# End-to-end reconstruction meets data-driven regularization for inverse problems

arXiv: https://arxiv.org/abs/2106.03538 · HTML: https://mlsanity.com/paper/2106.03538/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 49 references that are in the map.
- First posted on arXiv 7 Jun 2021.

## Builds on (most cited first, 25 of 25)

- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing](https://mlsanity.com/paper/1912.10557/) (2019)
- [The Little Engine that Could: Regularization by Denoising (RED)](https://mlsanity.com/paper/1611.02862/) (2016)
- [Deep Unfolding Network for Image Super-Resolution](https://mlsanity.com/paper/2003.10428/) (2020)
- [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)
- [Image reconstruction by domain transform manifold learning](https://mlsanity.com/paper/1704.08841/) (2017)
- [Plug-and-Play Methods Provably Converge with Properly Trained Denoisers](https://mlsanity.com/paper/1905.05406/) (2019)
- [Solving ill-posed inverse problems using iterative deep neural networks](https://mlsanity.com/paper/1704.04058/) (2017)
- [Learning Proximal Operators: Using Denoising Networks for Regularizing Inverse Imaging Problems](https://mlsanity.com/paper/1704.03488/) (2017)
- [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)
- [NETT: Solving Inverse Problems with Deep Neural Networks](https://mlsanity.com/paper/1803.00092/) (2018)
- [An Online Plug-and-Play Algorithm for Regularized Image Reconstruction](https://mlsanity.com/paper/1809.04693/) (2018)
- [RARE: Image Reconstruction using Deep Priors Learned without Ground Truth](https://mlsanity.com/paper/1912.05854/) (2019)
- [Learning Maximally Monotone Operators for Image Recovery](https://mlsanity.com/paper/2012.13247/) (2020)
- [Learned convex regularizers for inverse problems](https://mlsanity.com/paper/2008.02839/) (2020)
- [Total Deep Variation for Linear Inverse Problems](https://mlsanity.com/paper/2001.05005/) (2020)
- [Unsupervised MRI Reconstruction with Generative Adversarial Networks](https://mlsanity.com/paper/2008.13065/) (2020)
- [SIMBA: Scalable Inversion in Optical Tomography using Deep Denoising Priors](https://mlsanity.com/paper/1911.13241/) (2019)
- [Wasserstein GANs for MR Imaging: from Paired to Unpaired Training](https://mlsanity.com/paper/1910.07048/) (2019)
- [Unsupervised Deep Basis Pursuit: Learning inverse problems without ground-truth data](https://mlsanity.com/paper/1910.13110/) (2019)
- [Self-supervised learning of inverse problem solvers in medical imaging](https://mlsanity.com/paper/1905.09325/) (2019)

