We propose a scaled adaptive version of the Fast Iterative Soft-Thresholding Algorithm, named S-FISTA, for the efficient solution of convex optimization problems with sparsity-enforcing regularization. S-FISTA couples a non-monotone backtracking procedure with a scaling strategy for the proximal–gradient step, which is particularly effective in situations where signal-dependent noise is present in the data. The proposed algorithm is tested on some image super-resolution problems where a sparsity-promoting regularization term is coupled with a weighted- ℓ2 data fidelity. Our numerical experiments show that S-FISTA allows for faster convergence in function values with respect to standard FISTA, as well as being an efficient inner solver for iteratively reweighted ℓ1 algorithms, thus reducing the overall computational times.
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