2012/11/09 by Martin Benning, Martin Burger, Benning, Martin +1 · 2 citations
Engineering · Mathematics · #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Sparse and Compressive Sensing Techniques #Statistical and numerical algorithms
paper · pdf · doi:10.48550/arxiv.1211.2057
openalex publication_date 2012/11/09 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
Singular value decomposition is the key tool in the analysis and\nunderstanding of linear regularization methods. In the last decade nonlinear\nvariational approaches such as \ℓ1 or total variation regularizations\nbecame quite prominent regularization techniques with certain properties being\nsuperior to standard methods. In the analysis of those, singular values and\nvectors did not play any role so far, for the obvious reason that these\nproblems are nonlinear, together with the issue of defining singular values and\nsingular vectors. In this paper however we want to start a study of singular\nvalues and vectors for nonlinear variational regularization of linear inverse\nproblems, with particular focus on singular one-homogeneous regularization\nfunctionals. A major role is played by the smallest singular value, which we\ndefine as the ground state of an appropriate functional combining the\n(semi-)norm introduced by the forward operator and the regularization\nfunctional. The optimality condition for the ground state further yields a\nnatural generalization to higher singular values and vectors involving the\nsubdifferential of the regularization functional. We carry over two main\nproperties from the world of linear regularization. The first one is gaining\ninformation about scale, respectively the behavior of regularization techniques\nat different scales. This also leads to novel estimates at different scales,\ngeneralizing the estimates for the coefficients in the linear singular value\nexpansion. The second one is to provide exact solutions for variational\nregularization methods. We will show that all singular vectors can be\nreconstructed up to a scalar factor by the standard Tikhonov-type\nregularization approach even in the presence of (small) noise. Moreover, we\nwill show that they can even be reconstructed without any bias by the recently\npopularized inverse scale space method.\n