The aim of this paper is to provide a theoretically founded investigation of state-of-the-art learning approaches for inverse problems. We give an extended definition of regularization methods and their convergence in terms of the underlying data distributions, which paves the way for future theoretical studies. Based on a simple spectral learning model previously introduced for supervised learning, we investigate some key properties of different learning paradigms for inverse problems, which can be formulated independently of specific architectures. In particular we investigate the regularization properties, bias, and critical dependence on training data distributions. Moreover, our framework allows to highlight and compare the specific behavior of the different paradigms in the infinite-dimensional limit.

翻译：本文旨在对反问题中的前沿学习方法进行理论基础的探究。我们给出了正则化方法的扩展定义及其在数据分布下的收敛性，这为未来的理论研究铺平了道路。基于先前为监督学习引入的简单光谱学习模型，我们研究了反问题中不同学习范式的一些关键特性，这些特性可以独立于具体架构进行表述。特别地，我们探究了正则化性质、偏差以及对训练数据分布的关键依赖。此外，我们的框架能够突出并比较不同范式在无穷维极限下的特定行为。