A Predual Proximal Point Algorithm solving a Non Negative Basis Pursuit Denoising model

Abstract

International audienceThis paper develops an implementation of a Predual Proximal Point Algorithm (PPPA) solving a Non Negative Basis Pursuit Denoising model. The model imposes a constraint on the l2 norm of the residual, instead of penalizing it. The PPPA solves the predual of the problem with a Proximal Point Algorithm (PPA). Moreover, the minimization that needs to be performed at each iteration of PPA is solved with a dual method. We can prove that these dual variables converge to a solution of the initial problem. Our analysis proves that we turn a constrained non differentiable con- vex problem into a short sequence of nice concave maximization problems. By nice, we mean that the functions which are maximized are differen- tiable and their gradient is Lipschitz. The algorithm is easy to implement, easier to tune and more general than the algorithms found in the literature. In particular, it can be ap- plied to the Basis Pursuit Denoising (BPDN) and the Non Negative Basis Pursuit Denoising (NNBPDN) and it does not make any assumption on the dictionary. We prove its convergence to the set of solutions of the model and provide some convergence rates. Experiments on image approximation show that the performances of the PPPA are at the current state of the art for the BPDN