Adaptive Gaussian Inverse Regression with Partially Unknown Operator

• Published in: Communications in statistics. Theory and methods Vol. 42; no. 7; pp. 1343 - 1362
• Main Authors: Johannes, Jan, Schwarz, Maik
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis Group 01.04.2013; Taylor & Francis Ltd
• Subjects: Adaptive nonparametric estimation; Estimators; Eulers equations; Gaussian; Gaussian sequence space model; Inverse problems; Lepski's method; Mathematical models; Mildly and severely ill-posed inverse problems; Minimax technique; Minimax theory; Model selection; Noise levels; Operators; Optimization; Regression analysis; Sobolev spaces
• ISSN: 0361-0926; 1532-415X
• DOI: 10.1080/03610926.2012.731548

This work deals with the ill-posed inverse problem of reconstructing a function f given implicitly as the solution of g = Af, where A is a compact linear operator with unknown singular values and known eigenfunctions. We observe the function g and the singular values of the operator subject to Gaussian white noise with respective noise levels ϵ and σ. We develop a minimax theory in terms of both noise levels and propose an orthogonal series estimator attaining the minimax rates. This estimator requires the optimal choice of a dimension parameter depending on certain characteristics of f and A. This work addresses the fully data-driven choice of the dimension parameter combining model selection with Lepski's method. We show that the fully data-driven estimator preserves minimax optimality over a wide range of classes for f and A and noise levels ϵ and σ. The results are illustrated considering Sobolev spaces and mildly and severely ill-posed inverse problems.