发表机构
Imagination Corp.; Fictional University; School of Mathematics, China University of Mining and Technology（Imagination公司; 虚构大学; 中国矿业大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对双重噪声线性系统，首次分析了REK、RBK、RDBK等随机Kaczmarz型算法的收敛性，证明其收敛到无噪声系统最小二乘解邻域，并通过数值实验验证了理论结果。
AI 中文摘要
随机Kaczmarz（RK）方法是一种高效的迭代投影算法，计算复杂度低，用于求解相容线性系统。然而，在实际应用中噪声不可避免，系数矩阵和右端向量都可能被噪声污染。针对系统矩阵和测量向量均受扰动的双重噪声线性系统，RK型方法的收敛分析仍相对有限。本文研究了在不施加任何额外初始假设的情况下，RK算法求解双重噪声不相容线性系统的极限行为。此外，据我们所知，本工作首次对随机扩展Kaczmarz（REK）、随机块Kaczmarz（RBK）和随机双块Kaczmarz（RDBK）算法在双重噪声线性系统上的收敛性进行了分析。我们证明这些算法收敛到基础无噪声系统最小二乘解的邻域。与现有理论估计相比，所提出的界能有效表征这些算法应用于双重噪声线性系统时的收敛行为。最后，开展数值实验以验证理论结果。
英文摘要
In this paper, we consider the convergence of some popular Kaczmarz-type methods, including the randomized Kaczmarz (RK) method, the randomized extended Kaczmarz (REK) method, the randomized block Kaczmarz (RBK) method, and the randomized double block Kaczmarz (RDBK) method for doubly noisy linear system. To the best of our knowledge, this is the first work that considers and compares the convergence of the Kaczmarz-type methods for consistent linear systems, as well as that for inconsistent linear systems to solve double noisy linear system. We prove that these methods converge to a neighborhood of the smallest norm least squares solution of the underlying noiseless system. Specifically, RK and RBK that are designed for consistent linear systems, often outperform REK and RDBK that are designed for inconsistent linear systems, respectively, in terms of the convergence rate or the convergence horizon. Moreover, we establish the convergence of the more general sketch-and-project framework for the doubly noisy linear system. The analysis provides more comprehensive results without imposing any additional restrictions. Numerical experiments are performed to validate our theoretical results.
