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.
Highlights
Linear inverse problems are concerned with the reconstruction of a quantity u ∈ U from indirect measurements y ∈ Y which are related by the linear forward operator A : U → Y
We demonstrate that regularisation by projection and variational regularisation can be formulated by using the training data only and without making use of the forward operator
We demonstrate that the amount of training data plays the role of a regularisation parameter, for noisy training data the size of the training set should be chosen in agreement with the noise level
Summary
Linear inverse problems are concerned with the reconstruction of a quantity u ∈ U from indirect measurements y ∈ Y which are related by the linear forward operator A : U → Y. Our main goal in this article is to carry over some classical results in regularisation theory to the model-free setting, with an overarching theme of using projections on subspaces defined by training data in the framework of regularisation by projection. This perspective requires new, data driven regularity conditions, for which we show a relation to source conditions in special cases, whilst in general such relationship remains an open question. This is in accordance with the results in [24, Thm. 4.2] on training neural networks from noisy data, where the number of neurons in the network plays a role similar to the number of training inputs in our setting
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