Empirical risk minimization in inverse problems

Details Empirical risk minimization in inverse problems Empirical risk minimization in inverse problems

2010-01-13

arxiv.org/abs/1001.2089v1

Annals of Statistics 2010, Vol. 38, No. 1, 482-511

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/09-AOS726 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of

Scientific paper

10.1214/09-AOS726

We study estimation of a multivariate function $f:\mathbf{R}^d\to\mathbf{R}$ 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 $L_2$-empirical risk functional which is used to define a $\delta$-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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