Learned reconstruction methods for inverse problems: sample error estimates

The mathematical treatment of inverse problems has proved to be a lively and attractive research field, driven and motivated by a wide variety of applications and by the theoretical challenges induced by their ill-posed nature. In order to provide more accurate and reliable strategies, especially for the reconstruction task, the scientific research in the field has shown a growing interest in the topic of learned reconstruction, or data-driven, methods, to combine classical, model-based approaches with valuable information of statistical nature. This has represented a natural outcome and development of the analysis of inverse problems, both on a numerical and on a theoretical side: indeed, the idea of leveraging prior knowledge on the solution has traditionally been considered to mitigate ill-posedness, as a regularization tool as much as a support for the reconstruction. We have now witnessed the emergence of several learning-based approaches to inverse problems, providing, in many cases, striking numerical results in terms of accuracy and efficiency. Moreover, significant interest has grown in the direction of theoretical guarantees for such techniques, spanning from the demand of interpretability and reliability, up to the issues of stability and convergence [8, 55]. Despite several distinct avenues have emerged, which are now well-established and are developing independently (to name a few: generative models, unrolled techniques, Plug and Play schemes), it is possible to provide a unifying overview of them from the point of view of statistical learning theory [20]. In this context, the goal pursued by this paper is twofold. On the one side, it aims to provide a general theoretical framework in statistical learning that is able to comprehend a large family of data-driven reconstruction methods.