  In this paper we introduce the class of infinite infimal convolution
functionals and apply these functionals to the regularization of ill-posed
inverse problems. The proposed regularization involves an infimal convolution
of a continuously parametrized family of convex, positively one-homogeneous
functionals defined on a common Banach space $X$. We show that, under mild
assumptions, this functional admits an equivalent convex lifting in the space
of measures with values in $X$. This reformulation allows us to prove
well-posedness of a Tikhonov regularized inverse problem and opens the door to
a sparse analysis of the solutions. In the case of finite-dimensional
measurements we prove a representer theorem, showing that there exists a
solution of the inverse problem that is sparse, in the sense that it can be
represented as a linear combination of the extremal points of the ball of the
lifted infinite infimal convolution functional. Then, we design a generalized
conditional gradient method for computing solutions of the inverse problem
without relying on an a priori discretization of the parameter space and of the
Banach space $X$. The iterates are constructed as linear combinations of the
extremal points of the lifted infinite infimal convolution functional. We prove
a sublinear rate of convergence for our algorithm and apply it to denoising of
signals and images using, as regularizer, infinite infimal convolutions of
fractional-Laplacian-type operators with adaptive orders of smoothness and
anisotropies.
