Abstract
In this paper we characterize sparse solutions for variational problems of the form min _{uin X} phi (u) + F(mathcal {A}u), where X is a locally convex space, mathcal {A} is a linear continuous operator that maps into a finite dimensional Hilbert space and phi 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 phi (possibly translated by an element in the null space of phi ). 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.
Highlights
One of the fundamental tasks of inverse problems is to reconstruct data from a small number of usually noisy observations
Where X is a locally convex space, φ : X → [0, +∞] is a lower semi-continuous seminorm, A : X → H is a linear continuous map with values in a finite-dimensional Hilbert space H and F is a proper, convex, lower semi-continuous functional. (Notice that this generality allows problems of the type (1) for noise-free data as well as soft constraints in case of noisy data.) we ask that A(dom φ) = H and that φ is coercive when restricted to the quotient space of X with the null-space of φ that we denote by N. Under these hypotheses we prove that there exists a sparse minimizer of (3), namely a minimizer that can be written as a linear combination of extremal points of the unit ball associated to φ
The abstract main result of this paper contained in Theorem 3.3 about the structure of a minimizer of a variational problem with finite dimensional data appears to be widely applicable, thanks to its generality
Summary
One of the fundamental tasks of inverse problems is to reconstruct data from a small number of usually noisy observations This is of capital importance in a huge variety of fields in science and engineering, where typically one has access only to a fixed and small number of measurements of the sought unknown. In general, this type of problem is underdetermined and the recovery of the true data is practically impossible.
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