The special importance of regularization has been recognized in recent studies on sparse modeling (particularly on compressed sensing). The 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 regularization, we propose an iterative thresholding algorithm for fast solution of regularization, corresponding to the well-known iterative thresholding algorithm for regularization, and the iterative thresholding algorithm for regularization. We prove the existence of the resolvent of gradient of , calculate its analytic expression, and establish an alternative feature theorem on solutions of regularization, based on which a thresholding representation of solutions of 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 regularization case. We verify the convergence of the iterative thresholding algorithm and provide a series of experiments to assess performance of the algorithm. The experiments show that the algorithm is effective, efficient, and can be accepted as a fast solver for regularization. With the new algorithm, we conduct a phase diagram study to further demonstrate the superiority of regularization over regularization.