On Randomized Sampling Kaczmarz Method with Application in Compressed Sensing

Abstract

We propose a randomized sampling Kaczmarz algorithm for the solution of very large systems of linear equations by introducing a maximal sampling probability control criterion, which is aimed at grasping the largest entry of the absolute sampling residual vector at each iteration. This new method differs from the greedy randomized Kaczmarz algorithm, which needs not to compute the residual vector of the whole linear system to determine the working rows. Numerical experiments show that the proposed algorithm has the most significant effect when the selected row number, i.e, the size of samples, is equal to the logarithm of all rows. Finally, we extend the randomized sampling Kaczmarz to signal reconstruction problems in compressed sensing. Signal experiments show that the new extended algorithm is more effective than the randomized sparse Kaczmarz method for online compressed sensing.