  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 new network structure in a two-step approach. For that purpose, we
propose so called null space networks and introduce the concept of
M-regularization. Combined with a standard regularization method as
reconstruction layer, the proposed deep null space learning approach is shown
to be a M-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.
