Convergence bounds for empirical nonlinear least-squares

dc.contributor.authorEigel, Martin
dc.contributor.authorSchneider, Reinhold
dc.contributor.authorTrunschke, Philipp
dc.date.accessioned2022-09-14T13:27:16Z
dc.date.available2022-09-14T13:27:16Z
dc.date.issued2022-02-07
dc.description.abstractWe consider best approximation problems in a nonlinear subset ā„³ of a Banach space of functions (š’±,āˆ„ā€¢āˆ„). The norm is assumed to be a generalization of the L 2-norm for which only a weighted Monte Carlo estimate āˆ„ā€¢āˆ„n can be computed. The objective is to obtain an approximation vā€„āˆˆā€„ā„³ of an unknown function uā€„āˆˆā€„š’± by minimizing the empirical norm āˆ„uā€…āˆ’ā€…vāˆ„n. We consider this problem for general nonlinear subsets and establish error bounds for the empirical best approximation error. Our results are based on a restricted isometry property (RIP) which holds in probability and is independent of the specified nonlinear least squares setting. Several model classes are examined and the analytical statements about the RIP are compared to existing sample complexity bounds from the literature. We find that for well-studied model classes our general bound is weaker but exhibits many of the same properties as these specialized bounds. Notably, we demonstrate the advantage of an optimal sampling density (as known for linear spaces) for sets of functions with sparse representations.en
dc.identifier.eissn2804-7214
dc.identifier.issn2822-7840
dc.identifier.urihttps://depositonce.tu-berlin.de/handle/11303/17460
dc.identifier.urihttp://dx.doi.org/10.14279/depositonce-16241
dc.language.isoenen
dc.relation.ispartof10.14279/depositonce-12854
dc.rights.urihttps://creativecommons.org/licenses/by/4.0/en
dc.subject.ddc518 Numerische Analysisde
dc.subject.ddc519 Wahrscheinlichkeiten, angewandte Mathematikde
dc.subject.otherweighted nonlinear least squaresen
dc.subject.othererror analysisen
dc.subject.otherconvergence ratesen
dc.subject.otherweighted sparsityen
dc.subject.othertensor networksen
dc.titleConvergence bounds for empirical nonlinear least-squaresen
dc.typeArticleen
dc.type.versionpublishedVersionen
dcterms.bibliographicCitation.doi10.1051/m2an/2021070en
dcterms.bibliographicCitation.issue1en
dcterms.bibliographicCitation.journaltitleMathematical modelling and numerical analysisen
dcterms.bibliographicCitation.originalpublishernameEDP Sciencesen
dcterms.bibliographicCitation.originalpublisherplaceLes Ulisen
dcterms.bibliographicCitation.pageend104en
dcterms.bibliographicCitation.pagestart79en
dcterms.bibliographicCitation.volume56en
tub.accessrights.dnbfreeen
tub.affiliationFak. 2 Mathematik und Naturwissenschaften::Inst. Mathematik::FG Modellierung, Simulation und Optimierung in Natur- und Ingenieurwissenschaftende
tub.affiliation.facultyFak. 2 Mathematik und Naturwissenschaftende
tub.affiliation.groupFG Modellierung, Simulation und Optimierung in Natur- und Ingenieurwissenschaftende
tub.affiliation.instituteInst. Mathematikde
tub.publisher.universityorinstitutionTechnische UniversitƤt Berlinen

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