  Recently, a large number of efficient deep learning methods for solving
inverse problems have been developed and show outstanding numerical
performance. For these deep learning methods, however, a solid theoretical
foundation in the form of reconstruction guarantees is missing. In contrast,
for classical reconstruction methods, such as convex variational and
frame-based regularization, theoretical convergence and convergence rate
results are well established. In this paper, we introduce deep synthesis
regularization (DESYRE) using neural networks as nonlinear synthesis operator
bridging the gap between these two worlds. The proposed method allows to
exploit the deep learning benefits of being well adjustable to available
training data and on the other hand comes with a solid mathematical foundation.
We present a complete convergence analysis with convergence rates for the
proposed deep synthesis regularization. We present a strategy for constructing
a synthesis network as part of an analysis-synthesis sequence together with an
appropriate training strategy. Numerical results show the plausibility of our
approach.
