# Learning convex regularizers satisfying the variational source condition for inverse problems

arXiv: https://arxiv.org/abs/2110.12520 · HTML: https://mlsanity.com/paper/2110.12520/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 13 references that are in the map.
- First posted on arXiv 24 Oct 2021.

## Builds on (most cited first, 7 of 7)

- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [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)
- [Learned convex regularizers for inverse problems](https://mlsanity.com/paper/2008.02839/) (2020)
- [Total Deep Variation for Linear Inverse Problems](https://mlsanity.com/paper/2001.05005/) (2020)

## Built on by (most cited first, 8 of 8)

- [A Neural-Network-Based Convex Regularizer for Inverse Problems](https://mlsanity.com/paper/2211.12461/) (2022)
- [Learning Weakly Convex Regularizers for Convergent Image-Reconstruction Algorithms](https://mlsanity.com/paper/2308.10542/) (2023)
- [Learning Regularization Functionals for Inverse Problems: A Comparative Study](https://mlsanity.com/paper/2510.01755/) (2025)
- [An Over Complete Deep Learning Method for Inverse Problems](https://mlsanity.com/paper/2402.04653/) (2024)
- [Learned Regularization for Inverse Problems: Insights from a Spectral Model](https://mlsanity.com/paper/2312.09845/) (2023)
- [Invertible ResNets for Inverse Imaging Problems: Competitive Performance with Provable Regularization Properties](https://mlsanity.com/paper/2409.13482/) (2024)
- [DAWN-FM: Data-Aware and Noise-Informed Flow Matching for Solving Inverse Problems](https://mlsanity.com/paper/2412.04766/) (2024)
- [Why the noise model matters: A performance gap in learned regularization](https://mlsanity.com/paper/2510.12521/) (2025)

## Abstract (arXiv)

Variational regularization has remained one of the most successful approaches for reconstruction in imaging inverse problems for several decades. With the emergence and astonishing success of deep learning in recent years, a considerable amount of research has gone into data-driven modeling of the regularizer in the variational setting. Our work extends a recently proposed method, referred to as adversarial convex regularization (ACR), that seeks to learn data-driven convex regularizers via adversarial training in an attempt to combine the power of data with the classical convex regularization theory. Specifically, we leverage the variational source condition (SC) during training to enforce that the ground-truth images minimize the variational loss corresponding to the learned convex regularizer. This is achieved by adding an appropriate penalty term to the ACR training objective. The resulting regularizer (abbreviated as ACR-SC) performs on par with the ACR, but unlike ACR, comes with a quantitative convergence rate estimate.