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Sharp adaptation for inverse problems with random noise
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  • Published: July 2002

Sharp adaptation for inverse problems with random noise

  • Laurent Cavalier1 &
  • Alexandre Tsybakov2 

Probability Theory and Related Fields volume 123, pages 323–354 (2002)Cite this article

Abstract.

 We consider a heteroscedastic sequence space setup with polynomially increasing variances of observations that allows to treat a number of inverse problems, in particular multivariate ones. We propose an adaptive estimator that attains simultaneously exact asymptotic minimax constants on every ellipsoid of functions within a wide scale (that includes ellipoids with polynomially and exponentially decreasing axes) and, at the same time, satisfies asymptotically exact oracle inequalities within any class of linear estimates having monotone non-increasing weights. The construction of the estimator is based on a properly penalized blockwise Stein's rule, with weakly geometically increasing blocks. As an application, we construct sharp adaptive estimators in the problems of deconvolution and tomography.

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Authors and Affiliations

  1. CMI, Université de Provence, 39, rue F. Joliot-Curie, F-13453 Marseille, France. e-mail: cavalier@cmi.univ-mrs.fr, , , , , , FR

    Laurent Cavalier

  2. Laboratoire de Probabilités et Modèles Aléatoires, Université Paris 6, 4, place. Jussieu, BP 188, F-75252 Paris, France. e-mail: tsybakov@ccr.jussieu.fr, , , , , , FR

    Alexandre Tsybakov

Authors
  1. Laurent Cavalier
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  2. Alexandre Tsybakov
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Received: 19 January 2000 / Revised version: 30 April 2001 / Published online: 14 June 2002

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Cavalier, L., Tsybakov, A. Sharp adaptation for inverse problems with random noise. Probab Theory Relat Fields 123, 323–354 (2002). https://doi.org/10.1007/s004400100169

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  • Issue Date: July 2002

  • DOI: https://doi.org/10.1007/s004400100169

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Keywords

  • Stein
  • Inverse Problem
  • Deconvolution
  • Linear Estimate
  • Random Noise
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