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.

翻译：基于学习与数据驱动的技术近期已成为逆问题重建与正则化领域的主要关注对象。除开发在多项应用中取得优异成果的新方法外，其理论探索亦日益引发学界兴趣，例如在可靠性、稳定性和可解释性等议题上。本文描述了一个通用框架，使我们能够在统计学习的背景下理解众多此类技术。本文无意对现有方法进行全面综述，而是将其置于一个便于理论处理的实践视角中。因此，本文的主要目标是探讨学习型重建方法的泛化性质，特别是对其样本误差进行分析。这一在统计学习中已充分发展的任务，旨在估计学习算子对其训练所用数据的依赖性。本文提出了一种相当通用的策略，其假设适用于一大类逆问题及学习型方法——正如通过一系列实例所展示的那样。