Adaptive Gaussian Inverse Regression with Partially Unknown Operator
| Autor: | Jan Johannes, Maik Schwarz |
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| Rok vydání: | 2013 |
| Předmět: | |
| Zdroj: | Communications in Statistics - Theory and Methods. 42:1343-1362 |
| ISSN: | 1532-415X 0361-0926 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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