A fast algorithm for group square-root Lasso based group-sparse regression

Abstract 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.