Orthogonally weighted $\ell_{2,1}$ regularization for rank-aware joint sparse recovery: algorithm and analysis

We propose and analyze an efficient algorithm for solving the joint sparse recovery problem using a new regularization-based method, named orthogonally weighted $\ell_{2,1}$ ($\mathit{ow}\ell_{2,1}$), which is specifically designed to take into account the rank of the solution matrix. This method has applications in feature extraction, matrix column selection, and dictionary learning, and it is distinct from commonly used $\ell_{2,1}$ regularization and other existing regularization-based approaches because it can exploit the full rank of the row-sparse solution matrix, a key feature in many applications. We provide a proof of the method's rank-awareness, establish the existence of solutions to the proposed optimization problem, and develop an efficient algorithm for solving it, whose convergence is analyzed. We also present numerical experiments to illustrate the theory and demonstrate the effectiveness of our method on real-life problems.