Adversarial Computation of Optimal Transport Maps

Abstract

Computing optimal transport maps between high-dimensional and continuous distributions is a challenging problem in optimal transport (OT). Generative adversarial networks (GANs) are powerful generative models which have been successfully applied to learn maps across high-dimensional domains. However, little is known about the nature of the map learned with a GAN objective. To address this problem, we propose a generative adversarial model in which the discriminator's objective is the $2$-Wasserstein metric. We show that during training, our generator follows the $W_2$-geodesic between the initial and the target distributions. As a consequence, it reproduces an optimal map at the end of training. We validate our approach empirically in both low-dimensional and high-dimensional continuous settings, and show that it outperforms prior methods on image data.

References in corpus (7)

• Fast Computation of Wasserstein Barycenters
• On Convergence and Stability of GANs
• On the regularization of Wasserstein GANs
• Smooth and Sparse Optimal Transport
• Learning Generative Models with Sinkhorn Divergences
• Improving GANs Using Optimal Transport
• Implicit Manifold Learning on Generative Adversarial Networks

Cited by in corpus (8)

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• Potential Flow Generator with $L_2$ Optimal Transport Regularity for Generative Models
• Improving Mini-batch Optimal Transport via Partial Transportation
• On Transportation of Mini-batches: A Hierarchical Approach
• Large-Scale Optimal Transport via Adversarial Training with Cycle-Consistency
• Continual Learning of Generative Models with Limited Data: From Wasserstein-1 Barycenter to Adaptive Coalescence