We investigate the randomized Kaczmarz method that adaptively updates the stepsize using readily available information for solving inconsistent linear systems. A novel geometric interpretation is provided which shows that the proposed method can be viewed as an orthogonal projection method in some sense. We prove that this method converges linearly in expectation to the unique minimum Euclidean norm least-squares solution of the linear system. Numerical experiments are given to illustrate the theoretical results.

翻译：本文研究利用现有信息自适应更新步长的随机Kaczmarz方法，以求解不相容线性系统。我们提供了一种新的几何解释，表明该方法在某种意义上可被视为正交投影法。我们证明该方法以期望线性收敛到线性系统的唯一最小欧几里得范数最小二乘解。数值实验验证了理论结果。