# Preconditioning Kaczmarz method by sketching

## TL;DR

This paper introduces a novel preconditioning technique for the Kaczmarz method using sketching, which aims to improve convergence speed while reducing computational complexity through random sketching and QR decomposition.

## Contribution

The paper presents a new sketching-based preconditioning approach for the Kaczmarz method, offering a computationally efficient alternative to QR-based preconditioning.

## Key findings

- Sketching-based preconditioning improves convergence speed.
- The method is effective on both random and real systems.
- Numerical experiments validate the approach.

## Abstract

We propose a new method for preconditioning Kaczmarz method by sketching. Kaczmarz method is a stochastic method for solving overdetermined linear systems based on a sampling of matrix rows. The standard approach to speed up convergence of iterative methods is using preconditioner. As we show the best possible preconditioner for this method can be constructed from QR decomposition of the system matrix, but the complexity of this procedure is too high. Therefore, to reduce this complexity, we use random sketching and compare it with the Kaczmarz method without preconditioning. The developed method is applicable for different modifications of classical Kaczmarz method that were proposed recently. We provide numerical experiments to show the performance of the developed methods on solving both random and real overdetermined linear systems.