Convergence estimates in probability and in expectation for discrete least squares with noisy evaluations at random points

Handle URI:
http://hdl.handle.net/10754/576075
Title:
Convergence estimates in probability and in expectation for discrete least squares with noisy evaluations at random points
Authors:
Migliorati, Giovanni; Nobile, Fabio; Tempone, Raul ( 0000-0003-1967-4446 )
Abstract:
We 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.
KAUST Department:
Applied Mathematics and Computational Science Program; Center for Uncertainty Quantification in Computational Science and Engineering (SRI-UQ)
Citation:
Convergence estimates in probability and in expectation for discrete least squares with noisy evaluations at random points 2015 Journal of Multivariate Analysis
Publisher:
Elsevier BV
Journal:
Journal of Multivariate Analysis
Issue Date:
28-Aug-2015
DOI:
10.1016/j.jmva.2015.08.009
Type:
Article
ISSN:
0047259X
Additional Links:
http://linkinghub.elsevier.com/retrieve/pii/S0047259X15001931
Appears in Collections:
Articles; Applied Mathematics and Computational Science Program

Full metadata record

DC FieldValue Language
dc.contributor.authorMigliorati, Giovannien
dc.contributor.authorNobile, Fabioen
dc.contributor.authorTempone, Raulen
dc.date.accessioned2015-08-30T11:13:34Zen
dc.date.available2015-08-30T11:13:34Zen
dc.date.issued2015-08-28en
dc.identifier.citationConvergence estimates in probability and in expectation for discrete least squares with noisy evaluations at random points 2015 Journal of Multivariate Analysisen
dc.identifier.issn0047259Xen
dc.identifier.doi10.1016/j.jmva.2015.08.009en
dc.identifier.urihttp://hdl.handle.net/10754/576075en
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.en
dc.language.isoenen
dc.publisherElsevier BVen
dc.relation.urlhttp://linkinghub.elsevier.com/retrieve/pii/S0047259X15001931en
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.2015.08.009en
dc.subjectApproximation theoryen
dc.subjectDiscrete least squaresen
dc.subjectNoisy evaluationsen
dc.subjectError analysisen
dc.subjectConvergence ratesen
dc.subjectLarge deviationsen
dc.subjectLearning theoryen
dc.subjectMultivariate polynomial approximationen
dc.titleConvergence estimates in probability and in expectation for discrete least squares with noisy evaluations at random pointsen
dc.typeArticleen
dc.contributor.departmentApplied Mathematics and Computational Science Programen
dc.contributor.departmentCenter for Uncertainty Quantification in Computational Science and Engineering (SRI-UQ)en
dc.identifier.journalJournal of Multivariate Analysisen
dc.eprint.versionPost-printen
dc.contributor.institutionMATHICSE-CSQI, École Polytechnique Fédérale de Lausanne, Lausanne CH-1015, Switzerlanden
dc.contributor.affiliationKing Abdullah University of Science and Technology (KAUST)en
kaust.authorTempone, Raulen
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