TY - JOUR
T1 - Sparsity of solutions for variational inverse problems with finite-dimensional data
AU - Bredies, Kristian
AU - Carioni, Marcello
N1 - Funding Information:
Open access funding provided by University of Graz. The authors gratefully acknowledge the funding of this work by the Austrian Science Fund (FWF) within the Project P 29192. We also thank Professor Luigi Ambrosio for the useful remarks regarding [1].
Funding Information:
Open access funding provided by University of Graz. The authors gratefully acknowledge the funding of this work by the Austrian Science Fund (FWF) within the Project P 29192. We also thank Professor Luigi Ambrosio for the useful remarks regarding [ 1 ]. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Publisher Copyright:
© 2019, The Author(s).
PY - 2020/2/1
Y1 - 2020/2/1
N2 - In this paper we characterize sparse solutions for variational problems of the form min u ∈ Xϕ(u) + F(Au) , where X is a locally convex space, A is a linear continuous operator 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 differential 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 Unser et al. (SIAM Rev 59(4):769–793, 2017) under weaker hypotheses.
AB - In this paper we characterize sparse solutions for variational problems of the form min u ∈ Xϕ(u) + F(Au) , where X is a locally convex space, A is a linear continuous operator 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 differential 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 Unser et al. (SIAM Rev 59(4):769–793, 2017) under weaker hypotheses.
UR - http://www.scopus.com/inward/record.url?scp=85075861500&partnerID=8YFLogxK
U2 - 10.1007/s00526-019-1658-1
DO - 10.1007/s00526-019-1658-1
M3 - Article
AN - SCOPUS:85075861500
SN - 0944-2669
VL - 59
JO - Calculus of Variations and Partial Differential Equations
JF - Calculus of Variations and Partial Differential Equations
IS - 1
M1 - 14
ER -