Preasymptotic Convergence of Randomized Kaczmarz Method
Provides theoretical insights into the fast convergence of randomized Kaczmarz for practitioners in inverse problems, though the analysis is incremental.
The paper analyzes the preasymptotic convergence of the randomized Kaczmarz method, showing that low-frequency error decays faster than high-frequency error in early iterations, explaining its fast empirical convergence for smooth inverse solutions. A variance reduction strategy is proposed to stabilize asymptotic convergence.
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.