Sparse Regularization With Reverse Sorted Sum of Squares via an Unrolled Difference-of-Convex Approach

Abstract

This paper proposes a sparse regularization method with a novel sorted regularization function. Sparse regularization is commonly used to solve underdetermined inverse problems. Traditional sparse regularization functions, such as $L_{1}$-norm, suffer from problems like amplitude underestimation and vanishing perturbations. The reverse ordered weighted $L_{1}$-norm (ROWL) addresses these issues but introduces new challenges. These include developing an algorithm grounded in theory, not heuristics, reducing computational complexity, enabling the automatic determination of numerous parameters, and ensuring the number of iterations remains feasible. In this study, we propose a novel sparse regularization function called the reverse sorted sum of squares (RSSS) and then construct an unrolled algorithm that addresses both the aforementioned two problems and these four challenges. The core idea of our proposed method lies in transforming the optimization problem into a difference-of-convex programming problem, for which solutions are known. In experiments, we apply the RSSS regularization method to image deblurring and super-resolution tasks and confirmed its superior performance compared to conventional methods, all with feasible computational complexity.