Learned reconstruction methods for inverse problems: sample error estimates

Learning-based and data-driven techniques have recently become a subject of primary interest in the field of reconstruction and regularization of inverse problems. Besides the development of novel methods, yielding excellent results in several applications, their theoretical investigation has attracted growing interest, e. g., on the topics of reliability, stability, and interpretability. In this chapter, a general framework is described, allowing us to interpret many of these techniques in the context of statistical learning. This is not intended to provide a complete survey of existing methods, but rather to put them in a working perspective, which naturally allows their theoretical treatment. The main goal of this dissertation is thereby to address the generalization properties of learned reconstruction methods, and specifically to perform their sample error analysis. This task, well developed in statistical learning, consists of estimating the dependence of the learned operators with respect to the data employed for their training. A rather general strategy is proposed, whose assumptions are met for a large class of inverse problems and learned methods, as depicted via a selection of examples.