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 (notably on compressed sensing). The $L_1/2$ regularization, however, leads to a nonconvex, nonsmooth, and non-Lipschitz optimization downside that is tough to solve quick and efficiently. In this paper, through developing a threshoding illustration theory for $L_1/2$ regularization, we have a tendency to propose an iterative $0.5$ thresholding algorithm for quick answer of $L_1/2$ regularization, admire the well-known iterative $soft$ thresholding algorithm for $L_1$ regularization, and therefore the iterative $arduous$ thresholding algorithm for $L_0$ regularization. We prove the existence of the resolvent of gradient of $Vert xVert^1/2_1/2$, calculate its analytic expression, and establish an alternative feature theorem on solutions of $L_1/2$ regularization, primarily based on that a thresholding illustration of solutions of $L_1/2$ regularization is derived and an optimal regularization parameter setting rule is formulated. The developed theory provides a successful follow 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 $0.5$ thresholding algorithm and offer 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 quick solver for $L_1/2$ regularization. With the new algorithm, we tend to conduct a part diagram study to more demonstrate the superiority of $L_1/2$ regularization over $L_1$ regularization.

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