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：逆问题本质上是不适定的，因此需要正则化技术来获得稳定解。传统变分方法具有完善的理论基础，而近年来基于机器学习的先进方法在实际应用中展现出显著性能。然而，关于基于学习方法在正则化框架中的理论根基仍待深入探索。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现存差距。我们特别强调数据一致性在逆问题求解中的关键作用，并提出数据近端零空间网络作为其求解的核心组件。通过在视觉部分引入数据邻近性扩展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，针对有限角度计算机断层扫描问题呈现的数值结果验证了该框架的有效性。