A Variational Equation and Lower Bound for the Linear Least-Squares Backward Error

TL;DR

This paper introduces a new variational equation for the linear least-squares backward error, linking it to eigenvalue problems and proposing a sketching-based lower bound with practical applications.

Contribution

It derives a novel variational equation for backward error, decomposes it for multiple right-hand sides, and proposes a new sketching-based lower bound with theoretical guarantees.

Findings

Backward error expressed via generalized eigenvalue problem.

Decomposition of backward error for multiple right-hand sides.

Proposed lower bound is comparable to existing sketched estimates.

Abstract

This paper derives a new variational equation for the linear least-squares backward error by expressing the backward error in terms of a generalized eigenvalue problem and using results from indefinite linear algebra. For problems with multiple right-hand sides, the variational equation also shows that the backward error can be decomposed as a sum of smaller backward error problems. Applications to stopping criteria for iterative methods are considered, and a new sketching-based lower bound is proposed which is provably of quality comparable to the sketched Karlson-Wald\'{e}n estimate.