$L_{1/2}$ Regularization: Convergence of Iterative Half Thresholding Algorithm

In recent studies on sparse modeling, the nonconvex regularization approaches (particularly, L_q regularization with q ∈ (0,1)) have been demonstrated to possess capability of gaining much benefit in sparsity-inducing and efficiency. As compared with the convex regularization approaches (say, L₁ regularization), however, the convergence issue of the corresponding algorithms are more difficult to tackle. In this paper, we deal with this difficult issue for a specific but typical nonconvex regularization scheme, the L_1/2 regularization, which has been successfully used to many applications. More specifically, we study the convergence of the iterative half thresholding algorithm (the half algorithm for short), one of the most efficient and important algorithms for solution to the L_1/2 regularization. As the main result, we show that under certain conditions, the half algorithm converges to a local minimizer of the L_1/2 regularization, with an eventually linear convergence rate. The established result provides a theoretical guarantee for a wide range of applications of the half algorithm. We provide also a set of simulations to support the correctness of theoretical assertions and compare the time efficiency of the half algorithm with other known typical algorithms forL_1/2 regularization like the iteratively reweighted least squares (IRLS) algorithm and the iteratively reweighted l₁ minimization (IRL1) algorithm.