# Structured Low-Rank Algorithms: Theory, MR Applications, and Links to Machine Learning

arXiv: https://arxiv.org/abs/1910.12162 · HTML: https://mlsanity.com/paper/1910.12162/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Oct 2019.

## Built on by (most cited first, 11 of 11)

- [One-dimensional Deep Low-rank and Sparse Network for Accelerated MRI](https://mlsanity.com/paper/2112.04721/) (2021)
- [High-Dimensional MR Reconstruction Integrating Subspace and Adaptive Generative Models](https://mlsanity.com/paper/2306.08630/) (2023)
- [Deep Separable Spatiotemporal Learning for Fast Dynamic Cardiac MRI](https://mlsanity.com/paper/2402.15939/) (2024)
- [Accelerating Ill-conditioned Hankel Matrix Recovery via Structured Newton-like Descent](https://mlsanity.com/paper/2406.07409/) (2024)
- [Physics-Informed DeepMRI: Bridging the Gap from Heat Diffusion to k-Space Interpolation](https://mlsanity.com/paper/2308.15918/) (2023)
- [MCU-Net: A Multi-prior Collaborative Deep Unfolding Network with Gates-controlled Spatial Attention for Accelerated MR Image Reconstruction](https://mlsanity.com/paper/2402.03383/) (2024)
- [Reconstruction and Segmentation of Parallel MR Data using Image Domain DEEP-SLR](https://mlsanity.com/paper/2102.01172/) (2021)
- [Magnetic Resonance Spectroscopy Deep Learning Denoising Using Few In Vivo Data](https://mlsanity.com/paper/2101.11442/) (2021)
- [A Structured Sparse Neural Network and Its Matrix Calculations Algorithm](https://mlsanity.com/paper/2207.00903/) (2022)
- [Towards Globally Predictable k-Space Interpolation: A White-box Transformer Approach](https://mlsanity.com/paper/2508.04051/) (2025)
- [Efficient Generative Modeling with Unitary Matrix Product States Using Riemannian Optimization](https://mlsanity.com/paper/2603.12026/) (2026)

## Abstract (arXiv)

In this survey, we provide a detailed review of recent advances in the recovery of continuous domain multidimensional signals from their few non-uniform (multichannel) measurements using structured low-rank matrix completion formulation. This framework is centered on the fundamental duality between the compactness (e.g., sparsity) of the continuous signal and the rank of a structured matrix, whose entries are functions of the signal. This property enables the reformulation of the signal recovery as a low-rank structured matrix completion, which comes with performance guarantees. We will also review fast algorithms that are comparable in complexity to current compressed sensing methods, which enables the application of the framework to large-scale magnetic resonance (MR) recovery problems. The remarkable flexibility of the formulation can be used to exploit signal properties that are difficult to capture by current sparse and low-rank optimization strategies. We demonstrate the utility of the framework in a wide range of MR imaging (MRI) applications, including highly accelerated imaging, calibration-free acquisition, MR artifact correction, and ungated dynamic MRI.