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The Extended Randomized Kaczmarz method is a well known iterative schemewhich can find the Moore-Penrose inverse solution of a possibly inconsistentlinear system and requires only one additional column of the system matrix ineach iteration in comparison with the standard randomized Kaczmarz method.Also, the Sparse Randomized Kaczmarz method has been shown to converge linearlyto a sparse solution of a consistent linear system. Here, we combine both ideasand propose an Extended Sparse Randomized Kaczmarz method. We show linearexpected convergence to a sparse least squares solution in the sense that anextended variant of the regularized basis pursuit problem is solved. Moreover,we generalize the additional step in the method and prove convergence to a moreabstract optimization problem. We demonstrate numerically that our method canfind sparse least squares solutions of real and complex systems if the noise isconcentrated in the complement of the range of the system matrix and that ourgeneralization can handle impulsive noise.

Extended Randomized Kaczmarz Method for Sparse Least Squares and Impulsive Noise Problems