Nuclear Norms of Rank 2 Matrices for Spectral Condition Numbers of Full Rank Linear Least Squares Solutions

TL;DR

This paper derives bounds on the spectral condition numbers of full rank linear least squares solutions using nuclear norms of rank 2 matrices, highlighting the factors influencing ill-conditioning and improving understanding of these bounds.

Contribution

It introduces a nuclear norm-based approach to derive bounds on the condition number, clarifying why a closed-form formula is unlikely.

Findings

Provides the best known lower and upper bounds on the condition number

Shows the condition number depends on three contributing quantities

Highlights limitations of existing estimates and overestimates

Abstract

The condition number of solutions to full rank linear least-squares problem are shown to be given by an optimization problem that involves nuclear norms of rank 2 matrices. The condition number is with respect to the least-squares coefficient matrix and 2-norms. It depends on three quantities each of which can contribute ill-conditioning. The literature presents several estimates for this condition number with varying results; even standard reference texts contain serious overestimates. The use of the nuclear norm affords a single derivation of the best known lower and upper bounds on the condition number and shows why there is unlikely to be a closed formula.