2014 Ground states and singular vectors of convex variational regularization methods. Methods and Applications of Analysis

Singular value decomposition is the key tool in the analysis and understanding of lin-ear regularization methods in Hilbert spaces. Besides simplifying computations it allows to provide a good understanding of properties of the forward problem compared to the prior in-formation introduced by the regularization methods. In the last decade nonlinear variational approaches such as `1 or total variation regularizations became quite prominent regulariza-tion techniques with certain properties being superior to standard methods. In the analysis of those, singular values and vectors did not play any role so far, for the obvious reason that these problems are nonlinear, together with the issue of defining singular values and singular vectors in the first place. In this paper however we want to start a study of singular values and vectors for nonlinear variational regularization of linear inverse problems, with particular focus on singular one-homogeneous regularization functionals. A major role is played by the smallest singular value, which we define as the ground state of an appropriate functional combining the (semi-)norm introduced by the forward operator and the regularization functional. The optimality condition for the ground state further yields a natural generalization to higher singular value