# Multiscale Total Variation Estimators for Regression and Inverse Problems

 dc.contributor.advisor Munk, Axel Prof. Dr. dc.contributor.author Álamo, Miguel del dc.date.accessioned 2019-06-28T08:59:28Z dc.date.available 2019-06-28T08:59:28Z dc.date.issued 2019-06-28 dc.identifier.uri http://hdl.handle.net/21.11130/00-1735-0000-0003-C148-2 dc.identifier.uri http://dx.doi.org/10.53846/goediss-7531 dc.language.iso eng de dc.rights.uri http://creativecommons.org/licenses/by-nc-nd/4.0/ dc.subject.ddc 510 de dc.title Multiscale Total Variation Estimators for Regression and Inverse Problems de dc.type doctoralThesis de dc.contributor.referee Munk, Axel Prof. Dr. dc.date.examination 2019-05-24 dc.description.abstracteng In the context of nonparametric regression and inverse problems, variational multiscale methods combine multiscale dictionaries with regularization functionals in a variational framework. de In recent years, these methods have gained popularity in nonparametric statistics due to their good reconstruction properties. Nevertheless, their theoretical performance is, with few exceptions, poorly understood. In this thesis we apply variational multiscale methods to the estimation of functions of bounded variation ($BV$). $BV$ functions are relevant in many applications, since they involve minimal smoothness assumptions and give simplified and interpretable reconstructions. These functions are however remarkably difficult to analyze, and there is to date no statistical theory for the estimation of $BV$ functions in dimension $d\geq 2$. The main theoretical contribution of this thesis is the proof that a class of multiscale estimators with a $BV$ penalty is minimax optimal up to logarithms for the estimation of $BV$ functions in regression and inverse problems in any dimension. Conceptually, our proof exploits a connection between multiscale dictionaries and Besov spaces. Besides the theoretical analysis, in this thesis we consider the efficient implementation and computation of the estimator, and illustrate it in a simulation study. dc.contributor.coReferee Hohage, Thorsten Prof. Dr. dc.subject.eng Minimax estimation de dc.subject.eng Bounded Variation de dc.subject.eng Inverse problems de dc.subject.eng Wavelet methods de dc.subject.eng Nonparametric regression de dc.subject.eng White noise model de dc.identifier.urn urn:nbn:de:gbv:7-21.11130/00-1735-0000-0003-C148-2-2 dc.affiliation.institute Fakultät für Mathematik und Informatik de dc.subject.gokfull Mathematics (PPN61756535X) de dc.identifier.ppn 1668124556
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