# Learned reconstruction methods with convergence guarantees

arXiv: https://arxiv.org/abs/2206.05431 · HTML: https://mlsanity.com/paper/2206.05431/

## Key facts

- Cited by 46 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 64 references that are in the map.
- First posted on arXiv 11 Jun 2022.

## Builds on (most cited first, 30 of 37)

- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Learning a Variational Network for Reconstruction of Accelerated MRI Data](https://mlsanity.com/paper/1704.00447/) (2017)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [Learned Primal-dual Reconstruction](https://mlsanity.com/paper/1707.06474/) (2017)
- [Plug-and-Play ADMM for Image Restoration: Fixed Point Convergence and Applications](https://mlsanity.com/paper/1605.01710/) (2016)
- [On instabilities of deep learning in image reconstruction - Does AI come at a cost?](https://mlsanity.com/paper/1902.05300/) (2019)
- [Plug-and-Play Methods Provably Converge with Properly Trained Denoisers](https://mlsanity.com/paper/1905.05406/) (2019)
- [Deep Equilibrium Architectures for Inverse Problems in Imaging](https://mlsanity.com/paper/2102.07944/) (2021)
- [Gradient Step Denoiser for convergent Plug-and-Play](https://mlsanity.com/paper/2110.03220/) (2021)
- [Adversarial Regularizers in Inverse Problems](https://mlsanity.com/paper/1805.11572/) (2018)
- [Regularization by Denoising: Clarifications and New Interpretations](https://mlsanity.com/paper/1806.02296/) (2018)
- [Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie](https://mlsanity.com/paper/2103.04715/) (2021)
- [NETT: Solving Inverse Problems with Deep Neural Networks](https://mlsanity.com/paper/1803.00092/) (2018)
- [JFB: Jacobian-Free Backpropagation for Implicit Networks](https://mlsanity.com/paper/2103.12803/) (2021)
- [Plug-and-Play Priors for Bright Field Electron Tomography and Sparse Interpolation](https://mlsanity.com/paper/1512.07331/) (2015)
- [Learning Maximally Monotone Operators for Image Recovery](https://mlsanity.com/paper/2012.13247/) (2020)
- [Deep Bayesian Inversion](https://mlsanity.com/paper/1811.05910/) (2018)
- [Solving Inverse Problems With Deep Neural Networks -- Robustness Included?](https://mlsanity.com/paper/2011.04268/) (2020)
- [Learned convex regularizers for inverse problems](https://mlsanity.com/paper/2008.02839/) (2020)
- [Algorithmic Guarantees for Inverse Imaging with Untrained Network Priors](https://mlsanity.com/paper/1906.08763/) (2019)
- [End-to-end reconstruction meets data-driven regularization for inverse problems](https://mlsanity.com/paper/2106.03538/) (2021)
- [SHINE: SHaring the INverse Estimate from the forward pass for bi-level optimization and implicit models](https://mlsanity.com/paper/2106.00553/) (2021)
- [Solving Inverse Problems by Joint Posterior Maximization with Autoencoding Prior](https://mlsanity.com/paper/2103.01648/) (2021)
- [Bayesian Imaging With Data-Driven Priors Encoded by Neural Networks: Theory, Methods, and Algorithms](https://mlsanity.com/paper/2103.10182/) (2021)
- [On the well-posedness of Bayesian inverse problems](https://mlsanity.com/paper/1902.10257/) (2019)
- [Fixed-Point and Objective Convergence of Plug-and-Play Algorithms](https://mlsanity.com/paper/2104.10348/) (2021)
- [Inference with Deep Generative Priors in High Dimensions](https://mlsanity.com/paper/1911.03409/) (2019)

