To be or not to be stable, that is the question: understanding neural networks for inverse problems
This addresses the problem of instability in deep learning methods for inverse problems, which is crucial for applications like signal and image processing, though it is incremental as it builds on existing neural network approaches.
The paper tackles the instability of neural networks in solving linear imaging inverse problems, such as image deblurring, by theoretically analyzing the trade-off between stability and accuracy and proposing supervised and unsupervised solutions to enhance stability while maintaining good accuracy, with extensive numerical experiments confirming effectiveness.
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.