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逆问题的主流方法之一。尽管大量此类方法通过经验手段解决了逆问题，但目前仍缺乏清晰的理论保证。另一方面，许多研究通过过参数化来控制神经正切核，在更广泛的框架下证明了神经网络收敛至最优解。本文旨在探索如何弥合这两个领域，并为求解逆问题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。此外，我们推导出在何种过参数化边界条件下，采用光滑激活函数的两层深度逆先验网络能够受益于这些保证。