# Regularization: A Thresholding Representation Theory and a Fast Solver

PROJECT TITLE :

Regularization: A Thresholding Representation Theory and a Fast Solver

ABSTRACT:

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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