# Deep Equilibrium Learning of Explicit Regularizers for Imaging Inverse Problems

arXiv: https://arxiv.org/abs/2303.05386 · HTML: https://mlsanity.com/paper/2303.05386/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Mar 2023.

## Built on by (most cited first, 16 of 16)

- [Learning Regularization Functionals for Inverse Problems: A Comparative Study](https://mlsanity.com/paper/2510.01755/) (2025)
- [Unfolded proximal neural networks for robust image Gaussian denoising](https://mlsanity.com/paper/2308.03139/) (2023)
- [Positive concave deep equilibrium models](https://mlsanity.com/paper/2402.04029/) (2024)
- [MRI Super-Resolution with Deep Learning: A Comprehensive Survey](https://mlsanity.com/paper/2511.16854/) (2025)
- [Test like you Train in Implicit Deep Learning](https://mlsanity.com/paper/2305.15042/) (2023)
- [No-Clean-Reference Image Super-Resolution: Application to Electron Microscopy](https://mlsanity.com/paper/2401.08115/) (2024)
- [Pruning Unrolled Networks (PUN) at Initialization for MRI Reconstruction Improves Generalization](https://mlsanity.com/paper/2412.18668/) (2024)
- [Recovering Pulse Waves from Video Using Deep Unrolling and Deep Equilibrium Models](https://mlsanity.com/paper/2503.17269/) (2025)
- [Restarted contractive operators to learn at equilibrium](https://mlsanity.com/paper/2506.13239/) (2025)
- [A Stability Benchmark of Generative Regularizers for Inverse Problems](https://mlsanity.com/paper/2605.10076/) (2026)
- [Learning Binary Sampling Patterns for Single-Pixel Imaging using Bilevel Optimisation](https://mlsanity.com/paper/2508.19068/) (2025)
- [Equivariant Deep Equilibrium Models for Imaging Inverse Problems](https://mlsanity.com/paper/2511.18667/) (2025)
- [Weakly Convex Ridge Regularization for 3D Non-Cartesian MRI Reconstruction](https://mlsanity.com/paper/2603.27158/) (2026)
- [Monotone Optimisation with Learned Projections](https://mlsanity.com/paper/2601.20983/) (2026)
- [A geometry-based deep equilibrium model for image restoration under multiplicative Gamma noise](https://mlsanity.com/paper/2608.04944/) (2026)
- [Taming Nonconvexity with Epigraph Lifting](https://mlsanity.com/paper/2610.16429/) (2026)

## Abstract (arXiv)

There has been significant recent interest in the use of deep learning for regularizing imaging inverse problems. Most work in the area has focused on regularization imposed implicitly by convolutional neural networks (CNNs) pre-trained for image reconstruction. In this work, we follow an alternative line of work based on learning explicit regularization functionals that promote preferred solutions. We develop the Explicit Learned Deep Equilibrium Regularizer (ELDER) method for learning explicit regularizers that minimize a mean-squared error (MSE) metric. ELDER is based on a regularization functional parameterized by a CNN and a deep equilibrium learning (DEQ) method for training the functional to be MSE-optimal at the fixed points of the reconstruction algorithm. The explicit regularizer enables ELDER to directly inherit fundamental convergence results from optimization theory. On the other hand, DEQ training enables ELDER to improve over existing explicit regularizers without prohibitive memory complexity during training. We use ELDER to train several approaches to parameterizing explicit regularizers and test their performance on three distinct imaging inverse problems. Our results show that ELDER can greatly improve the quality of explicit regularizers compared to existing methods, and show that learning explicit regularizers does not compromise performance relative to methods based on implicit regularization.