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February 2010 Empirical risk minimization in inverse problems
Jussi Klemelä, Enno Mammen
Ann. Statist. 38(1): 482-511 (February 2010). DOI: 10.1214/09-AOS726


We study estimation of a multivariate function f : RdR when the observations are available from the function Af, where A is a known linear operator. Both the Gaussian white noise model and density estimation are studied. We define an L2-empirical risk functional which is used to define a δ-net minimizer and a dense empirical risk minimizer. Upper bounds for the mean integrated squared error of the estimators are given. The upper bounds show how the difficulty of the estimation depends on the operator through the norm of the adjoint of the inverse of the operator and on the underlying function class through the entropy of the class. Corresponding lower bounds are also derived. As examples, we consider convolution operators and the Radon transform. In these examples, the estimators achieve the optimal rates of convergence. Furthermore, a new type of oracle inequality is given for inverse problems in additive models.


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Jussi Klemelä. Enno Mammen. "Empirical risk minimization in inverse problems." Ann. Statist. 38 (1) 482 - 511, February 2010.


Published: February 2010
First available in Project Euclid: 31 December 2009

zbMATH: 1181.62044
MathSciNet: MR2589328
Digital Object Identifier: 10.1214/09-AOS726

Primary: 62G07

Keywords: Deconvolution , empirical risk minimization , multivariate density estimation , nonparametric function estimation , Radon transform , tomography

Rights: Copyright © 2010 Institute of Mathematical Statistics


Vol.38 • No. 1 • February 2010
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