ProPPA: A Fast Algorithm for ${\mathrm{\ell }}_1$ Minimization and Low-Rank Matrix Completion

We propose a Projected Proximal Point Algorithm (ProPPA) for solving a class of optimization problems. The algorithm iteratively computes the proximal point of the last estimated solution projected into an affine space which itself is parallel and approaching to the feasible set. We provide convergence analysis theoretically supporting the general algorithm, and then apply it for solving ${\mathrm{\ell }}_1$-minimization problems and the matrix completion problem. These problems arise in many applications including machine learning, image and signal processing. We compare our algorithm with the existing state-of-the-art algorithms. Experimental results on solving these problems show that our algorithm is very efficient and competitive.