text: Kaczmarz algorithm is an efficient iterative algorithm to solve overdetermined consistent system of linear equations. During each updating step, Kaczmarz chooses a hyperplane based on an individual equation and projects the current estimate for the exact solution onto that space to get a new estimate. Many vairants of Kaczmarz algorithms are proposed on how to choose better hyperplanes. Using the property of randomly sampled data in high-dimensional space, we propose an accelerated algorithm based on clustering information to improve block Kaczmarz and Kaczmarz via Johnson-Lindenstrauss lemma. Additionally, we theoretically demonstrate convergence improvement on block Kaczmarz algorithm.
  @cite_9: 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.
summary: The exact greedy idea has been highlighted in @cite_9 when they proposed RKA-JL algorithm. The idea is quite simple but reasonable. At the @math -th updating step, we can choose the hyperplane @math to project on where @math and update the estimate as Eq). But, after thinking of the practical issue, we may find that it is unaffordable to sweep through all rows of @math to pick the best one at each iteration. Thus, @cite_9 proposed a procedure to approximate the best hyperplane before performing each updating step, which is the key component of RKA-JL algorithm.