  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 not under-determined cases. 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 and pre-processing
stabilizing operator in the neural networks. 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.
