Neural networks have become a prominent approach to solve inverse problems in recent years. While a plethora of such methods was developed to solve inverse problems empirically, we are still lacking clear theoretical guarantees for these methods. On the other hand, many works proved convergence to optimal solutions of neural networks in a more general setting using overparametrization as a way to control the Neural Tangent Kernel. In this work we investigate how to bridge these two worlds and we provide deterministic convergence and recovery guarantees for the class of unsupervised feedforward multilayer neural networks trained to solve inverse problems. We also derive overparametrization bounds under which a two-layers Deep Inverse Prior network with smooth activation function will benefit from our guarantees.

翻译：近年来，神经网络已成为求解反问题的重要方法。尽管大量此类方法通过经验手段解决了反问题，但这些方法的理论保证仍显不足。另一方面，许多研究通过过参数化控制神经正切核，在更一般的设定下证明了神经网络收敛到最优解。本文旨在弥合这两个领域间的鸿沟，针对为求解反问题而训练的无监督前馈多层神经网络，提供确定性的收敛性与恢复保证。我们同时推导出过参数化边界条件，在此条件下，采用平滑激活函数的两层深度逆先验网络将受益于本文的保证。