GradNetOT: Learning Optimal Transport Maps with GradNets

TL;DR

GradNetOT introduces a neural network approach leveraging Monotone Gradient Networks to directly learn optimal transport maps, effectively solving Monge-Ampère equations in image and high-dimensional applications.

Contribution

The paper presents GradNetOT, a novel method that uses Monotone Gradient Networks to directly learn optimal transport maps by minimizing a Monge-Ampère based loss.

Findings

Effective in image morphing tasks

Performs well in high-dimensional OT problems

Structural bias aids learning of optimal transport maps

Abstract

Monotone gradient functions play a central role in solving the Monge formulation of the optimal transport (OT) problem, which arises in modern applications ranging from fluid dynamics to robot swarm control. When the transport cost is the squared Euclidean distance, Brenier's theorem guarantees that the unique optimal transport map satisfies a Monge-Amp\`ere equation and is the gradient of a convex function. In [arXiv:2301.10862] [arXiv:2404.07361], we proposed Monotone Gradient Networks (mGradNets), neural networks that directly parameterize the space of monotone gradient maps. In this work, we leverage mGradNets to directly learn the optimal transport mapping by minimizing a training loss function defined using the Monge-Amp\`ere equation. We empirically show that the structural bias of mGradNets facilitates the learning of optimal transport maps across both image morphing tasks and high-dimensional OT problems.