# Equilibrated Zeroth-Order Unrolled Deep Networks for Accelerated MRI

arXiv: https://arxiv.org/abs/2112.09891 · HTML: https://mlsanity.com/paper/2112.09891/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 50 references that are in the map.
- First posted on arXiv 18 Dec 2021.

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

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018)
- [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)
- [Learning a Variational Network for Reconstruction of Accelerated MRI Data](https://mlsanity.com/paper/1704.00447/) (2017)
- [Learned Primal-dual Reconstruction](https://mlsanity.com/paper/1707.06474/) (2017)
- [Image reconstruction by domain transform manifold learning](https://mlsanity.com/paper/1704.08841/) (2017)
- [On instabilities of deep learning in image reconstruction - Does AI come at a cost?](https://mlsanity.com/paper/1902.05300/) (2019)
- [Learning to Optimize: A Primer and A Benchmark](https://mlsanity.com/paper/2103.12828/) (2021)
- [Multiscale Deep Equilibrium Models](https://mlsanity.com/paper/2006.08656/) (2020)
- [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)
- [A Bayesian Perspective on the Deep Image Prior](https://mlsanity.com/paper/1904.07457/) (2019)
- [Momentum-Net: Fast and convergent iterative neural network for inverse problems](https://mlsanity.com/paper/1907.11818/) (2019)
- [Deep Low-rank plus Sparse Network for Dynamic MR Imaging](https://mlsanity.com/paper/2010.13677/) (2020)
- [Deep Generalization of Structured Low-Rank Algorithms (Deep-SLR)](https://mlsanity.com/paper/1912.03433/) (2019)
- [LORAKI: Autocalibrated Recurrent Neural Networks for Autoregressive MRI Reconstruction in k-Space](https://mlsanity.com/paper/1904.09390/) (2019)
- [Uncertainty Estimation in Medical Image Denoising with Bayesian Deep Image Prior](https://mlsanity.com/paper/2008.08837/) (2020)
- [Feasibility-based Fixed Point Networks](https://mlsanity.com/paper/2104.14090/) (2021)
- [On the Convergence of Learning-based Iterative Methods for Nonconvex Inverse Problems](https://mlsanity.com/paper/1808.05331/) (2018)
- [Deep Manifold Learning for Dynamic MR Imaging](https://mlsanity.com/paper/2104.01102/) (2021)

## Built on by (most cited first, 5 of 5)

- [High-Frequency Space Diffusion Models for Accelerated MRI](https://mlsanity.com/paper/2208.05481/) (2022)
- [Self-Score: Self-Supervised Learning on Score-Based Models for MRI Reconstruction](https://mlsanity.com/paper/2209.00835/) (2022)
- [NESTANets: Stable, accurate and efficient neural networks for analysis-sparse inverse problems](https://mlsanity.com/paper/2203.00804/) (2022)
- [Deep unfolding as iterative regularization for imaging inverse problems](https://mlsanity.com/paper/2211.13452/) (2022)
- [Self-Consistent Nested Diffusion Bridge for Accelerated MRI Reconstruction](https://mlsanity.com/paper/2412.09998/) (2024)

## Abstract (arXiv)

Recently, model-driven deep learning unrolls a certain iterative algorithm of a regularization model into a cascade network by replacing the first-order information (i.e., (sub)gradient or proximal operator) of the regularizer with a network module, which appears more explainable and predictable compared to common data-driven networks. Conversely, in theory, there is not necessarily such a functional regularizer whose first-order information matches the replaced network module, which means the network output may not be covered by the original regularization model. Moreover, up to now, there is also no theory to guarantee the global convergence and robustness (regularity) of unrolled networks under realistic assumptions. To bridge this gap, this paper propose to present a safeguarded methodology on network unrolling. Specifically, focusing on accelerated MRI, we unroll a zeroth-order algorithm, of which the network module represents the regularizer itself, so that the network output can be still covered by the regularization model. Furthermore, inspired by the ideal of deep equilibrium models, before backpropagating, we carry out the unrolled iterative network to converge to a fixed point to ensure the convergence. In case the measurement data contains noise, we prove that the proposed network is robust against noisy interference. Finally, numerical experiments show that the proposed network consistently outperforms the state-of-the-art MRI reconstruction methods including traditional regularization methods and other deep learning methods.