Implicitly-weighted total least squares

In a total least squares (TLS) problem, we estimate an optimal set of model parameters X, so that (A-ΔA)X=B-ΔB, where A is the model matrix, B is the observed data, and ΔA and ΔB are corresponding corrections. When B is a single vector, Rao (1997) and Paige and Strakoš (2002) suggested formulating standard least squares problems, for which ΔA=0, and data least squares problems, for which ΔB=0, as weighted and scaled TLS problems. In this work we define an implicitly-weighted TLS formulation (ITLS) that reparameterizes these formulations to make computation easier. We derive asymptotic properties of the estimates as the number of rows in the problem approaches infinity, handling the rank-deficient case as well. We discuss the role of the ratio between the variances of errors in A and B in choosing an appropriate parameter in ITLS. We also propose methods for computing the family of solutions efficiently and for choosing the appropriate solution if the ratio of variances is unknown. We provide experimental results on the usefulness of the ITLS family of solutions.