Neural networks have become a prominent approach to solve inverse problems in recent years. Amongst the different existing methods, the Deep Image/Inverse Priors (DIPs) technique is an unsupervised approach that optimizes a highly overparametrized neural network to transform a random input into an object whose image under the forward model matches the observation. However, the level of overparametrization necessary for such methods remains an open problem. In this work, we aim to investigate this question for a two-layers neural network with a smooth activation function. We provide overparametrization bounds under which such network trained via continuous-time gradient descent will converge exponentially fast with high probability which allows to derive recovery prediction bounds. This work is thus a first step towards a theoretical understanding of overparametrized DIP networks, and more broadly it participates to the theoretical understanding of neural networks in inverse problem settings.

翻译：近年来，神经网络已成为解决逆问题的重要方法。在现有不同技术中，深度图像/逆先验（DIP）方法是一种无监督方法，它通过优化一个高度过参数化的神经网络，将随机输入转换为前向模型下其图像与观测值匹配的目标。然而，此类方法所需的过参数化程度仍是一个未解问题。本文旨在针对具有光滑激活函数的两层神经网络研究该问题。我们给出了过参数化边界，在此边界下，通过连续时间梯度下降训练的网络将以高概率指数级快速收敛，从而推导出恢复预测边界。因此，本文是理解过参数化DIP网络理论的第一步，更广泛而言，它有助于在逆问题场景下建立神经网络的理论理解。