Optimal adaptation for early stopping in statistical inverse problems
- For linear inverse problems Y = A mu + zeta, it is classical to recover the unknown signal mu by iterative regularization methods ((mu) over cap,(m) = 0,1, . . .) and halt at a data-dependent iteration tau using some stopping rule, typically based on a discrepancy principle, so that the weak (or prediction) squared-error parallel to A((mu) over cap (()(tau)) - mu)parallel to(2) is controlled. In the context of statistical estimation with stochastic noise zeta, we study oracle adaptation (that is, compared to the best possible stopping iteration) in strong squared- error E[parallel to((mu) over cap (()(tau)) - mu)parallel to(2)]. For a residual-based stopping rule oracle adaptation bounds are established for general spectral regularization methods. The proofs use bias and variance transfer techniques from weak prediction error to strong L-2-error, as well as convexity arguments and concentration bounds for the stochastic part. Adaptive early stopping for the Landweber method is studied in further detail and illustrated numerically.
Author details: | Gilles BlanchardORCiDGND, Marc Hoffmann, Markus ReissGND |
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DOI: | https://doi.org/10.1137/17M1154096 |
ISSN: | 2166-2525 |
Title of parent work (English): | SIAM/ASA Journal on Uncertainty Quantification |
Publisher: | Society for Industrial and Applied Mathematics |
Place of publishing: | Philadelphia |
Publication type: | Article |
Language: | English |
Date of first publication: | 2018/07/19 |
Publication year: | 2018 |
Release date: | 2022/03/17 |
Tag: | Landweber iteration; adaptive estimation; discrepancy principle; early stopping; linear inverse problems; oracle inequality |
Volume: | 6 |
Issue: | 3 |
Number of pages: | 33 |
First page: | 1043 |
Last Page: | 1075 |
Funding institution: | DFG via Research Unit 1735 Structural Inference in Statistics; [SFB 1294] |
Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
DDC classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
Peer review: | Referiert |
Publishing method: | Open Access / Green Open-Access |