Ground States and Singular Vectors of Convex Variational Regularization Methods

Martin Benning* and Martin Burger

Singular value decomposition is the key tool in the analysis and understanding of linear regularization methods. In the last decade nonlinear variational approaches such as $\ell^1$ or total variation regularizations became quite prominent regularization 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 this talk 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.

Mathematics Subject Classification: 45Q05

Keywords: Inverse Problems, Variational Regularization, Singular Values, Ground States, Total Variation Regularization, Bregman Distance, Inverse Scale Space Method, Compressed Sensing.

Minisymposion: Dual Methods for Approaching Inverse Problems