Acceleration of Randomized Kaczmarz Method via the Johnson-Lindenstrauss Lemma
This work improves the convergence rate of the randomized Kaczmarz method for solving overdetermined linear systems, which is relevant for numerical linear algebra and large-scale optimization.
The authors propose a modified randomized Kaczmarz method that selects the optimal projection from a random set, using Johnson-Lindenstrauss dimension reduction to maintain runtime. Empirical studies show significant acceleration in convergence compared to the original method.
The Kaczmarz method is an algorithm for finding the solution to an overdetermined consistent system of linear equations Ax=b by iteratively projecting onto the solution spaces. The randomized version put forth by Strohmer and Vershynin yields provably exponential convergence in expectation, which for highly overdetermined systems even outperforms the conjugate gradient method. In this article we present a modified version of the randomized Kaczmarz method which at each iteration selects the optimal projection from a randomly chosen set, which in most cases significantly improves the convergence rate. We utilize a Johnson-Lindenstrauss dimension reduction technique to keep the runtime on the same order as the original randomized version, adding only extra preprocessing time. We present a series of empirical studies which demonstrate the remarkable acceleration in convergence to the solution using this modified approach.