On the convergence speed of iterative methods for linear inverse problems with sparsity constraints

Abstract

In this article we report on different iterative algorithms for the minimization of Tikhonov type functionals which involve sparsity constraints in form of lp-penalties. We summarize results on the well known iterated soft thresholding, the iterated hard thresholding and a semismooth Newton approach. While the first two converge globally but very slow, the last one converges only locally but superlinearly. Furthermore we propose a combination of soft and hard thresholding. While this method has theoretically the same convergence rate than the soft thresholding (namely it converges linearly), our numerical experiments show that the combined approach is almost always better than both methods alone.