In nonparametric regression and inve rse problems\, variational methods based on bounde d variation (BV) penalties are a well-known and es tablished tool for yielding edge-preserving recons tructions\, which is a desirable feature in many a pplications. Despite its practical success\, the t heory behind BV-regularization is poorly understoo d: most importantly\, there is a lack of convergen ce guarantees in spatial dimension \;d\\geq 2.

In this talk we present a variation al estimator that combines a BV penalty and a mult iscale constraint\, and prove that it converges to the truth at the optimal rate. Our theoretical an alysis relies on a proper analysis of the multisca le constraint\, which is motivated by the statisti cal properties of the noise\, and relates in a nat ural way to certain Besov spaces of negative smoot hness. Further\, the main novelty of our approach is the use of refined interpolation inequalities b etween function spaces. We also illustrate the per formance of these variational estimators in simula tions on signals and images.

LOCATION:Seminar Room 1\, Newton Institute CONTACT:INI IT END:VEVENT END:VCALENDAR