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dc.contributor.authorMigliorati, Giovanni
dc.contributor.authorNobile, Fabio
dc.contributor.authorTempone, Raul
dc.date.accessioned2015-08-30T11:13:34Z
dc.date.available2015-08-30T11:13:34Z
dc.date.issued2015-08-28
dc.identifier.citationConvergence estimates in probability and in expectation for discrete least squares with noisy evaluations at random points 2015 Journal of Multivariate Analysis
dc.identifier.issn0047259X
dc.identifier.doi10.1016/j.jmva.2015.08.009
dc.identifier.urihttp://hdl.handle.net/10754/576075
dc.description.abstractWe study the accuracy of the discrete least-squares approximation on a finite dimensional space of a real-valued target function from noisy pointwise evaluations at independent random points distributed according to a given sampling probability measure. The convergence estimates are given in mean-square sense with respect to the sampling measure. The noise may be correlated with the location of the evaluation and may have nonzero mean (offset). We consider both cases of bounded or square-integrable noise / offset. We prove conditions between the number of sampling points and the dimension of the underlying approximation space that ensure a stable and accurate approximation. Particular focus is on deriving estimates in probability within a given confidence level. We analyze how the best approximation error and the noise terms affect the convergence rate and the overall confidence level achieved by the convergence estimate. The proofs of our convergence estimates in probability use arguments from the theory of large deviations to bound the noise term. Finally we address the particular case of multivariate polynomial approximation spaces with any density in the beta family, including uniform and Chebyshev.
dc.language.isoen
dc.publisherElsevier BV
dc.relation.urlhttp://linkinghub.elsevier.com/retrieve/pii/S0047259X15001931
dc.rightsNOTICE: this is the author’s version of a work that was accepted for publication in Journal of Multivariate Analysis. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Multivariate Analysis, 28 August 2015. DOI: 10.1016/j.jmva.2015.08.009
dc.subjectApproximation theory
dc.subjectDiscrete least squares
dc.subjectNoisy evaluations
dc.subjectError analysis
dc.subjectConvergence rates
dc.subjectLarge deviations
dc.subjectLearning theory
dc.subjectMultivariate polynomial approximation
dc.titleConvergence estimates in probability and in expectation for discrete least squares with noisy evaluations at random points
dc.typeArticle
dc.contributor.departmentApplied Mathematics and Computational Science Program
dc.contributor.departmentCenter for Uncertainty Quantification in Computational Science and Engineering (SRI-UQ)
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
dc.identifier.journalJournal of Multivariate Analysis
dc.eprint.versionPost-print
dc.contributor.institutionMATHICSE-CSQI, École Polytechnique Fédérale de Lausanne, Lausanne CH-1015, Switzerland
dc.contributor.affiliationKing Abdullah University of Science and Technology (KAUST)
kaust.personTempone, Raul
refterms.dateFOA2017-08-28T00:00:00Z
dc.date.published-online2015-08-28
dc.date.published-print2015-12


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