L Regularization: A Thresholding Representation Theory and a Fast Solver - Xu, Zongben; Chang, Xiangyu; Xu, Fengmin; Zhang, Hai (IEEE transaction on neural networks and learning systems, 2012)
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 L_{1} regularization, and the iterative hard thresholding algorithm for L_{0} regularization. We prove the existence of the resolvent of gradient of \Vert x\Vert^{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 L_{1} regularization.