Nonconvex Regularization

Nonconvex regularization techniques aim to improve the accuracy and efficiency of solving inverse problems and machine learning tasks by employing nonconvex penalty functions to promote desirable properties like sparsity or low rank in solutions. Current research focuses on developing efficient algorithms, such as those based on the Difference of Convex functions Algorithm (DCA), iteratively reweighted methods, and alternating direction method of multipliers (ADMM), to handle the challenges posed by nonconvexity, often incorporating techniques like proximal operators and Moreau envelopes for improved tractability. These advancements are significant because they enable better solutions for various applications, including image denoising, magnetic resonance image reconstruction, and robust matrix/tensor completion, often surpassing the performance of convex methods. The development of provably convergent algorithms for nonconvex problems is a key area of ongoing investigation.