This works addresses the resolution of penalized least-squares problems using the proximal gradient algorithm (PGA). It is known that PGA can be accelerated by preconditioning strategies. However, typical effective choices of preconditioners may correspond to intricate matrices that are not easily inverted, and lead to an increased complexity in the computation of the proximity step. To relax these requirements, we propose a modified preconditioning approach where the metric used in the gradient step is no longer assumed to be the same as the one in the proximity step. We provide convergence conditions for this new iterative scheme and characterize its limit point. Simulations for tomographic image reconstruction from undersampled measurements show the benefits of our approach for various simple choices of metrics.
Technical report - December 2021