Block Kaczmarz–Motzkin Method via Mean Shift Clustering

Abstract

Solving systems of linear equations is a fundamental problem in mathematics. Combining mean shift clustering (MS) with greedy techniques, a novel block version of the Kaczmarz–Motzkin method (BKMS), where the blocks are predetermined by MS clustering, is proposed in this paper. Using a greedy strategy, which collects the row indices with the almost maximum distance of the linear subsystem per iteration, can be considered an efficient extension of the sampling Kaczmarz–Motzkin algorithm (SKM). The new method linearly converges to the least-norm solution when the system is consistent. Several examples show that the BKMS algorithm is more efficient compared with other methods (for example, RK, Motzkin, GRK, SKM, RBK, and GRBK).