Acceleration of Randomized Kaczmarz Method via the Johnson-Lindenstrauss Lemma

TL;DR

This paper introduces a modified randomized Kaczmarz method that employs Johnson-Lindenstrauss dimension reduction to significantly accelerate convergence while maintaining similar runtime complexity, supported by empirical results.

Contribution

It proposes a new variant of the randomized Kaczmarz method that selects optimal projections using Johnson-Lindenstrauss dimension reduction, improving convergence speed without increasing runtime.

Findings

Significant acceleration in convergence demonstrated empirically.

Maintains similar runtime complexity as the original method.

Effective dimension reduction with Johnson-Lindenstrauss technique.

Abstract

The Kaczmarz method is an algorithm for finding the solution to an overdetermined consistent system of linear equations Ax=b by iteratively projecting onto the solution spaces. The randomized version put forth by Strohmer and Vershynin yields provably exponential convergence in expectation, which for highly overdetermined systems even outperforms the conjugate gradient method. In this article we present a modified version of the randomized Kaczmarz method which at each iteration selects the optimal projection from a randomly chosen set, which in most cases significantly improves the convergence rate. We utilize a Johnson-Lindenstrauss dimension reduction technique to keep the runtime on the same order as the original randomized version, adding only extra preprocessing time. We present a series of empirical studies which demonstrate the remarkable acceleration in convergence to the solution using this modified approach.