To be or not to be stable, that is the question: understanding neural networks for inverse problems
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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 deep-learning based algorithms overwhelm the more traditional model-based approaches but they typically suffer from instability with respect to data perturbation. In this paper, we theoretically analyse 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 by maintaining 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 networks in solving inverse problems with stability with respect to noise.
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