Optimal transport mapping via input convex neural networks

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.

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Cited by in corpus (7)

• Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey
• Wasserstein-2 Generative Networks
• Transport away your problems: Calibrating stochastic simulations with optimal transport
• Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark
• Continuous Wasserstein-2 Barycenter Estimation without Minimax Optimization
• Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimization
• Large-Scale Optimal Transport via Adversarial Training with Cycle-Consistency