Early stopping for statistical inverse problems via truncated SVD estimation
- We consider truncated SVD (or spectral cut-off, projection) estimators for a prototypical statistical inverse problem in dimension D. Since calculating the singular value decomposition (SVD) only for the largest singular values is much less costly than the full SVD, our aim is to select a data-driven truncation level (m) over cap is an element of {1, . . . , D} only based on the knowledge of the first (m) over cap singular values and vectors. We analyse in detail whether sequential early stopping rules of this type can preserve statistical optimality. Information-constrained lower bounds and matching upper bounds for a residual based stopping rule are provided, which give a clear picture in which situation optimal sequential adaptation is feasible. Finally, a hybrid two-step approach is proposed which allows for classical oracle inequalities while considerably reducing numerical complexity.
Author details: | Gilles BlanchardGND, Marc Hoffmann, Markus ReissGND |
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DOI: | https://doi.org/10.1214/18-EJS1482 |
ISSN: | 1935-7524 |
Title of parent work (English): | Electronic journal of statistics |
Publisher: | Institute of Mathematical Statistics |
Place of publishing: | Cleveland |
Publication type: | Article |
Language: | English |
Date of first publication: | 2018/06/01 |
Publication year: | 2018 |
Release date: | 2022/02/24 |
Tag: | Linear inverse problems; adaptive estimation; discrepancy principle; early stopping; oracle inequalities; spectral cut-off; truncated SVD |
Volume: | 12 |
Issue: | 2 |
Number of pages: | 28 |
First page: | 3204 |
Last Page: | 3231 |
Funding institution: | DFGGerman Research Foundation (DFG) [SFB 1294]; DFG via Research Unit 1735 Structural Inference in Statistics |
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 / Gold Open-Access |
License (German): | CC-BY - Namensnennung 4.0 International |