The Annals of Statistics

Empirical risk minimization in inverse problems

Jussi Klemelä and Enno Mammen

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Abstract

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.

Article information

Source
Ann. Statist., Volume 38, Number 1 (2010), 482-511.

Dates
First available in Project Euclid: 31 December 2009

Permanent link to this document
https://projecteuclid.org/euclid.aos/1262271621

Digital Object Identifier
doi:10.1214/09-AOS726

Mathematical Reviews number (MathSciNet)
MR2589328

Zentralblatt MATH identifier
1181.62044

Subjects
Primary: 62G07: Density estimation

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

Citation

Klemelä, Jussi; Mammen, Enno. Empirical risk minimization in inverse problems. Ann. Statist. 38 (2010), no. 1, 482--511. doi:10.1214/09-AOS726. https://projecteuclid.org/euclid.aos/1262271621


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