Abstract.
A standard approach for the solution of several ill-posed signal and image processing problems consists in the minimisation of a composite (smooth + non-smooth) energy functional encoding prior information on the desired solution as well as on the degradation/noise model. In convex scenarios, a reference algorithm dealing with this type of problems is the popular proximal gradient algorithm which deals explicitly (i.e. by a gradient step) with the differentiable part and implicitly (i.e. by a proximal step) with the non-smooth component typically favouring sparsity w.r.t. to some representation. To overcome the slow the convergence properties of such algorithm, inertial techniques enforce accelerated convergence by minimal changes in the iteration steps. However, from a practical viewpoint, these algorithms are effective only whenever precise quantitative estimates of the function regularity are known.
In this talk, I will review some recent contributions on the automation of the well-known Fast Iterative Soft-Thresholding Algorithm (FISTA) under a quadratic growth condition of the composite cost functional including (but not limited to) strongly convex objectives. I will show, in particular, how linear convergence results can be obtained by using adaptive (i.e., non-monotone) backtracking strategies and restarting approaches providing as a byproduct quantitative estimations of regularity parameters along the iterations.
Some exemplar imaging problems will be be considered as for numerical verifications showing improved convergence w.r.t. iterations and CPU times, thus justifying the use of such algorithms in the context of large-scale optimisation.