On Convergence Behavior of Randomized Kaczmarz-type Methods for Solving Doubly Noisy Linear Systems

Yudan Gan; Gang Wu

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arXiv · 2608.22259

On Convergence Behavior of Randomized Kaczmarz-type Methods for Solving Doubly Noisy Linear Systems

Yudan Gan·Gang Wu

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Abstract

In this paper, we investigate the limiting behavior of the RK algorithm for solving doubly noisy inconsistent linear systems without imposing any additional initial assumptions. The proposed bounds effectively characterize the convergence behavior of these algorithms when applied to doubly noisy linear systems. Furthermore, to the best of our knowledge, this work provides the first convergence analysis of the randomized extended Kaczmarz (REK), randomized block Kaczmarz (RBK), and randomized double block Kaczmarz (RDBK) algorithms for doubly noisy linear systems. We prove that these algorithms converge to a neighborhood of the least-squares solution of the underlying noiseless system. These results show that RK and RBK, designed for consistent systems, outperform REK and RDBK, designed for inconsistent systems, in both the convergence rate and the convergence horizon. What's more, the analytical skills to the sketch-and-project method completely overcome the limitation when analyzing RK and RBK. Finally, numerical experiments are conducted to validate the theoretical results.

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Yudan Gan, Gang Wu. 2026-09-17. On Convergence Behavior of Randomized Kaczmarz-type Methods for Solving Doubly Noisy Linear Systems. https://arxiv.org/abs/2608.22259

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