Inverse problems are inherently ill-posed and therefore require regularization techniques to achieve a stable solution. While traditional variational methods have well-established theoretical foundations, recent advances in machine learning based approaches have shown remarkable practical performance. However, the theoretical foundations of learning-based methods in the context of regularization are still underexplored. In this paper, we propose a general framework that addresses the current gap between learning-based methods and regularization strategies. In particular, our approach emphasizes the crucial role of data consistency in the solution of inverse problems and introduces the concept of data-proximal null-space networks as a key component for their solution. We provide a complete convergence analysis by extending the concept of regularizing null-space networks with data proximity in the visual part. We present numerical results for limited-view computed tomography to illustrate the validity of our framework.

翻译：逆问题具有本质上的不适定性，因此需要正则化技术来获得稳定解。传统变分方法具备完善的理论基础，而基于机器学习的方法近年来展现出卓越的实践性能。然而，学习类方法在正则化框架下的理论基础仍待深入探索。本文提出一个通用框架，旨在弥合学习方法与正则化策略之间的理论鸿沟。具体而言，我们着重强调数据一致性在逆问题求解中的关键作用，并引入数据近端零空间网络作为解决方案的核心要素。通过将数据近端性融入视觉部分的正则化零空间网络概念，我们完成了完整的收敛性分析。最后，我们展示了有限视角计算机断层扫描的数值结果，以验证所提出框架的有效性。