# A Compound Gaussian Least Squares Algorithm and Unrolled Network for Linear Inverse Problems

arXiv: https://arxiv.org/abs/2305.11120 · HTML: https://mlsanity.com/paper/2305.11120/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 58 references that are in the map.
- First posted on arXiv 18 May 2023.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing](https://mlsanity.com/paper/1912.10557/) (2019)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [Learned Primal-dual Reconstruction](https://mlsanity.com/paper/1707.06474/) (2017)
- [Coordinate Descent Algorithms](https://mlsanity.com/paper/1502.04759/) (2015)
- [Learning Proximal Operators: Using Denoising Networks for Regularizing Inverse Imaging Problems](https://mlsanity.com/paper/1704.03488/) (2017)
- [Convolutional Recurrent Neural Networks for Dynamic MR Image Reconstruction](https://mlsanity.com/paper/1712.01751/) (2017)
- [FISTA-Net: Learning A Fast Iterative Shrinkage Thresholding Network for Inverse Problems in Imaging](https://mlsanity.com/paper/2008.02683/) (2020)
- [Deep Gaussian Scale Mixture Prior for Spectral Compressive Imaging](https://mlsanity.com/paper/2103.07152/) (2021)
- [Unrolled Optimization with Deep Priors](https://mlsanity.com/paper/1705.08041/) (2017)
- [Memory-Augmented Deep Unfolding Network for Compressive Sensing](https://mlsanity.com/paper/2110.09766/) (2021)
- [Radon Inversion via Deep Learning](https://mlsanity.com/paper/1808.03015/) (2018)
- [LEARN++: Recurrent Dual-Domain Reconstruction Network for Compressed Sensing CT](https://mlsanity.com/paper/2012.06983/) (2020)
- [Fast Stochastic Hierarchical Bayesian MAP for Tomographic Imaging](https://mlsanity.com/paper/1707.02336/) (2017)

## Built on by (most cited first, 2 of 2)

- [On Generalization Bounds for Deep Compound Gaussian Neural Networks](https://mlsanity.com/paper/2402.13106/) (2024)
- [Deep Regularized Compound Gaussian Network for Solving Linear Inverse Problems](https://mlsanity.com/paper/2311.17248/) (2023)

## Abstract (arXiv)

For solving linear inverse problems, particularly of the type that appears in tomographic imaging and compressive sensing, this paper develops two new approaches. The first approach is an iterative algorithm that minimizes a regularized least squares objective function where the regularization is based on a compound Gaussian prior distribution. The compound Gaussian prior subsumes many of the commonly used priors in image reconstruction, including those of sparsity-based approaches. The developed iterative algorithm gives rise to the paper's second new approach, which is a deep neural network that corresponds to an "unrolling" or "unfolding" of the iterative algorithm. Unrolled deep neural networks have interpretable layers and outperform standard deep learning methods. This paper includes a detailed computational theory that provides insight into the construction and performance of both algorithms. The conclusion is that both algorithms outperform other state-of-the-art approaches to tomographic image formation and compressive sensing, especially in the difficult regime of low training.