Learned reconstruction methods for inverse problems: sample error
  estimates
Luca Ratti
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 work, 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 in 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.