ipINPEEYInverse ProblemsIPInverse Problems0266-56111361-6420IOP Publishingipabb61b10.1088/1361-6420/abb61babb61bIP-102693.R1PaperData driven regularization by projectionAspriAndrea1andrea.aspri@ricam.oeaw.ac.at0000-0002-6339-652XKorolevYury2*y.korolev@damtp.cam.ac.uk0000-0001-9378-7452ScherzerOtmar13otmar.scherzer@univie.ac.at Johann Radon Institute for Computational and Applied Mathematics, Altenberger Straße 69, 4040 Linz, Austria Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom Department of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria

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We study linear inverse problems under the premise that the forward operator is not at hand but given indirectly through some input-output training pairs. We demonstrate that regularization by projection and variational regularization can be formulated by using the training data only and without making use of the forward operator. We study convergence and stability of the regularized solutions in view of Seidman (1980 J. Optim. Theory Appl. 30 535), who showed that regularization by projection is not convergent in general, by giving some insight on the generality of Seidman’s nonconvergence example. Moreover, we show, analytically and numerically, that regularization by projection is indeed capable of learning linear operators, such as the Radon transform.

data driven regularizationvariational regularizationregularization by projectioninverse problemsGram–Schmidt orthogonalizationRoyal Societyhttps://doi.org/10.13039/501100000288 NF170045Austrian Science Fundhttps://doi.org/10.13039/501100002428 I3661-N27 SFB F68 F6807-N36ccc1361-6420/20/125009+35$33.00printedPrinted in the UKcrossmarkyes