Efficient ℓq Minimization Algorithms for Compressive Sensing Based on Proximity Operator

This paper considers solving the unconstrained lq-norm (0 ≤ q < 1) regularized least squares (lq-LS) problem for recovering sparse signals in compressive sensing. We propose two highly efficient first-order algorithms via incorporating the proximity operator for nonconvex lq-norm functions into the fast iterative shrinkage/thresholding (FISTA) and the alternative direction method of multipliers (ADMM) frameworks, respectively. Furthermore, in solving the nonconvex lq-LS problem, a sequential minimization strategy is adopted in the new algorithms to gain better global convergence performance. Unlike most existing lqminimization algorithms, the new algorithms solve the lq-minimization problem without smoothing (approximating) the lq-norm. Meanwhile, the new algorithms scale well for large-scale problems, as often encountered in image processing. We show that the proposed algorithms are the fastest methods in solving the nonconvex lq-minimization problem, while offering competent performance in recovering sparse signals and compressible images compared with several state-of-the-art algorithms. Index Terms Compressive sensing (CS), sparse signal recovery, alternating direction method of multipliers, iterative shrinkage-thresholding, nonconvex lq regularization.