# Improved Model based Deep Learning using Monotone Operator Learning (MOL)

arXiv: https://arxiv.org/abs/2111.11380 · HTML: https://mlsanity.com/paper/2111.11380/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 6 references that are in the map.
- First posted on arXiv 22 Nov 2021.

## Builds on (most cited first, 5 of 5)

- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [MoDL: Model Based Deep Learning Architecture for Inverse Problems](https://mlsanity.com/paper/1712.02862/) (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)
- [Monotone operator equilibrium networks](https://mlsanity.com/paper/2006.08591/) (2020)

## Built on by (most cited first, 4 of 4)

- [A Neural-Network-Based Convex Regularizer for Inverse Problems](https://mlsanity.com/paper/2211.12461/) (2022)
- [Self-Supervised Deep Equilibrium Models for Inverse Problems with Theoretical Guarantees](https://mlsanity.com/paper/2210.03837/) (2022)
- [Lipschitz Multiscale Deep Equilibrium Models: A Theoretically Guaranteed and Accelerated Approach](https://mlsanity.com/paper/2602.03297/) (2026)
- [Accelerated parallel MRI using memory efficient and robust monotone operator learning (MOL)](https://mlsanity.com/paper/2304.01351/) (2023)

## Abstract (arXiv)

Model-based deep learning (MoDL) algorithms that rely on unrolling are emerging as powerful tools for image recovery. In this work, we introduce a novel monotone operator learning framework to overcome some of the challenges associated with current unrolled frameworks, including high memory cost, lack of guarantees on robustness to perturbations, and low interpretability. Unlike current unrolled architectures that use finite number of iterations, we use the deep equilibrium (DEQ) framework to iterate the algorithm to convergence and to evaluate the gradient of the convolutional neural network blocks using Jacobian iterations. This approach significantly reduces the memory demand, facilitating the extension of MoDL algorithms to high dimensional problems. We constrain the CNN to be a monotone operator, which allows us to introduce algorithms with guaranteed convergence properties and robustness guarantees. We demonstrate the utility of the proposed scheme in the context of parallel MRI.