On the extended randomized multiple row method for solving linear least-squares problems

TL;DR

This paper introduces an extended randomized multiple row method for efficiently solving overdetermined and inconsistent linear least-squares problems, with proven linear convergence and practical applications demonstrated.

Contribution

It extends the randomized row method to handle inconsistent systems and analyzes its convergence and complexity, providing new insights and methods for least-squares solutions.

Findings

The method converges linearly in mean square to the minimum norm least-squares solution.

Numerical studies confirm theoretical convergence and efficiency.

Applications include image reconstruction and noisy data fitting.

Abstract

The randomized row method is a popular representative of the iterative algorithm because of its efficiency in solving the overdetermined and consistent systems of linear equations. In this paper, we present an extended randomized multiple row method to solve a given overdetermined and inconsistent linear system and analyze its computational complexities at each iteration. We prove that the proposed method can linearly converge in the mean square to the least-squares solution with a minimum Euclidean norm. Several numerical studies are presented to corroborate our theoretical findings. The real-world applications, such as image reconstruction and large noisy data fitting in computer-aided geometric design, are also presented for illustration purposes.