Learning Affine-Equivariant Proximal Operators

2026-04-16

AI Summary

Traditional learned proximal operators lack equivariance to translation and scaling, which limits their generalization in out-of-distribution scenarios. This work proposes an affine-equivariant neural parameterization that, for the first time, enables provably exact computation while strictly adhering to the mathematical definition of proximal operators. By embedding affine-equivariant constraints into Learned Proximal Networks, the resulting model significantly outperforms non-equivariant baselines on both synthetic data and real-world image denoising tasks. Notably, it demonstrates markedly enhanced robustness and generalization when confronted with out-of-distribution transformations and noise levels.

Technology Category

• Computer Vision: Learning & Optimization for CV
• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization

Application Category

• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Proximal operators are fundamental across many applications in signal processing and machine learning, including solving ill-posed inverse problems. Recent work has introduced Learned Proximal Networks (LPNs), providing parametric functions that compute exact proximals for data-driven and potentially non-convex regularizers. However, in many settings it is important to include additional structure to these regularizers--and their corresponding proximals--such as shift and scale equivariance. In this work, we show how to obtain learned functions parametrized by neural networks that provably compute exact proximal operators while being equivariant to shifts and scaling, which we dub Affine-Equivariant Learned Proximal Networks (AE-LPNs). We demonstrate our results on synthetic, constructive examples, and then on real data via denoising in out-of-distribution settings. Our equivariant learned proximals enhance robustness to noise distributions and affine shifts far beyond training distributions, improving the practical utility of learned proximal operators

Problem

• proximal operators
• equivariance
• affine transformations
• learned regularization
• out-of-distribution robustness

Innovation

• affine-equivariance
• learned proximal operators
• neural proximal networks
• distributional robustness
• inverse problems