# Learning a Gaussian Mixture for Sparsity Regularization in Inverse Problems

arXiv: https://arxiv.org/abs/2401.16612 · HTML: https://mlsanity.com/paper/2401.16612/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 36 references that are in the map.
- First posted on arXiv 29 Jan 2024.

## Builds on (most cited first, 6 of 6)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adversarial Regularizers in Inverse Problems](https://mlsanity.com/paper/1805.11572/) (2018)
- [Convergence Rates for Learning Linear Operators from Noisy Data](https://mlsanity.com/paper/2108.12515/) (2021)
- [Robust and Efficient Subspace Segmentation via Least Squares Regression](https://mlsanity.com/paper/1404.6736/) (2014)
- [Learning the optimal Tikhonov regularizer for inverse problems](https://mlsanity.com/paper/2106.06513/) (2021)
- [Bayesian sparsity and class sparsity priors for dictionary learning and coding](https://mlsanity.com/paper/2309.00999/) (2023)

## Built on by (most cited first, 3 of 3)

- [Learning sparsity-promoting regularizers for linear inverse problems](https://mlsanity.com/paper/2412.16031/) (2024)
- [Why the noise model matters: A performance gap in learned regularization](https://mlsanity.com/paper/2510.12521/) (2025)
- [The learned range test method for the inverse inclusion problem](https://mlsanity.com/paper/2411.00463/) (2024)

## Abstract (arXiv)

In inverse problems, it is widely recognized that the incorporation of a sparsity prior yields a regularization effect on the solution. This approach is grounded on the a priori assumption that the unknown can be appropriately represented in a basis with a limited number of significant components, while most coefficients are close to zero. This occurrence is frequently observed in real-world scenarios, such as with piecewise smooth signals. In this study, we propose a probabilistic sparsity prior formulated as a mixture of degenerate Gaussians, capable of modeling sparsity with respect to a generic basis. Under this premise, we design a neural network that can be interpreted as the Bayes estimator for linear inverse problems. Additionally, we put forth both a supervised and an unsupervised training strategy to estimate the parameters of this network. To evaluate the effectiveness of our approach, we conduct a numerical comparison with commonly employed sparsity-promoting regularization techniques, namely LASSO, group LASSO, iterative hard thresholding, and sparse coding/dictionary learning. Notably, our reconstructions consistently exhibit lower mean square error values across all 1D datasets utilized for the comparisons, even in cases where the datasets significantly deviate from a Gaussian mixture model.