Convergence rates of general regularization methods for statistical inverse problems and applications
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Date
2007-02-21T14:34:55Z
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Abstract
During the past the convergence analysis for linear statistical inverse problems has mainly focused
on spectral cut-off and Tikhonov type estimators. Spectral cut-off estimators achieve minimax rates for a broad
range of smoothness classes and operators, but their practical usefulness is limited by the fact that they require
a complete spectral decomposition of the operator. Tikhonov estimators are simpler to compute, but still involve
the inversion of an operator and achieve minimax rates only in restricted smoothness classes. In this paper we
introduce a unifying technique to study the mean square error of a large class of regularization methods (spectral
methods) including the aforementioned estimators as well as many iterative methods, such as ν-methods and the
Landweber iteration. The latter estimators converge at the same rate as spectral cut-off, but only require matrixvector
products. Our results are applied to various problems, in particular we obtain precise convergence rates for
satellite gradiometry, L2-boosting, and errors in variable problems.
AMS subject classifications: 62G05, 62J05, 62P35, 65J10, 35R30
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Boosting, Errors in variable, Hilbert scales, Iterative regularization methods, Minimax convergence rates, Nonparametric regression, Satellite gradiometry, Statistical inverse problems, Tikhonov regularization