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Risk estimators for choosing regularization parameters in ill-posed problems - properties and limitations

Lucka, F; Proksch, T; Brune, C; Bissantz, N; Burger, M; Dette, H; Wubbeeling, F; (2018) Risk estimators for choosing regularization parameters in ill-posed problems - properties and limitations. Inverse Problems and Imaging , 12 (5) pp. 1121-1155. 10.3934/ipi.2018047. Green open access

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Abstract

This paper discusses the properties of certain risk estimators that recently regained popularity for choosing regularization parameters in ill-posed problems, in particular for sparsity regularization. They apply Stein's unbiased risk estimator (SURE) to estimate the risk in either the space of the unknown variables or in the data space. We will call the latter PSURE in order to distinguish the two different risk functions. It seems intuitive that SURE is more appropriate for ill-posed problems, since the properties in the data space do not tell much about the quality of the reconstruction. We provide theoretical studies of both approaches for linear Tikhonov regularization in a finite dimensional setting and estimate the quality of the risk estimators, which also leads to asymptotic convergence results as the dimension of the problem tends to infinity. Unlike previous works which studied single realizations of image processing problems with a very low degree of ill-posedness, we are interested in the statistical behaviour of the risk estimators for increasing ill-posedness. Interestingly, our theoretical results indicate that the quality of the SURE risk can deteriorate asymptotically for ill-posed problems, which is confirmed by an extensive numerical study. The latter shows that in many cases the SURE estimator leads to extremely small regularization parameters, which obviously cannot stabilize the reconstruction. Similar but less severe issues with respect to robustness also appear for the PSURE estimator, which in comparison to the rather conservative discrepancy principle leads to the conclusion that regularization parameter choice based on unbiased risk estimation is not a reliable procedure for ill-posed problems. A similar numerical study for sparsity regularization demonstrates that the same issue appears in non-linear variational regularization approaches.

Type: Article
Title: Risk estimators for choosing regularization parameters in ill-posed problems - properties and limitations
Open access status: An open access version is available from UCL Discovery
DOI: 10.3934/ipi.2018047
Publisher version: https://doi.org/10.3934/ipi.2018047
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: Science & Technology, Physical Sciences, Mathematics, Applied, Physics, Mathematical, Mathematics, Physics, Ill-posed problems, regularization parameter choice, risk estimators, Stein's method, discrepancy principle, GENERALIZED CROSS-VALIDATION, MONTE-CARLO SURE, IMAGE-RESTORATION, DISCREPANCY PRINCIPLE, HIERARCHICAL MODEL, L-CURVE, NOISE, RECONSTRUCTION, SELECTION, ALGORITHMS
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Computer Science
URI: https://discovery.ucl.ac.uk/id/eprint/10060934
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