Spectral Condition Numbers of Orthogonal Projections and Full Rank Linear Least Squares Residuals

TL;DR

This paper derives a tight, simple formula for the condition number of full rank linear least squares residuals, revealing key factors influencing ill-conditioning and improving upon previous estimates.

Contribution

It introduces a new, tight estimate for the condition number of least squares residuals, surpassing prior bounds and clarifying the factors affecting numerical stability.

Findings

The formula is tight and provides accurate condition number estimates.

Two of the three key quantities can cause ill-conditioning.

Previous estimates tend to overestimate the condition number significantly.

Abstract

A simple formula is proved to be a tight estimate for the condition number of the full rank linear least squares residual with respect to the matrix of least squares coefficients and scaled 2-norms. The tight estimate reveals that the condition number depends on three quantities, two of which can cause ill-conditioning. The numerical linear algebra literature presents several estimates of various instances of these condition numbers. All the prior values exceed the formula introduced here, sometimes by large factors.