A randomized sparse Kaczmarz solver for sparse signal recovery via minimax-concave penalty

The randomized sparse Kaczmarz (RSK) method is an algorithm used to calculate sparse solutions for the basis pursuit problem. In this paper, we propose an algorithm framework for computing sparse solutions of linear systems, which includes the sparse Kaczmarz and sparse block Kaczmarz algorithms. In order to overcome the limitations of the l1$$ {\ell}_1 $$ penalty, we design an effective and new randomized sparse Kaczmarz algorithm (RSK-MCP) based on the non-convex minimax-concave penalty (MCP) in sparse signal reconstruction. Additionally, we prove that the RSK-MCP algorithm is equivalent to the randomized coordinate descent method for the corresponding dual problem. Based on this result, we demonstrate that the RSK-MCP algorithm exhibits linear convergence, meaning it converges to a sparse solution of the MCP model when the regularization of MCP is a strongly convex function. Numerical experiments indicate that the RSK-MCP algorithm outperforms RSK-L1 in terms of both efficiency and accuracy.