A fast block sparse Kaczmarz algorithm for sparse signal recovery

The randomized sparse Kaczmarz (RSK) method is an iterative algorithm for computing sparse solutions of linear systems. Recently, Tondji and Lorenz analyzed the parallel version of the RSK method and established its linear expected convergence by implementing a randomized control scheme for subset selection at each iteration. Expanding upon this groundwork, we explore a natural extension of the randomized control scheme: greedy strategies such as the Motzkin criteria. Specifically, we propose a fast block sparse Kaczmarz algorithm based on the Motzkin criterion. It is proved that the proposed method converges linearly to the sparse solutions of the linear systems. Additionally, we offer error estimates for linear systems with noisy right-hand sides, and show that the proposed method converges within an error threshold of the noise level. Numerical results substantiate the feasibility of our proposed method and highlight its superior convergence rate compared to the parallel version of the RSK method.