Scaled, inexact and adaptive generalised FISTA for (strongly) convex imaging problems

Abstract: We consider an inexact, scaled generalised Fast Iterative Soft-Thresholding Algorithm (FISTA) for minimising the sum of two (possibly strongly) convex functions, which we name SAGE-FISTA. Here, the inexactness is explicitly taken into account so as to describe standard situations where proximal operators cannot be evaluated in closed form. The idea of considering data-dependent scaling in forward-backward splitting methods has furthermore been shown to be effective in incorporating Newton-type information along the optimisation via suitable variable-metric updates. Finally, in order to account for the adjustment of the algorithmic step-size along the iterations, we propose a non-monotone backtracking strategy which improves the convergence speed compared to standard Armijoo-type analogs. Analytically, linear convergence result for the function values is proved. The result depends on the strong convexity moduli of the two functions, the upper and lower bounds on the spectrum of the variable metric operators and the inexactness/backtracking parameters. The performance of SAGE-FISTA is validated on convex and strongly-convex exemplar image denoising, deblurring and super-resolution problems where sparsity-promoting regularisation is combined with data-dependent Kullback-Leibler-type fidelity terms.
This is joint work with S. Rebegoldi (University of Florence).

numerical analysis

Audience: researchers in the topic

Seminars on Numerics and Applications