Group square-root Lasso (GSRL) is a promising tool for group-sparse regression since the hyperparameter is independent of noise level. Recent works also reveal its connections to some statistically sound and hyperparameter-free methods, e.g., group-sparse iterative covariance-based estimation (GSPICE). However, the non-smoothness of the data-fitting term leads to the difficulty in solving the optimization problem of GSRL, and available solvers usually suffer either slow convergence or restrictions on the dictionary. In this paper, we propose a class of efficient solvers for GSRL in a block coordinate descent manner, including group-wise cyclic minimization (GCM) for group-wise orthonormal dictionary and generalized GCM (G-GCM) for general dictionary. Both strict descent property and global convergence are proved. To cope with signal processing applications, the complex-valued multiple measurement vectors (MMV) case is considered. The proposed algorithm can also be used for the fast implementation of methods with theoretical equivalence to GSRL, e.g., GSPICE. Significant superiority in computational efficiency is verified by simulation results.

平方根拉索（GSRL）组是用于组稀疏回归的有前途的工具，因为超参数与噪声水平无关。最近的工作还揭示了它与某些统计上合理且无超参数的方法的联系，例如，基于组稀疏迭代协方差的估计（GSPICE）。但是，数据拟合项的不平滑性导致难以解决GSRL的优化问题，并且可用的求解器通常会遇到收敛速度慢或字典受限制的问题。在本文中，我们提出了一种以块坐标下降的方式求解GSRL的有效解算器，其中包括用于分组正交字典的逐组循环最小化（GCM）和用于常规字典的广义GCM（G-GCM）。证明了严格的下降特性和全局收敛性。为了应对信号处理应用，考虑了复值多个测量向量（MMV）的情况。所提出的算法还可以用于与GSRL具有理论等效性的方法（例如GSPICE）的快速实现。仿真结果证明了计算效率的显着优势。