Foundations of Supervised Deep Learning for Inverse Problems

在过去的十年中，深度学习方法在各种数据处理任务中表现出色，包括求解不适定逆问题。虽然许多工作已经证明了这种深度网络在图像重建方面优于经典（例如变分）正则化方法，但真正理解深度网络作为正则化技术的理论基础在很大程度上是缺失的，这种正则化技术可以重建解决方案对数据的连续依赖。本提案的目标是分三步缩小这一差距：首先，我们将研究将数据的离散（有限维）表示映射到解决方案的离散表示的深度网络架构，我们以与经典正则化方法的离散化类似的方式建立数据一致性。其次，我们将研究如何设计、解释和训练深度网络作为无限维函数空间之间的映射。最后，我们将研究如何将有限维误差估计转移到无限维设置，通过对重建数据和训练网络的数据进行适当的假设，并利用适当的正则化方案对网络本身进行监督训练。我们将评估我们的网络，并以图像反卷积和计算机断层扫描作为常见的测试设置，对成像中的线性逆问题进行数值测试。

Over the last decade, deep learning methods have excelled at various data processing tasks including the solution of ill-posed inverse problems. While many works have demonstrated the superiority of such deep networks over classical (e.g. variational) regularization methods in image reconstruction, the theoretical foundation for truly understanding deep networks as regularization techniques, which can reestablish a continuous dependence of the solution on the data, is largely missing. The goal of this proposal is to close this gap in three step: First we will study deep network architectures that map a discrete (finite dimensional) representation of the data to a discrete representation of the solution in such a way, that we establish data consistency in a similar way as discretizations of classical regularization methods do. Secondly, we will study how to design, interpret and train deep networks as mappings between infinite-dimensional function spaces. Finally, we will investigate how to transfer finite-dimensional error estimates to the infinite-dimensional setting, by making suitable assumptions on the data to reconstruct as well as the data to train the network with, and utilizing suitable regularization schemes for the supervised training of the networks themselves. We will evaluate our networks and test our finding numerically on linear inverse problems in imaging using image deconvolution and computerized tomography as common test settings.