Foundations of Supervised Deep Learning for Inverse Problems

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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.