On the Condition Number of the Total Least Squares Problem

TL;DR

This paper introduces new formulas and bounds for the condition number of the Total Least Squares problem, requiring only the SVD of the augmented matrix, which improves efficiency especially for large-scale problems.

Contribution

A novel closed-form formula for the TLS condition number using only the SVD of [A, b], along with sharp bounds that enhance computational efficiency.

Findings

The new formula accurately estimates the condition number.

Derived bounds are sharp and involve only the smallest singular values.

Numerical experiments confirm the bounds outperform existing methods.

Abstract

This paper concerns singular value decomposition (SVD)-based computable formulas and bounds for the condition number of the Total Least Squares (TLS) problem. For the TLS problem with the coefficient matrix $A$ and the right-hand side $b$, a new closed formula is presented for the condition number. Unlike an important result in the literature that uses the SVDs of both $A$ and $[A,\ b]$, our formula only requires the SVD of $[A,\ b]$. Based on the closed formula, both lower and upper bounds for the condition number are derived. It is proved that they are always sharp and estimate the condition number accurately. A few lower and upper bounds are further established that involve at most the smallest two singular values of $A$ and of $[A,\ b]$. Tightness of these bounds is discussed, and numerical experiments are presented to confirm our theory and to demonstrate the improvement of our upper bounds over the two upper bounds due to Golub and Van Loan as well as Baboulin and Gratton. Such lower and upper bounds are particularly useful for large scale TLS problems since they require the computation of only a few singular values of $A$ and $[A, \ b]$ other than all the singular values of them.