## Built on by (most cited first, 30 of 32)

- [Learned reconstruction methods with convergence guarantees](https://mlsanity.com/paper/2206.05431/) (2022)
- [What's in a Prior? Learned Proximal Networks for Inverse Problems](https://mlsanity.com/paper/2310.14344/) (2023)
- [PatchNR: Learning from Very Few Images by Patch Normalizing Flow Regularization](https://mlsanity.com/paper/2205.12021/) (2022)
- [Online Deep Equilibrium Learning for Regularization by Denoising](https://mlsanity.com/paper/2205.13051/) (2022)
- [Deep Bayesian inference for seismic imaging with tasks](https://mlsanity.com/paper/2110.04825/) (2021)
- [ASPIRE: Iterative Amortized Posterior Inference for Bayesian Inverse Problems](https://mlsanity.com/paper/2405.05398/) (2024)
- [QN-Mixer: A Quasi-Newton MLP-Mixer Model for Sparse-View CT Reconstruction](https://mlsanity.com/paper/2402.17951/) (2024)
- [CoRRECT: A Deep Unfolding Framework for Motion-Corrected Quantitative R2* Mapping](https://mlsanity.com/paper/2210.06330/) (2022)
- [Solving Low-Dose CT Reconstruction via GAN with Local Coherence](https://mlsanity.com/paper/2309.13584/) (2023)
- [Unsupervised Imaging Inverse Problems with Diffusion Distribution Matching](https://mlsanity.com/paper/2506.14605/) (2025)
- [Stochastic Primal-Dual Deep Unrolling](https://mlsanity.com/paper/2110.10093/) (2021)
- [NF-ULA: Langevin Monte Carlo with Normalizing Flow Prior for Imaging Inverse Problems](https://mlsanity.com/paper/2304.08342/) (2023)
- [Provably Convergent Data-Driven Convex-Nonconvex Regularization](https://mlsanity.com/paper/2310.05812/) (2023)
- [LCOT: Linear circular optimal transport](https://mlsanity.com/paper/2310.06002/) (2023)
- [Unsupervised approaches based on optimal transport and convex analysis for inverse problems in imaging](https://mlsanity.com/paper/2311.08972/) (2023)
- [Benchmarking Self-Supervised Learning Methods for Accelerated MRI Reconstruction](https://mlsanity.com/paper/2502.14009/) (2025)
- [Unsupervised Training of Convex Regularizers using Maximum Likelihood Estimation](https://mlsanity.com/paper/2404.05445/) (2024)
- [Neural Incremental Data Assimilation](https://mlsanity.com/paper/2406.15076/) (2024)
- [Swap-Net: A Memory-Efficient 2.5D Network for Sparse-View 3D Cone Beam CT Reconstruction](https://mlsanity.com/paper/2410.10836/) (2024)
- [Estimating a potential without the agony of the partition function](https://mlsanity.com/paper/2208.09433/) (2022)
- [A new method for determining Wasserstein 1 optimal transport maps from Kantorovich potentials, with deep learning applications](https://mlsanity.com/paper/2211.00820/) (2022)
- [A Lipschitz spaces view of infinitely wide shallow neural networks](https://mlsanity.com/paper/2410.14591/) (2024)
- [An invertible generative model for forward and inverse problems](https://mlsanity.com/paper/2509.03910/) (2025)
- [Trust the Critics: Generatorless and Multipurpose WGANs with Initial Convergence Guarantees](https://mlsanity.com/paper/2111.15099/) (2021)
- [Learned Interferometric Imaging for the SPIDER Instrument](https://mlsanity.com/paper/2301.10260/) (2023)
- [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)
- [Debiasing Machine Learning Models by Using Weakly Supervised Learning](https://mlsanity.com/paper/2402.15477/) (2024)
- [Towards Prospective Medical Image Reconstruction via Knowledge-Informed Dynamic Optimal Transport](https://mlsanity.com/paper/2505.17644/) (2025)
- [Why the noise model matters: A performance gap in learned regularization](https://mlsanity.com/paper/2510.12521/) (2025)

## Abstract (arXiv)

We propose an unsupervised approach for learning end-to-end reconstruction operators for ill-posed inverse problems. The proposed method combines the classical variational framework with iterative unrolling, which essentially seeks to minimize a weighted combination of the expected distortion in the measurement space and the Wasserstein-1 distance between the distributions of the reconstruction and ground-truth. More specifically, the regularizer in the variational setting is parametrized by a deep neural network and learned simultaneously with the unrolled reconstruction operator. The variational problem is then initialized with the reconstruction of the unrolled operator and solved iteratively till convergence. Notably, it takes significantly fewer iterations to converge, thanks to the excellent initialization obtained via the unrolled operator. The resulting approach combines the computational efficiency of end-to-end unrolled reconstruction with the well-posedness and noise-stability guarantees of the variational setting. Moreover, we demonstrate with the example of X-ray computed tomography (CT) that our approach outperforms state-of-the-art unsupervised methods, and that it outperforms or is on par with state-of-the-art supervised learned reconstruction approaches.