EMPIRICAL RISK MINIMIZATION IN INVERSE PROBLEMS

• 成果类型: Article
• 署名作者: Klemela, Jussi; Mammen, Enno
• 署名单位: University of Oulu; University of Mannheim
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/09-AOS726
• 发表日期: 2010
• 页码: 482-511
• 关键词: convergence; rates; pointwise

摘要

We study estimation of a multivariate function f : R-d -> 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.