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Optimal rates for regularization of statistical inverse learning problems

  • We consider a statistical inverse learning problem, where we observe the image of a function f through a linear operator A at i.i.d. random design points X_i, superposed with an additional noise. The distribution of the design points is unknown and can be very general. We analyze simultaneously the direct (estimation of Af) and the inverse (estimation of f) learning problems. In this general framework, we obtain strong and weak minimax optimal rates of convergence (as the number of observations n grows large) for a large class of spectral regularization methods over regularity classes defined through appropriate source conditions. This improves on or completes previous results obtained in related settings. The optimality of the obtained rates is shown not only in the exponent in n but also in the explicit dependence of the constant factor in the variance of the noise and the radius of the source condition set.

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Metadaten
Author:Gilles BlanchardGND, Nicole MückeORCiDGND
URN:urn:nbn:de:kobv:517-opus4-89782
ISSN:2193-6943 (online)
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Potsdam (5 (2016) 5)
Publisher:Universitätsverlag Potsdam
Place of publication:Potsdam
Document Type:Preprint
Language:English
Year of first Publication:2016
Year of Completion:2016
Publishing Institution:Universität Potsdam
Publishing Institution:Universitätsverlag Potsdam
Release Date:2016/04/15
Tag:kernel method; minimax rate; statistical inverse problem
Volume:5
Issue:5
Pagenumber:36
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
MSC Classification:62-XX STATISTICS / 62Cxx Decision theory [See also 90B50, 91B06; for game theory, see 91A35] / 62C20 Minimax procedures
62-XX STATISTICS / 62Gxx Nonparametric inference / 62G05 Estimation
62-XX STATISTICS / 62Gxx Nonparametric inference / 62G20 Asymptotic properties
65-XX NUMERICAL ANALYSIS / 65Jxx Numerical analysis in abstract spaces / 65J22 Inverse problems
Publication Way:Universitätsverlag Potsdam
Collections:Universität Potsdam / Schriftenreihen / Preprints des Instituts für Mathematik der Universität Potsdam, ISSN 2193-6943 / 2016
Licence (German):License LogoKeine Nutzungslizenz vergeben - es gilt das deutsche Urheberrecht