Data-proximal null-space networks for inverse problems

Abstract: 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.