Flexible sparse regularization. (5th December 2016) - Lorenz, Dirk A; Resmerita, Elena
Abstract: The seminal paper of Daubechies, Defrise, DeMol made clear that ℓ p spaces with p ∊ [ 1, 2 ) and p -powers of the corresponding norms are appropriate settings for dealing with reconstruction of sparse solutions of ill-posed problems by regularization. It seems that the case p = 1 provides the best results in most of the situations compared to the cases p ∊ ( 1, 2 ) . An extensive literature gives great credit also to using ℓ p spaces with p ∊ ( 0, 1 ) together with the corresponding quasi-norms, although one has to tackle challenging numerical problems raised by the non-convexity of the quasi-norms. In any of these settings, either superlinear, linear or sublinear, the question of how to choose the exponent p has been not only a numerical issue, but also a philosophical one. In this work we introduce a more flexible way of sparse regularization by varying exponents. We introduce the corresponding functional analytic framework, that leaves the setting of normed spaces but works with so-called F-norms. One curious result is that there are F-norms which generate the ℓ 1 space, but they are strictly convex, while the ℓ 1 -norm is just convex.