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We show that many nonlinear observation models in signal recovery can be represented using firmly nonexpansive operators. To address problems with inaccurate measurements, we propose solving a vari- ational inequality relaxation which is guaranteed to possess solutions under mild conditions and which coincides with the original problem if it happens to be consistent. We then present an efficient algorithm for its solution, as well as numerical applications in signal and im- age recovery, including an experimental operator-theoretic method of promoting sparsity. 

Under consideration are multicomponent minimization problems in- volving a separable nonsmooth convex function penalizing the com- ponents individually, and nonsmooth convex coupling terms penal- izing linear mixtures of the components. We investigate the appli- cation of block-activated proximal algorithms for solving such prob- lems, i.e., algorithms which, at each iteration, need to use only a block of the underlying functions, as opposed to all of them as in standard methods. For smooth coupling functions, several block- activated algorithms exist and they are well understood. By con- trast, in the fully nonsmooth case, few block-activated methods are available and little effort has been devoted to assessing them. Our goal is to shed more light on the implementation, the features, and the behavior of these algorithms, compare their merits, and provide machine learning and image recovery experiments illustrating their performance.