Optimal transport mapping via input convex neural networks

TL;DR

This paper introduces a new neural network-based method for learning optimal transport maps between distributions using input convex neural networks, ensuring optimality and handling discontinuous targets.

Contribution

It proposes a novel minimax framework leveraging input convex neural networks to learn the optimal transport map directly from samples.

Findings

Successfully learns optimal transport maps from data.

Handles discontinuous target distributions effectively.

Ensures transport map optimality regardless of initialization.

Abstract

In this paper, we present a novel and principled approach to learn the optimal transport between two distributions, from samples. Guided by the optimal transport theory, we learn the optimal Kantorovich potential which induces the optimal transport map. This involves learning two convex functions, by solving a novel minimax optimization. Building upon recent advances in the field of input convex neural networks, we propose a new framework where the gradient of one convex function represents the optimal transport mapping. Numerical experiments confirm that we learn the optimal transport mapping. This approach ensures that the transport mapping we find is optimal independent of how we initialize the neural networks. Further, target distributions from a discontinuous support can be easily captured, as gradient of a convex function naturally models a {\em discontinuous} transport mapping.