Learning Gradients of Convex Functions with Monotone Gradient Networks

TL;DR

This paper introduces two neural network architectures, C-MGN and M-MGN, designed to learn the gradients of convex functions, demonstrating improved training efficiency and accuracy, with applications in optimal transport mapping.

Contribution

The paper proposes novel monotone gradient neural networks that effectively learn convex function gradients with fewer parameters and better accuracy than existing methods.

Findings

Easier to train than state-of-the-art methods

More accurate in learning monotone gradient fields

Successfully applied to optimal transport mappings

Abstract

While much effort has been devoted to deriving and analyzing effective convex formulations of signal processing problems, the gradients of convex functions also have critical applications ranging from gradient-based optimization to optimal transport. Recent works have explored data-driven methods for learning convex objective functions, but learning their monotone gradients is seldom studied. In this work, we propose C-MGN and M-MGN, two monotone gradient neural network architectures for directly learning the gradients of convex functions. We show that, compared to state of the art methods, our networks are easier to train, learn monotone gradient fields more accurately, and use significantly fewer parameters. We further demonstrate their ability to learn optimal transport mappings to augment driving image data.