Title: Data-driven regularization and infinite dimensional learning architectures
Abstract: In recent years, numerous data-driven strategies to solve inverse problems have emerged. In this context, the inverse problem is usually formulated in a probabilistic setting as it is known from the field of Bayesian inverse problems. The reconstruction approaches themselves however, are often deterministic and more related to classical methods. In this talk, we transfer the classical concept of regularization to a typical self-supervised learning setting, in which access to ground-truth data and the forward operator is assumed. To take into account instabilities arising from an infinite dimensional problem formulation, the theoretical framework is based on infinite dimensional reconstruction operators. Therefore, we first study linear, data-driven reconstruction operators based on spectral filtering and derive conditions under which convergent data-driven regularization methods can be obtained. We then discuss possibilities and challenges to implement such reconstruction operators in practice on the basis of Fourier neural operators and their connection to convolutional neural networks.