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.

翻译：本文提出并分析了一种基于新型正则化方法的高效算法，用于解决联合稀疏恢复问题。该方法命名为正交加权$\ell_{2,1}$（$\mathit{ow}\ell_{2,1}$），其设计充分考虑了解矩阵的秩。该算法可应用于特征提取、矩阵列选择及字典学习，与常用的$\ell_{2,1}$正则化及其他现有正则化方法不同，它能利用行稀疏解矩阵的满秩特性——这一特性在许多应用中至关重要。我们给出了该方法秩感知特性的理论证明，建立了所提优化问题解的存在性，并开发了高效求解算法，分析了其收敛性。最后通过数值实验验证理论结果，并展示该方法在实际问题中的有效性。