# Learning sparsity-promoting regularizers for linear inverse problems

arXiv: https://arxiv.org/abs/2412.16031 · HTML: https://mlsanity.com/paper/2412.16031/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 50 references that are in the map.
- First posted on arXiv 20 Dec 2024.

## Builds on (most cited first, 24 of 24)

- [Learned Primal-dual Reconstruction](https://mlsanity.com/paper/1707.06474/) (2017)
- [Solving ill-posed inverse problems using iterative deep neural networks](https://mlsanity.com/paper/1704.04058/) (2017)
- [Adversarial Regularizers in Inverse Problems](https://mlsanity.com/paper/1805.11572/) (2018)
- [NETT: Solving Inverse Problems with Deep Neural Networks](https://mlsanity.com/paper/1803.00092/) (2018)
- [A Mathematical Guide to Operator Learning](https://mlsanity.com/paper/2312.14688/) (2023)
- [Optimal Rates For Regularization Of Statistical Inverse Learning Problems](https://mlsanity.com/paper/1604.04054/) (2016)
- [Learned convex regularizers for inverse problems](https://mlsanity.com/paper/2008.02839/) (2020)
- [Convergence Rates for Learning Linear Operators from Noisy Data](https://mlsanity.com/paper/2108.12515/) (2021)
- [Learning the optimal Tikhonov regularizer for inverse problems](https://mlsanity.com/paper/2106.06513/) (2021)
- [Bilevel approaches for learning of variational imaging models](https://mlsanity.com/paper/1505.02120/) (2015)
- [Bilevel parameter learning for higher-order total variation regularisation models](https://mlsanity.com/paper/1508.07243/) (2015)
- [Convergent regularization in inverse problems and linear plug-and-play denoisers](https://mlsanity.com/paper/2307.09441/) (2023)
- [Sample Complexity of Dictionary Learning and other Matrix Factorizations](https://mlsanity.com/paper/1312.3790/) (2013)
- [Convex regularization in statistical inverse learning problems](https://mlsanity.com/paper/2102.09526/) (2021)
- [Learning Sparse Wavelet Representations](https://mlsanity.com/paper/1802.02961/) (2018)
- [Recovery and Generalization in Over-Realized Dictionary Learning](https://mlsanity.com/paper/2006.06179/) (2020)
- [Adversarially learned iterative reconstruction for imaging inverse problems](https://mlsanity.com/paper/2103.16151/) (2021)
- [Learnable Wavelet Packet Transform for Data-Adapted Spectrograms](https://mlsanity.com/paper/2201.11069/) (2022)
- [On Learning the Optimal Regularization Parameter in Inverse Problems](https://mlsanity.com/paper/2311.15845/) (2023)
- [Learning Sparsity-Promoting Regularizers using Bilevel Optimization](https://mlsanity.com/paper/2207.08939/) (2022)
- [Learning Variational Models with Unrolling and Bilevel Optimization](https://mlsanity.com/paper/2209.12651/) (2022)
- [Learning a Gaussian Mixture for Sparsity Regularization in Inverse Problems](https://mlsanity.com/paper/2401.16612/) (2024)
- [Learned reconstruction methods for inverse problems: sample error estimates](https://mlsanity.com/paper/2312.14078/) (2023)
- [Statistical inverse learning problems with random observations](https://mlsanity.com/paper/2312.15341/) (2023)

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

- [On the Sample Complexity of Learning for Blind Inverse Problems](https://mlsanity.com/paper/2512.23405/) (2025)
- [A Deep Risk Estimator for Known Operator Learning](https://mlsanity.com/paper/2605.08517/) (2026)
- [Regularization of Statistical Inverse Problems on Non-Reflexive Banach Spaces](https://mlsanity.com/paper/2608.17533/) (2026)

## Abstract (arXiv)

This paper introduces a novel approach to learning sparsity-promoting regularizers for solving linear inverse problems. We develop a bilevel optimization framework to select an optimal synthesis operator, denoted as B, which regularizes the inverse problem while promoting sparsity in the solution. The method leverages statistical properties of the underlying data and incorporates prior knowledge through the choice of B. We establish the well-posedness of the optimization problem, provide theoretical guarantees for the learning process, and present sample complexity bounds. The approach is demonstrated through theoretical infinite-dimensional examples, including compact perturbations of a known operator and the problem of learning the mother wavelet, and through extensive numerical simulations. This work extends previous efforts in Tikhonov regularization by addressing non-differentiable norms and proposing a data-driven approach for sparse regularization in infinite dimensions.