Deep null space learning for inverse problems: convergence analysis and rates

Recently, deep learning based methods appeared as a new paradigm for solving inverse problems. These methods empirically show excellent performance but lack of theoretical justification; in particular, no results on the regularization properties are available. In particular, this is the case for two-step deep learning approaches, where a classical reconstruction method is applied to the data in a first step and a trained deep neural network is applied to improve results in a second step. In this paper, we close the gap between practice and theory for a particular network structure in a two-step approach. For that purpose, we propose using so-called null space networks and introduce the concept of Phi-regularization. Combined with a standard regularization method as reconstruction layer. the proposed deep null space learning approach is shown to be a Phi-regularization method; convergence rates are also derived. The proposed null space network structure naturally preserves data consistency which is considered as key property of neural networks for solving inverse problems.