Randomized Block Kaczmarz Method with Projection for Solving Least Squares

TL;DR

This paper introduces two block randomized Kaczmarz methods with projections that efficiently compute least squares solutions for inconsistent linear systems, improving convergence speed through matrix paving techniques.

Contribution

It extends the randomized Kaczmarz method to block versions that converge to least squares solutions and demonstrates accelerated convergence via matrix paving.

Findings

Methods converge exponentially in expectation

Paving improves performance in certain regimes

Numerical experiments show practical advantages

Abstract

The Kaczmarz method is an iterative method for solving overcomplete linear systems of equations Ax=b. The randomized version of the Kaczmarz method put forth by Strohmer and Vershynin iteratively projects onto a randomly chosen solution space given by a single row of the matrix A and converges exponentially in expectation to the solution of a consistent system. In this paper we analyze two block versions of the method each with a randomized projection, that converge in expectation to the least squares solution of inconsistent systems. Our approach utilizes a paving of the matrix A to guarantee exponential convergence, and suggests that paving yields a significant improvement in performance in certain regimes. The proposed method is an extension of the block Kaczmarz method analyzed by Needell and Tropp and the Randomized Extended Kaczmarz method of Zouzias and Freris. The contribution is thus two-fold; unlike the standard Kaczmarz method, our methods converge to the least-squares solution of inconsistent systems, and by using appropriate blocks of the matrix this convergence can be significantly accelerated. Numerical experiments suggest that the proposed algorithm can indeed lead to advantages in practice.