Fast L1-L2 minimization via a proximal operator

This paper aims to develop new and fast algorithms for recovering a sparse vector from a small number of measurements, which is a fundamental problem in the field of compressive sensing (CS). Currently, CS favors incoherent systems, in which any two measurements are as little correlated as possible. In reality, however, many problems are coherent, and conventional methods such as L1 minimization do not work well. Recently, the difference of the L1 and L2 norms, denoted as L1-L2, is shown to have superior performance over the classic L1 method, but it is computationally expensive. We derive an analytical solution for the proximal operator of the L1-L2 metric, and it makes some fast L1 solvers such as forward-backward splitting (FBS) and alternating direction method of multipliers (ADMM) applicable for L1-L2. We describe in details how to incorporate the proximal operator into FBS and ADMM and show that the resulting algorithms are convergent under mild conditions. Both algorithms are shown to be much more efficient than the original implementation of L1-L2 based on a difference-of-convex approach in the numerical experiments.