A Unified Class of DC-type Convexity-Preserving Regularizers for Improved Sparse Regularization

Recently, a novel class of nonconvex sparse regularizers which can preserve the convexity of the cost function has gained considerable attention. These regularizers are expressed as difference of two convex functions, where the parameterized subtrahend function can be adjusted flexibly to maintain the overall convexity. In this paper we propose a unified class of such DC-type (Difference-of-Convex) convexity-preserving regularizers. By selecting proper kernel functions, the proposed regularizer reproduces existing convexity-preserving models and opens the way to a large number of promising new regularizers. In order to solve the convex but involved regularization model, we propose a novel iterative algorithm based on DC programming. Unlike normal DC algorithms, the proposed method is guaranteed to converge to a global minimizer of the cost function. In addition, compared to algorithms for existing convexity-preserving models, the proposed algorithm makes less stringent assumptions on the kernel functions, thus is more general. Moreover, the proposed algorithm can be interpreted as a nested forward-backward splitting method with overrelaxed step size, which leads to its fast convergence. Numerical experiments are conducted to verify the efficiency of our algorithm with comparisons to existing methods.