# Self-Supervised Deep Equilibrium Models for Inverse Problems with Theoretical Guarantees

arXiv: https://arxiv.org/abs/2210.03837 · HTML: https://mlsanity.com/paper/2210.03837/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 42 references that are in the map.
- First posted on arXiv 7 Oct 2022.

## Builds on (most cited first, 26 of 26)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Noise2Noise: Learning Image Restoration without Clean Data](https://mlsanity.com/paper/1803.04189/) (2018)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [MoDL: Model Based Deep Learning Architecture for Inverse Problems](https://mlsanity.com/paper/1712.02862/) (2017)
- [Noise2Void - Learning Denoising from Single Noisy Images](https://mlsanity.com/paper/1811.10980/) (2018)
- [Learning a Variational Network for Reconstruction of Accelerated MRI Data](https://mlsanity.com/paper/1704.00447/) (2017)
- [The Little Engine that Could: Regularization by Denoising (RED)](https://mlsanity.com/paper/1611.02862/) (2016)
- [Deep Learning Techniques for Inverse Problems in Imaging](https://mlsanity.com/paper/2005.06001/) (2020)
- [A Deep Cascade of Convolutional Neural Networks for Dynamic MR Image Reconstruction](https://mlsanity.com/paper/1704.02422/) (2017)
- [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)
- [Deep Equilibrium Architectures for Inverse Problems in Imaging](https://mlsanity.com/paper/2102.07944/) (2021)
- [Noisier2Noise: Learning to Denoise from Unpaired Noisy Data](https://mlsanity.com/paper/1910.11908/) (2019)
- [JFB: Jacobian-Free Backpropagation for Implicit Networks](https://mlsanity.com/paper/2103.12803/) (2021)
- [Equivariant Imaging: Learning Beyond the Range Space](https://mlsanity.com/paper/2103.14756/) (2021)
- [Learning Differential Equations that are Easy to Solve](https://mlsanity.com/paper/2007.04504/) (2020)
- [RARE: Image Reconstruction using Deep Priors Learned without Ground Truth](https://mlsanity.com/paper/1912.05854/) (2019)
- [Online Deep Equilibrium Learning for Regularization by Denoising](https://mlsanity.com/paper/2205.13051/) (2022)
- [Unsupervised Deep Learning Methods for Biological Image Reconstruction and Enhancement](https://mlsanity.com/paper/2105.08040/) (2021)
- [Deformation-Compensated Learning for Image Reconstruction without Ground Truth](https://mlsanity.com/paper/2107.05533/) (2021)
- [Deep Equilibrium Models for Video Snapshot Compressive Imaging](https://mlsanity.com/paper/2201.06931/) (2022)
- [Improved Model based Deep Learning using Monotone Operator Learning (MOL)](https://mlsanity.com/paper/2111.11380/) (2021)

## Built on by (most cited first, 1 of 1)

- [Recovering Pulse Waves from Video Using Deep Unrolling and Deep Equilibrium Models](https://mlsanity.com/paper/2503.17269/) (2025)

## Abstract (arXiv)

Deep equilibrium models (DEQ) have emerged as a powerful alternative to deep unfolding (DU) for image reconstruction. DEQ models-implicit neural networks with effectively infinite number of layers-were shown to achieve state-of-the-art image reconstruction without the memory complexity associated with DU. While the performance of DEQ has been widely investigated, the existing work has primarily focused on the settings where groundtruth data is available for training. We present self-supervised deep equilibrium model (SelfDEQ) as the first self-supervised reconstruction framework for training model-based implicit networks from undersampled and noisy MRI measurements. Our theoretical results show that SelfDEQ can compensate for unbalanced sampling across multiple acquisitions and match the performance of fully supervised DEQ. Our numerical results on in-vivo MRI data show that SelfDEQ leads to state-of-the-art performance using only undersampled and noisy training data.