L1/2 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 hal f thresholding algorithm for fast solution of L-1/2 regularization, corresponding to the well-known iterative sof t thresholding algorithm for L-1 regularization, and the iterative hard thresholding algorithm for L-0 regularization. We prove the existence of the resolvent of gradient of parallel to x parallel to(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 hal f thresholding algorithm and provide a series of experiments to assess performance of the algorithm. The experiments show that the hal f 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 L-1 regularization.