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 the noise on the data. Recently introduced algorithms based on deep learning overwhelm the more traditional model-based approaches, but they typically suffer from instability with respect to data perturbation. In this paper, we theoretically analyze the trade-off between neural networks stability and accuracy in the solution of linear inverse problems. Moreover, we propose different supervised and unsupervised solutions to increase network stability that maintains good accuracy by inheriting, in the network training, regularization from a model-based iterative scheme. Extensive numerical experiments on image deblurring confirm the theoretical results and the effectiveness of the proposed deep learning-based solutions to stably solve noisy inverse problems.