total Least Squares problems, requiring only the solution of a rank-one perturbed ordinary least squares problem per iteration.
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.