Recovery Guarantees of Unsupervised Neural Networks for Inverse Problems
  trained with Gradient Descent
Nathan Buskulic Jalal Fadili Yvain Quéau
Advanced machine learning methods, and more prominently neural networks, have
become standard to solve inverse problems over the last years. However, the
theoretical recovery guarantees of such methods are still scarce and difficult
to achieve. Only recently did unsupervised methods such as Deep Image Prior
(DIP) get equipped with convergence and recovery guarantees for generic loss
functions when trained through gradient flow with an appropriate
initialization. In this paper, we extend these results by proving that these
guarantees hold true when using gradient descent with an appropriately chosen
step-size/learning rate. We also show that the discretization only affects the
overparametrization bound for a two-layer DIP network by a constant and thus
that the different guarantees found for the gradient flow will hold for
gradient descent.