# Preasymptotic Convergence of Randomized Kaczmarz Method

arXiv: https://arxiv.org/abs/1706.08459 · HTML: https://mlsanity.com/paper/1706.08459/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Jun 2017.

## Built on by (most cited first, 1 of 1)

- [From Order to Distribution: An Exact Operator Framework for Forgetting in Continual Learning](https://mlsanity.com/paper/2604.13460/) (2026)

## Abstract (arXiv)

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.