# Learned convex regularizers for inverse problems

arXiv: https://arxiv.org/abs/2008.02839 · HTML: https://mlsanity.com/paper/2008.02839/

## Key facts

- Cited by 55 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Aug 2020.

## Built on by (most cited first, 30 of 55)

- [Learning to Optimize: A Primer and A Benchmark](https://mlsanity.com/paper/2103.12828/) (2021)
- [Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie](https://mlsanity.com/paper/2103.04715/) (2021)
- [Learned reconstruction methods with convergence guarantees](https://mlsanity.com/paper/2206.05431/) (2022)
- [A Neural-Network-Based Convex Regularizer for Inverse Problems](https://mlsanity.com/paper/2211.12461/) (2022)
- [End-to-end reconstruction meets data-driven regularization for inverse problems](https://mlsanity.com/paper/2106.03538/) (2021)
- [Learning Weakly Convex Regularizers for Convergent Image-Reconstruction Algorithms](https://mlsanity.com/paper/2308.10542/) (2023)
- [Learning the optimal Tikhonov regularizer for inverse problems](https://mlsanity.com/paper/2106.06513/) (2021)
- [What's in a Prior? Learned Proximal Networks for Inverse Problems](https://mlsanity.com/paper/2310.14344/) (2023)
- [Improving Lipschitz-Constrained Neural Networks by Learning Activation Functions](https://mlsanity.com/paper/2210.16222/) (2022)
- [The Geometry of Adversarial Training in Binary Classification](https://mlsanity.com/paper/2111.13613/) (2021)
- [Feasibility-based Fixed Point Networks](https://mlsanity.com/paper/2104.14090/) (2021)
- [Input Convex Gradient Networks](https://mlsanity.com/paper/2111.12187/) (2021)
- [Variational Bayesian Imaging with an Efficient Surrogate Score-based Prior](https://mlsanity.com/paper/2309.01949/) (2023)
- [A study of why we need to reassess full reference image quality assessment with medical images](https://mlsanity.com/paper/2405.19097/) (2024)
- [Provably Convergent Plug-and-Play Quasi-Newton Methods](https://mlsanity.com/paper/2303.07271/) (2023)
- [Learning Regularization Functionals for Inverse Problems: A Comparative Study](https://mlsanity.com/paper/2510.01755/) (2025)
- [Weakly Convex Regularisers for Inverse Problems: Convergence of Critical Points and Primal-Dual Optimisation](https://mlsanity.com/paper/2402.01052/) (2024)
- [Convergent regularization in inverse problems and linear plug-and-play denoisers](https://mlsanity.com/paper/2307.09441/) (2023)
- [Learning convex regularizers satisfying the variational source condition for inverse problems](https://mlsanity.com/paper/2110.12520/) (2021)
- [Analyzing Inexact Hypergradients for Bilevel Learning](https://mlsanity.com/paper/2301.04764/) (2023)
- [Model-corrected learned primal-dual models for fast limited-view photoacoustic tomography](https://mlsanity.com/paper/2304.01963/) (2023)
- [On Optimal Regularization Parameters via Bilevel Learning](https://mlsanity.com/paper/2305.18394/) (2023)
- [Benchmarking learned algorithms for computed tomography image reconstruction tasks](https://mlsanity.com/paper/2412.08350/) (2024)
- [Deep unfolding as iterative regularization for imaging inverse problems](https://mlsanity.com/paper/2211.13452/) (2022)
- [Stability of Data-Dependent Ridge-Regularization for Inverse Problems](https://mlsanity.com/paper/2406.12289/) (2024)
- [Good Things Come in Pairs: Paired Autoencoders for Inverse Problems](https://mlsanity.com/paper/2505.06549/) (2025)
- [Input Convex Kolmogorov Arnold Networks](https://mlsanity.com/paper/2505.21208/) (2025)
- [Stochastic Primal-Dual Deep Unrolling](https://mlsanity.com/paper/2110.10093/) (2021)
- [NF-ULA: Langevin Monte Carlo with Normalizing Flow Prior for Imaging Inverse Problems](https://mlsanity.com/paper/2304.08342/) (2023)
- [Accelerated Bayesian imaging by relaxed proximal-point Langevin sampling](https://mlsanity.com/paper/2308.09460/) (2023)

## Abstract (arXiv)

We consider the variational reconstruction framework for inverse problems and propose to learn a data-adaptive input-convex neural network (ICNN) as the regularization functional. The ICNN-based convex regularizer is trained adversarially to discern ground-truth images from unregularized reconstructions. Convexity of the regularizer is desirable since (i) one can establish analytical convergence guarantees for the corresponding variational reconstruction problem and (ii) devise efficient and provable algorithms for reconstruction. In particular, we show that the optimal solution to the variational problem converges to the ground-truth if the penalty parameter decays sub-linearly with respect to the norm of the noise. Further, we prove the existence of a sub-gradient-based algorithm that leads to a monotonically decreasing error in the parameter space with iterations. To demonstrate the performance of our approach for solving inverse problems, we consider the tasks of deblurring natural images and reconstructing images in computed tomography (CT), and show that the proposed convex regularizer is at least competitive with and sometimes superior to state-of-the-art data-driven techniques for inverse problems.