# Matrix Completion-Informed Deep Unfolded Equilibrium Models for Self-Supervised k-Space Interpolation in MRI

arXiv: https://arxiv.org/abs/2309.13571 · HTML: https://mlsanity.com/paper/2309.13571/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 34 references that are in the map.
- First posted on arXiv 24 Sep 2023.

## Builds on (most cited first, 16 of 16)

- [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)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [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)
- [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)
- [Zero-Shot Self-Supervised Learning for MRI Reconstruction](https://mlsanity.com/paper/2102.07737/) (2021)
- [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)
- [Self-Supervised Learning for MRI Reconstruction with a Parallel Network Training Framework](https://mlsanity.com/paper/2109.12502/) (2021)
- [Self-Supervised Deep Active Accelerated MRI](https://mlsanity.com/paper/1901.04547/) (2019)
- [Deep unfolding as iterative regularization for imaging inverse problems](https://mlsanity.com/paper/2211.13452/) (2022)
- [Convex Latent-Optimized Adversarial Regularizers for Imaging Inverse Problems](https://mlsanity.com/paper/2309.09250/) (2023)

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

- [Score-based Diffusion Models With Self-supervised Learning For Accelerated 3D Multi-contrast Cardiac Magnetic Resonance Imaging](https://mlsanity.com/paper/2310.04669/) (2023)
- [Lipschitz Multiscale Deep Equilibrium Models: A Theoretically Guaranteed and Accelerated Approach](https://mlsanity.com/paper/2602.03297/) (2026)
- [Towards Globally Predictable k-Space Interpolation: A White-box Transformer Approach](https://mlsanity.com/paper/2508.04051/) (2025)
- [Hankel Subspace Self-Supervised Learning for Parallel MRI Reconstruction](https://mlsanity.com/paper/2610.11560/) (2026)

## Abstract (arXiv)

Recently, regularization model-driven deep learning (DL) has gained significant attention due to its ability to leverage the potent representational capabilities of DL while retaining the theoretical guarantees of regularization models. However, most of these methods are tailored for supervised learning scenarios that necessitate fully sampled labels, which can pose challenges in practical MRI applications. To tackle this challenge, we propose a self-supervised DL approach for accelerated MRI that is theoretically guaranteed and does not rely on fully sampled labels. Specifically, we achieve neural network structure regularization by exploiting the inherent structural low-rankness of the k-space data. Simultaneously, we constrain the network structure to resemble a nonexpansive mapping, ensuring the network's convergence to a fixed point. Thanks to this well-defined network structure, this fixed point can completely reconstruct the missing k-space data based on matrix completion theory, even in situations where full-sampled labels are unavailable. Experiments validate the effectiveness of our proposed method and demonstrate its superiority over existing self-supervised approaches and traditional regularization methods, achieving performance comparable to that of supervised learning methods in certain scenarios.