# Universal Architectures for the Learning of Polyhedral Norms and Convex Regularizers

arXiv: https://arxiv.org/abs/2503.19190 · HTML: https://mlsanity.com/paper/2503.19190/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 85 references that are in the map.
- First posted on arXiv 24 Mar 2025.

## Builds on (most cited first, 29 of 29)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [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)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [MoDL: Model Based Deep Learning Architecture for Inverse Problems](https://mlsanity.com/paper/1712.02862/) (2017)
- [Learning a Variational Network for Reconstruction of Accelerated MRI Data](https://mlsanity.com/paper/1704.00447/) (2017)
- [Plug-and-Play ADMM for Image Restoration: Fixed Point Convergence and Applications](https://mlsanity.com/paper/1605.01710/) (2016)
- [On instabilities of deep learning in image reconstruction - Does AI come at a cost?](https://mlsanity.com/paper/1902.05300/) (2019)
- [Plug-and-Play Methods Provably Converge with Properly Trained Denoisers](https://mlsanity.com/paper/1905.05406/) (2019)
- [Trainable Nonlinear Reaction Diffusion: A Flexible Framework for Fast and Effective Image Restoration](https://mlsanity.com/paper/1508.02848/) (2015)
- [Deep Equilibrium Architectures for Inverse Problems in Imaging](https://mlsanity.com/paper/2102.07944/) (2021)
- [Plug-and-Play Methods for Integrating Physical and Learned Models in Computational Imaging](https://mlsanity.com/paper/2203.17061/) (2022)
- [A Review of Convolutional Neural Networks for Inverse Problems in Imaging](https://mlsanity.com/paper/1710.04011/) (2017)
- [Image Reconstruction: From Sparsity to Data-adaptive Methods and Machine Learning](https://mlsanity.com/paper/1904.02816/) (2019)
- [Learned convex regularizers for inverse problems](https://mlsanity.com/paper/2008.02839/) (2020)
- [Scalable Plug-and-Play ADMM with Convergence Guarantees](https://mlsanity.com/paper/2006.03224/) (2020)
- [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)
- [Convolutional Proximal Neural Networks and Plug-and-Play Algorithms](https://mlsanity.com/paper/2011.02281/) (2020)
- [A characterization of proximity operators](https://mlsanity.com/paper/1807.04014/) (2018)
- [Learning Weakly Convex Regularizers for Convergent Image-Reconstruction Algorithms](https://mlsanity.com/paper/2308.10542/) (2023)
- [Rethinking the CSC Model for Natural Images](https://mlsanity.com/paper/1909.05742/) (2019)
- [Proximal Splitting Algorithms for Convex Optimization: A Tour of Recent Advances, with New Twists](https://mlsanity.com/paper/1912.00137/) (2019)
- [Stability of Image-Reconstruction Algorithms](https://mlsanity.com/paper/2206.07128/) (2022)
- [Learning ℓ1-based analysis and synthesis sparsity priors using bi-level optimization](https://mlsanity.com/paper/1401.4105/) (2014)
- [Learning Convex Regularizers for Optimal Bayesian Denoising](https://mlsanity.com/paper/1705.05591/) (2017)
- [A Note on the Forward-Douglas--Rachford Splitting for Monotone Inclusion and Convex Optimization](https://mlsanity.com/paper/1704.06948/) (2017)
- [Parseval Convolution Operators and Neural Networks](https://mlsanity.com/paper/2408.09981/) (2024)
- [Controlled Learning of Pointwise Nonlinearities in Neural-Network-Like Architectures](https://mlsanity.com/paper/2408.13114/) (2024)

## Built on by (most cited first, 1 of 1)

- [Multivariate Fields of Experts for Convergent Image Reconstruction](https://mlsanity.com/paper/2508.06490/) (2025)

## Abstract (arXiv)

This paper addresses the task of learning convex regularizers to guide the reconstruction of images from limited data. By imposing that the reconstruction be amplitude-equivariant, we narrow down the class of admissible functionals to those that can be expressed as a power of a seminorm. We then show that such functionals can be approximated to arbitrary precision with the help of polyhedral norms. In particular, we identify two dual parameterizations of such systems: (i) a synthesis form with an ℓ1-penalty that involves some learnable dictionary; and (ii) an analysis form with an ℓ_∞-penalty that involves a trainable regularization operator. After having provided geometric insights and proved that the two forms are universal, we propose an implementation that relies on a specific architecture (tight frame with a weighted ℓ1 penalty) that is easy to train. We illustrate its use for denoising and the reconstruction of biomedical images. We find that the proposed framework outperforms the sparsity-based methods of compressed sensing, while it offers essentially the same convergence and robustness guarantees.