## Built on by (most cited first, 30 of 46)

- [Learning Weakly Convex Regularizers for Convergent Image-Reconstruction Algorithms](https://mlsanity.com/paper/2308.10542/) (2023)
- [Equivariant Bootstrapping for Uncertainty Quantification in Imaging Inverse Problems](https://mlsanity.com/paper/2310.11838/) (2023)
- [Provably Convergent Plug-and-Play Quasi-Newton Methods](https://mlsanity.com/paper/2303.07271/) (2023)
- [Learning Regularization Functionals for Inverse Problems: A Comparative Study](https://mlsanity.com/paper/2510.01755/) (2025)
- [Weakly Convex Regularisers for Inverse Problems: Convergence of Critical Points and Primal-Dual Optimisation](https://mlsanity.com/paper/2402.01052/) (2024)
- [Convergent regularization in inverse problems and linear plug-and-play denoisers](https://mlsanity.com/paper/2307.09441/) (2023)
- [Learning from small data sets: Patch-based regularizers in inverse problems for image reconstruction](https://mlsanity.com/paper/2312.16611/) (2023)
- [The Perception-Robustness Tradeoff in Deterministic Image Restoration](https://mlsanity.com/paper/2311.09253/) (2023)
- [Model-corrected learned primal-dual models for fast limited-view photoacoustic tomography](https://mlsanity.com/paper/2304.01963/) (2023)
- [Do Bayesian imaging methods report trustworthy probabilities?](https://mlsanity.com/paper/2405.08179/) (2024)
- [Learning few-step posterior samplers by unfolding and distillation of diffusion models](https://mlsanity.com/paper/2507.02686/) (2025)
- [Stability of Data-Dependent Ridge-Regularization for Inverse Problems](https://mlsanity.com/paper/2406.12289/) (2024)
- [Good Things Come in Pairs: Paired Autoencoders for Inverse Problems](https://mlsanity.com/paper/2505.06549/) (2025)
- [Provably Convergent Data-Driven Convex-Nonconvex Regularization](https://mlsanity.com/paper/2310.05812/) (2023)
- [Unsupervised approaches based on optimal transport and convex analysis for inverse problems in imaging](https://mlsanity.com/paper/2311.08972/) (2023)
- [Inverse Problems with Learned Forward Operators](https://mlsanity.com/paper/2311.12528/) (2023)
- [Convergence and Recovery Guarantees of Unsupervised Neural Networks for Inverse Problems](https://mlsanity.com/paper/2309.12128/) (2023)
- [Unsupervised Training of Convex Regularizers using Maximum Likelihood Estimation](https://mlsanity.com/paper/2404.05445/) (2024)
- [Bayesian computation with generative diffusion models by Multilevel Monte Carlo](https://mlsanity.com/paper/2409.15511/) (2024)
- [Convergence Guarantees of Overparametrized Wide Deep Inverse Prior](https://mlsanity.com/paper/2303.11265/) (2023)
- [Learned Regularization for Inverse Problems: Insights from a Spectral Model](https://mlsanity.com/paper/2312.09845/) (2023)
- [Invertible ResNets for Inverse Imaging Problems: Competitive Performance with Provable Regularization Properties](https://mlsanity.com/paper/2409.13482/) (2024)
- [Efficient Bayesian Computation Using Plug-and-Play Priors for Poisson Inverse Problems](https://mlsanity.com/paper/2503.16222/) (2025)
- [Prompting Lipschitz-constrained network for multiple-in-one sparse-view CT reconstruction](https://mlsanity.com/paper/2511.20296/) (2025)
- [Error Analysis of Bayesian Inverse Problems with Generative Priors](https://mlsanity.com/paper/2601.17374/) (2026)
- [Learned reconstruction methods for inverse problems: sample error estimates](https://mlsanity.com/paper/2312.14078/) (2023)
- [Robustness and Exploration of Variational and Machine Learning Approaches to Inverse Problems: An Overview](https://mlsanity.com/paper/2402.12072/) (2024)
- [Robust width: A lightweight and certifiable adversarial defense](https://mlsanity.com/paper/2405.15971/) (2024)
- [Universal Architectures for the Learning of Polyhedral Norms and Convex Regularizers](https://mlsanity.com/paper/2503.19190/) (2025)
- [Graph Neural Regularizers for PDE Inverse Problems](https://mlsanity.com/paper/2510.21012/) (2025)

## Abstract (arXiv)

In recent years, deep learning has achieved remarkable empirical success for image reconstruction. This has catalyzed an ongoing quest for precise characterization of correctness and reliability of data-driven methods in critical use-cases, for instance in medical imaging. Notwithstanding the excellent performance and efficacy of deep learning-based methods, concerns have been raised regarding their stability, or lack thereof, with serious practical implications. Significant advances have been made in recent years to unravel the inner workings of data-driven image recovery methods, challenging their widely perceived black-box nature. In this article, we will specify relevant notions of convergence for data-driven image reconstruction, which will form the basis of a survey of learned methods with mathematically rigorous reconstruction guarantees. An example that is highlighted is the role of ICNN, offering the possibility to combine the power of deep learning with classical convex regularization theory for devising methods that are provably convergent. This survey article is aimed at both methodological researchers seeking to advance the frontiers of our understanding of data-driven image reconstruction methods as well as practitioners, by providing an accessible description of useful convergence concepts and by placing some of the existing empirical practices on a solid mathematical foundation.