Abstract

Co-authors: Housen Li (University of Goettingen), Axel Munk (University of Goettingen)

In nonparametric regression and inverse problems, variational methods based on bounded variation (BV) penalties are a well-known and established tool for yielding edge-preserving reconstructions, which is a desirable feature in many applications. Despite its practical success, the theory behind BV-regularization is poorly understood: most importantly, there is a lack of convergence guarantees in spatial dimension d\geq 2.

In this talk we present a variational estimator that combines a BV penalty and a multiscale constraint, and prove that it converges to the truth at the optimal rate. Our theoretical analysis relies on a proper analysis of the multiscale constraint, which is motivated by the statistical properties of the noise, and relates in a natural way to certain Besov spaces of negative smoothness. Further, the main novelty of our approach is the use of refined interpolation inequalities between function spaces. We also illustrate the performance of these variational estimators in simulations on signals and images.