Adaptive Gaussian inverse regression with partially unknown operator

• Published in: arXiv.org
• Main Authors: Johannes, Jan, Schwarz, Maik
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 05.04.2012
• Subjects: Eigenvectors; Ill posed problems; Inverse problems; Linear operators; Mathematics - Statistics Theory; Minimax technique; Noise; Noise levels; Optimization; Parameters; Sobolev space; Statistics - Theory; White noise
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1204.1226

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 \(\varepsilon\) and \(\sigma\). 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 \(\varepsilon\) and \(\sigma\). The results are illustrated considering Sobolev spaces and mildly and severely ill-posed inverse problems.