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In this paper we introduce the class of infinite infimal convolutionfunctionals and apply these functionals to the regularization of ill-posedinverse problems. The proposed regularization involves an infimal convolutionof a continuously parametrized family of convex, positively one-homogeneousfunctionals defined on a common Banach space $X$. We show that, under mildassumptions, this functional admits an equivalent convex lifting in the spaceof measures with values in $X$. This reformulation allows us to provewell-posedness of a Tikhonov regularized inverse problem and opens the door toa sparse analysis of the solutions. In the case of finite-dimensionalmeasurements we prove a representer theorem, showing that there exists asolution of the inverse problem that is sparse, in the sense that it can berepresented as a linear combination of the extremal points of the ball of thelifted infinite infimal convolution functional. Then, we design a generalizedconditional gradient method for computing solutions of the inverse problemwithout relying on an a priori discretization of the parameter space and of theBanach space $X$. The iterates are constructed as linear combinations of theextremal points of the lifted infinite infimal convolution functional. We provea sublinear rate of convergence for our algorithm and apply it to denoising ofsignals and images using, as regularizer, infinite infimal convolutions offractional-Laplacian-type operators with adaptive orders of smoothness andanisotropies.

A sparse optimization approach to infinite infimal convolution regularization