# Preasymptotic Convergence of Randomized Kaczmarz Method

## TL;DR

This paper investigates the initial convergence behavior of the randomized Kaczmarz method, revealing that low-frequency errors diminish faster initially, and proposes a variance reduction strategy to stabilize long-term convergence, supported by numerical experiments.

## Contribution

It provides the first detailed analysis of preasymptotic convergence of the randomized Kaczmarz method and introduces a variance reduction technique for improved stability.

## Key findings

- Low-frequency errors decay faster initially
- Smooth inverse solutions lead to rapid early convergence
- Variance reduction stabilizes asymptotic convergence

## Abstract

Kaczmarz method is one popular iterative method for solving inverse problems, especially in computed tomography. Recently, it was established that a randomized version of the method enjoys an exponential convergence for well-posed problems, and the convergence rate is determined by a variant of the condition number. In this work, we analyze the preasymptotic convergence behavior of the randomized Kaczmarz method, and show that the low-frequency error (with respect to the right singular vectors) decays faster during first iterations than the high-frequency error. Under the assumption that the inverse solution is smooth (e.g., sourcewise representation), the result explains the fast empirical convergence behavior, thereby shedding new insights into the excellent performance of the randomized Kaczmarz method in practice. Further, we propose a simple strategy to stabilize the asymptotic convergence of the iteration by means of variance reduction. We provide extensive numerical experiments to confirm the analysis and to elucidate the behavior of the algorithms.