The Annals of Statistics

Nonlinear estimation for linear inverse problems with error in the operator

Marc Hoffmann and Markus Reiss

Full-text: Open access

Abstract

We study two nonlinear methods for statistical linear inverse problems when the operator is not known. The two constructions combine Galerkin regularization and wavelet thresholding. Their performances depend on the underlying structure of the operator, quantified by an index of sparsity. We prove their rate-optimality and adaptivity properties over Besov classes.

Article information

Source
Ann. Statist., Volume 36, Number 1 (2008), 310-336.

Dates
First available in Project Euclid: 1 February 2008

Permanent link to this document
https://projecteuclid.org/euclid.aos/1201877303

Digital Object Identifier
doi:10.1214/009053607000000721

Mathematical Reviews number (MathSciNet)
MR2387973

Zentralblatt MATH identifier
1134.65038

Subjects
Primary: 65J20: Improperly posed problems; regularization 62G07: Density estimation

Keywords
Statistical inverse problem Galerkin projection method wavelet thresholding minimax rate degree of ill-posedness matrix compression

Citation

Hoffmann, Marc; Reiss, Markus. Nonlinear estimation for linear inverse problems with error in the operator. Ann. Statist. 36 (2008), no. 1, 310--336. doi:10.1214/009053607000000721. https://projecteuclid.org/euclid.aos/1201877303


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