$L_{1/2}$ Regularization: A Thresholding Representation Theory and a Fast Solver

The special importance of L_1/2 regularization has been recognized in recent studies on sparse modeling (particularly on compressed sensing). The L_1/2 regularization, however, leads to a nonconvex, nonsmooth, and non-Lipschitz optimization problem that is difficult to solve fast and efficiently. In this paper, through developing a threshoding representation theory for L_1/2 regularization, we propose an iterative half thresholding algorithm for fast solution of L_1/2 regularization, corresponding to the well-known iterative soft thresholding algorithm for L₁ regularization, and the iterative hard thresholding algorithm for L₀ regularization. We prove the existence of the resolvent of gradient of ||x||_1/2^1/2, calculate its analytic expression, and establish an alternative feature theorem on solutions of L_1/2 regularization, based on which a thresholding representation of solutions of L_1/2 regularization is derived and an optimal regularization parameter setting rule is formulated. The developed theory provides a successful practice of extension of the well- known Moreau´s proximity forward-backward splitting theory to the L_1/2 regularization case. We verify the convergence of the iterative half thresholding algorithm and provide a series of experiments to assess performance of the algorithm. The experiments show that the half algorithm is effective, efficient, and can be accepted as a fast solver for L_1/2 regularization. With the new algorithm, we conduct a phase diagram study to further demonstrate the superiority of L_1/2 regularization over L₁ regularization.