SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 2, Page 608-615, January 2021.
Randomized Kaczmarz is a simple iterative method for finding solutions of linear systems $Ax = b$. We point out that the arising sequence $(x_k)_{k=1}^{\infty}$ tends to converge to the solution $x$ in an interesting way: generically, as $k \rightarrow \infty$, $x_k - x$ tends to the singular vector of $A$ corresponding to the smallest singular value. This has interesting consequences: in particular, the error analysis of Strohmer and Vershynin is optimal. It also quantifies the “preconvergence” phenomenon where the method initially seems to converge faster. This fact also allows for a fast computation of vectors $x$ for which the Rayleigh quotient $\|Ax\|/\|x\|$ is small: solve $Ax = 0$ via randomized Kaczmarz.

中文翻译：

---

随机 Kaczmarz 沿小奇异向量收敛

SIAM Journal on Matrix Analysis and Applications，第 42 卷，第 2 期，第 608-615 页，2021 年 1 月。
随机 Kaczmarz 是一种用于寻找线性系统 $Ax = b$ 解的简单迭代方法。我们指出，出现的序列 $(x_k)_{k=1}^{\infty}$ 倾向于以一种有趣的方式收敛到解 $x$：一般来说，作为 $k \rightarrow \infty$, $x_k - x$ 趋向于$A$ 的奇异向量对应的最小奇异值。这产生了有趣的结果：特别是 Strohmer 和 Vershynin 的错误分析是最优的。它还量化了“预收敛”现象，其中该方法最初似乎收敛得更快。这一事实还允许快速计算向量 $x$，其中瑞利商 $\|Ax\|/\|x\|$ 很小：通过随机化 Kaczmarz 求解 $Ax = 0$。