To Be or Not to Be Stable, That Is the Question: Understanding Neural Networks for Inverse Problems

The solution of linear inverse problems arising, for example, in signal and image processing is a challenging problem since the ill-conditioning amplifies, in the solution, the noise present in the data. Recently introduced algorithms based on deep learning overwhelm the more traditional model-based approaches in performance, but they typically suffer from instability with respect to data perturbation. In this paper, we theoretically analyze the trade-off between stability and accuracy of neural networks when used to solve linear imaging inverse problems for cases that are not underdetermined. Moreover, we propose different supervised and unsupervised solutions to increase the network stability and maintain a good accuracy, by means of regularization properties inherited from a model-based iterative scheme during the network training. Extensive numerical experiments on image deblurring confirm the theoretical results and the effectiveness of the proposed deep learning--based approaches to handle noise on the data.