Total least squares problems on infinite dimensional spaces

TL;DR

This paper investigates weighted total least squares problems in infinite dimensional spaces, revealing non-existence issues and proposing regularization techniques with conditions for solutions and subset restrictions to ensure solvability.

Contribution

It introduces regularization methods for infinite dimensional total least squares problems and provides conditions under which solutions exist.

Findings

Most infinite dimensional weighted total least squares problems lack solutions

Regularization can enable solutions in certain cases

Restricting to specific subsets guarantees solvability

Abstract

In this work we study weighted total least squares problems on infinite dimensional spaces. We show that in most cases this problem does not admit a solution (except in the trivial case) and then, we consider a regularization on the problem. We present necessary conditions for the regularized problem to have a solution. We also show that, by restricting the regularized minimization problem to special subsets, the existence of a solution may be assured.