通过Johnson-Lindenstrauss引理加速随机Kaczmarz方法
Acceleration of Randomized Kaczmarz Method via the Johnson-Lindenstrauss Lemma
AI总结：
本文提出一种改进的随机Kaczmarz方法，通过从随机选择的集合中选择最优投影，并利用Johnson-Lindenstrauss降维技术保持运行时间，显著加速收敛。
摘要：
Kaczmarz方法是一种通过迭代投影到解空间来求解超定一致线性方程组Ax=b的算法。Strohmer和Vershynin提出的随机版本在期望上具有可证明的指数收敛速度，对于高度超定的系统甚至优于共轭梯度法。本文提出了一种改进的随机Kaczmarz方法，每次迭代从随机选择的集合中选择最优投影，在大多数情况下显著提高了收敛速度。我们利用Johnson-Lindenstrauss降维技术将运行时间保持在原始随机版本的同一量级，仅增加了额外的预处理时间。我们通过一系列实证研究展示了这种改进方法在收敛到解方面的显著加速。
英文摘要：
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.
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