Calculus of Variations and Geometric Measure Theory
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K. Bredies - M. Carioni

Sparsity of solutions for variational inverse problems with finite-dimensional data

created by carioni on 19 Sep 2018

[BibTeX]

Preprint

Inserted: 19 sep 2018

Year: 2018

Abstract:

In this paper we characterize sparse solutions for variational problems of the form minu∈X φ(u) + F (Au), where X is a locally convex space, A is a linear continuous oper- ator that maps into a finite dimensional Hilbert space and φ is a seminorm. More precisely, we prove that there exists a minimizer that is “sparse” in the sense that it is represented as a linear combination of the extremal points of the unit ball associated with the regularizer φ (possibly translated by an element in the null space of φ). We apply this result to relevant regularizers such as the total variation seminorm and the Radon norm of a scalar linear dif- ferential operator. In the first example, we provide a theoretical justification of the so-called staircase effect and in the second one, we recover the result in 31 under weaker hypotheses.

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