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 networks for regularizing inverse problems (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.

近来，已经开发了许多用于解决反问题的有效深度学习方法，并且这些方法具有出色的数值性能。但是，对于这些深度学习方法，缺少以重建保证形式的扎实的理论基础。相比之下，对于经典的重建方法（例如凸变分和基于帧的正则化），理论上的收敛性和收敛速度结果得到了很好的确立。在本文中，我们介绍了使用神经网络作为非线性综合算子弥合这两个世界之间差距的，用于正则化反问题的深度综合网络（DESYRE）。提出的方法可以利用深度学习的好处，即可以很好地适应可用的训练数据，另一方面具有扎实的数学基础。我们针对提出的深度综合正则化提出了具有收敛速度的完整收敛分析。我们提出了一种构建综合网络的策略，并将其作为分析综合序列的一部分，并提供了适当的培训策略。数值结果表明了我们方法的合理性。