Data driven regularization by projection. (3rd December 2020)

• Record Type: Journal Article
• Authors: Aspri, Andrea; Korolev, Yury; Scherzer, Otmar
• Is Part Of: Inverse problems. Volume 36:Number 12(2020)
• Publication Date: 2020-12-03
• Subjects: data driven regularization -- variational regularization -- regularization by projection -- inverse problems -- Gram–Schmidt orthogonalization; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/abb61b
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: 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.