# To be or not to be stable, that is the question: understanding neural networks for inverse problems

arXiv: https://arxiv.org/abs/2211.13692 · HTML: https://mlsanity.com/paper/2211.13692/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 62 references that are in the map.
- First posted on arXiv 24 Nov 2022.

## Builds on (most cited first, 20 of 20)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [DeepFool: a simple and accurate method to fool deep neural networks](https://mlsanity.com/paper/1511.04599/) (2015)
- [Deep Multi-scale Convolutional Neural Network for Dynamic Scene Deblurring](https://mlsanity.com/paper/1612.02177/) (2016)
- [On instabilities of deep learning in image reconstruction - Does AI come at a cost?](https://mlsanity.com/paper/1902.05300/) (2019)
- [Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't](https://mlsanity.com/paper/2009.10713/) (2020)
- [CNN-Based Projected Gradient Descent for Consistent Image Reconstruction](https://mlsanity.com/paper/1709.01809/) (2017)
- [The Modern Mathematics of Deep Learning](https://mlsanity.com/paper/2105.04026/) (2021)
- [Solving Inverse Problems With Deep Neural Networks -- Robustness Included?](https://mlsanity.com/paper/2011.04268/) (2020)
- [The gap between theory and practice in function approximation with deep neural networks](https://mlsanity.com/paper/2001.07523/) (2020)
- [Can stable and accurate neural networks be computed? -- On the barriers of deep learning and Smale's 18th problem](https://mlsanity.com/paper/2101.08286/) (2021)
- [Measuring Robustness in Deep Learning Based Compressive Sensing](https://mlsanity.com/paper/2102.06103/) (2021)
- [Data augmentation for deep learning based accelerated MRI reconstruction with limited data](https://mlsanity.com/paper/2106.14947/) (2021)
- [The mathematics of adversarial attacks in AI -- Why deep learning is unstable despite the existence of stable neural networks](https://mlsanity.com/paper/2109.06098/) (2021)
- [Deep Learning-Based Solvability of Underdetermined Inverse Problems in Medical Imaging](https://mlsanity.com/paper/2001.01432/) (2020)
- [A review and experimental evaluation of deep learning methods for MRI reconstruction](https://mlsanity.com/paper/2109.08618/) (2021)
- [Data Consistent Artifact Reduction for Limited Angle Tomography with Deep Learning Prior](https://mlsanity.com/paper/1908.06792/) (2019)
- [Adversarial Robustness of MR Image Reconstruction under Realistic Perturbations](https://mlsanity.com/paper/2208.03161/) (2022)
- [Validation and Generalizability of Self-Supervised Image Reconstruction Methods for Undersampled MRI](https://mlsanity.com/paper/2201.12535/) (2022)

## Built on by (most cited first, 8 of 8)

- [Ambiguity in solving imaging inverse problems with deep learning based operators](https://mlsanity.com/paper/2305.19774/) (2023)
- [Transformer learns the cross-task prior and regularization for in-context learning](https://mlsanity.com/paper/2505.12138/) (2025)
- [Learning Causal Graphs at Scale: A Foundation Model Approach](https://mlsanity.com/paper/2506.18285/) (2025)
- [Space-Variant Total Variation boosted by learning techniques in few-view tomographic imaging](https://mlsanity.com/paper/2404.16900/) (2024)
- [A Diffusion-Based Generative Prior Approach to Sparse-view Computed Tomography](https://mlsanity.com/paper/2602.10722/) (2026)
- [A Deep Risk Estimator for Known Operator Learning](https://mlsanity.com/paper/2605.08517/) (2026)
- [An incremental algorithm for non-convex AI-enhanced medical image processing](https://mlsanity.com/paper/2505.08324/) (2025)
- [Learning Generalizable Neural Operators for Inverse Problems](https://mlsanity.com/paper/2512.18120/) (2025)

## Abstract (arXiv)

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.