Orthogonally weighted $\ell_{2,1}$ regularization for rank-aware joint
  sparse recovery: algorithm and analysis
Armenak Petrosyan Konstantin Pieper Hoang Tran
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.