Fast nonnegative least squares through flexible Krylov subspaces

TL;DR

This paper introduces a novel, efficient method for solving nonnegative least squares problems using flexible Krylov subspaces, significantly improving speed while maintaining or enhancing solution quality, especially in image reconstruction applications.

Contribution

The paper presents a new approach leveraging KKT conditions and flexible Krylov subspaces for faster nonnegative least squares solutions, with theoretical analysis and practical validation.

Findings

Achieves equal or better solution quality than existing methods.

Provides significant computational speedup in numerical experiments.

Effectively handles Gaussian and Poisson noise in image reconstruction.

Abstract

Constrained least squares problems arise in a variety of applications, and many iterative methods are already available to compute their solutions. This paper proposes a new efficient approach to solve nonnegative linear least squares problems. The associated KKT conditions are leveraged to form an adaptively preconditioned linear system, which is then solved by a flexible Krylov subspace method. The new method can be easily applied to image reconstruction problems affected by both Gaussian and Poisson noise, where the components of the solution represent nonnegative intensities. {Theoretical insight is given, and} numerical experiments and comparisons are displayed in order to validate the new method, which delivers results of equal or better quality than many state-of-the-art methods for nonnegative least squares solvers, with a significant speedup.