Flexible sparse regularization

Abstract

The seminal paper of Daubechies, Defrise, DeMol made clear that $\ell^p$ spaces with $p\in [1,2)$ 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 $p\in (1,2)$. An extensive literature gives great credit also to using $\ell^p$ spaces with $p\in (0,1)$ together with the corresponding quasinorms, although one has to tackle challenging numerical problems raised by the non-convexity of the quasi-norms. In any of these settings, either super, 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 $\ell^1$ space, but they are strictly convex, while the $\ell^1$-norm is just convex.