# Local monotone operator learning using non-monotone operators: MnM-MOL

arXiv: https://arxiv.org/abs/2312.00386 · HTML: https://mlsanity.com/paper/2312.00386/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 33 references that are in the map.
- First posted on arXiv 1 Dec 2023.

## Builds on (most cited first, 23 of 23)

- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [fastMRI: An Open Dataset and Benchmarks for Accelerated MRI](https://mlsanity.com/paper/1811.08839/) (2018)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [MoDL: Model Based Deep Learning Architecture for Inverse Problems](https://mlsanity.com/paper/1712.02862/) (2017)
- [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)
- [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)
- [FISTA-Net: Learning A Fast Iterative Shrinkage Thresholding Network for Inverse Problems in Imaging](https://mlsanity.com/paper/2008.02683/) (2020)
- [An Online Plug-and-Play Algorithm for Regularized Image Reconstruction](https://mlsanity.com/paper/1809.04693/) (2018)
- [Learning Maximally Monotone Operators for Image Recovery](https://mlsanity.com/paper/2012.13247/) (2020)
- [Plug-and-Play Unplugged: Optimization Free Reconstruction using Consensus Equilibrium](https://mlsanity.com/paper/1705.08983/) (2017)
- [Solving Inverse Problems With Deep Neural Networks -- Robustness Included?](https://mlsanity.com/paper/2011.04268/) (2020)
- [Measuring Robustness in Deep Learning Based Compressive Sensing](https://mlsanity.com/paper/2102.06103/) (2021)
- [Scalable Plug-and-Play ADMM with Convergence Guarantees](https://mlsanity.com/paper/2006.03224/) (2020)
- [A Neural-Network-Based Convex Regularizer for Inverse Problems](https://mlsanity.com/paper/2211.12461/) (2022)
- [CLIP: Cheap Lipschitz Training of Neural Networks](https://mlsanity.com/paper/2103.12531/) (2021)
- [Memory-efficient model-based deep learning with convergence and robustness guarantees](https://mlsanity.com/paper/2206.04797/) (2022)
- [On the Robustness of deep learning-based MRI Reconstruction to image transformations](https://mlsanity.com/paper/2211.04930/) (2022)
- [Convex Denoising using Non-Convex Tight Frame Regularization](https://mlsanity.com/paper/1504.00976/) (2015)

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

- [Deep Learning for Accelerated and Robust MRI Reconstruction: a Review](https://mlsanity.com/paper/2404.15692/) (2024)

## Abstract (arXiv)

The recovery of magnetic resonance (MR) images from undersampled measurements is a key problem that has seen extensive research in recent years. Unrolled approaches, which rely on end-to-end training of convolutional neural network (CNN) blocks within iterative reconstruction algorithms, offer state-of-the-art performance. These algorithms require a large amount of memory during training, making them difficult to employ in high-dimensional applications. Deep equilibrium (DEQ) models and the recent monotone operator learning (MOL) approach were introduced to eliminate the need for unrolling, thus reducing the memory demand during training. Both approaches require a Lipschitz constraint on the network to ensure that the forward and backpropagation iterations converge. Unfortunately, the constraint often results in reduced performance compared to unrolled methods. The main focus of this work is to relax the constraint on the CNN block in two different ways. Inspired by convex-non-convex regularization strategies, we now impose the monotone constraint on the sum of the gradient of the data term and the CNN block, rather than constrain the CNN itself to be a monotone operator. This approach enables the CNN to learn possibly non-monotone score functions, which can translate to improved performance. In addition, we only restrict the operator to be monotone in a local neighborhood around the image manifold. Our theoretical results show that the proposed algorithm is guaranteed to converge to the fixed point and that the solution is robust to input perturbations, provided that it is initialized close to the true solution. Our empirical results show that the relaxed constraints translate to improved performance and that the approach enjoys robustness to input perturbations similar to MOL.