Learning Affine-Equivariant Proximal Operators

TL;DR

This paper introduces Affine-Equivariant Learned Proximal Networks (AE-LPNs), neural network-based proximal operators that are shift and scale equivariant, improving robustness in signal processing tasks.

Contribution

It develops neural network-based proximal operators that are provably affine-equivariant, enhancing robustness and applicability in non-convex and data-driven settings.

Findings

AE-LPNs compute exact proximal operators with affine equivariance.

AE-LPNs improve robustness to noise and shifts in out-of-distribution data.

Experimental results show enhanced performance in denoising tasks.

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