# A Fast Algorithm for Convolutional Structured Low-Rank Matrix Recovery

arXiv: https://arxiv.org/abs/1609.07429 · HTML: https://mlsanity.com/paper/1609.07429/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Sep 2016.

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

- [One-dimensional Deep Low-rank and Sparse Network for Accelerated MRI](https://mlsanity.com/paper/2112.04721/) (2021)
- [Deep Algorithm Unrolling for Biomedical Imaging](https://mlsanity.com/paper/2108.06637/) (2021)
- [Adapting model-based deep learning to multiple acquisition conditions: Ada-MoDL](https://mlsanity.com/paper/2304.11238/) (2023)
- [Hankel low-rank approximation and completion in time series analysis and forecasting: a brief review](https://mlsanity.com/paper/2206.05103/) (2022)
- [Learning Transition Operators From Sparse Space-Time Samples](https://mlsanity.com/paper/2212.00746/) (2022)

## Abstract (arXiv)

Fourier domain structured low-rank matrix priors are emerging as powerful alternatives to traditional image recovery methods such as total variation and wavelet regularization. These priors specify that a convolutional structured matrix, i.e., Toeplitz, Hankel, or their multi-level generalizations, built from Fourier data of the image should be low-rank. The main challenge in applying these schemes to large-scale problems is the computational complexity and memory demand resulting from lifting the image data to a large scale matrix. We introduce a fast and memory efficient approach called the Generic Iterative Reweighted Annihilation Filter (GIRAF) algorithm that exploits the convolutional structure of the lifted matrix to work in the original un-lifted domain, thus considerably reducing the complexity. Our experiments on the recovery of images from undersampled Fourier measurements show that the resulting algorithm is considerably faster than previously proposed algorithms, and can accommodate much larger problem sizes than previously studied.