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The least-square regression problems or inverse problems have been widely studied in many fields such as compressive sensing, signal processing, and image processing. To solve this kind of ill-posed problems, a regularization term (i.e., regularizer) should be introduced, under the assumption that the solutions have some specific properties, such as sparsity and group sparsity. Widely used regularizers include the l1 norm, total variation (TV) semi-norm, and so on. Recently, a new regularization term with overlapping group sparsity has been considered. Majorization minimization iteration method or variable duplication methods are often applied to solve them. However, there have been no direct methods for solving the relevant problems due to the difficulty of overlapping. In this paper, we proposed new inexact explicit shrinkage formulas for one class of these relevant problems, whose regularization terms have translation invariant overlapping groups. Moreover, we apply our results to TV deblurring and denoising problems with overlapping group sparsity. We use alternating direction method of multipliers to iteratively solve them. Numerical results verify the validity and effectiveness of our new inexact explicit shrinkage formulas.

New inexact explicit thresholding/shrinkage formulas for inverse problems with overlapping group sparsity