Large-Scale Optimal Transport and Mapping Estimation

Abstract

This paper presents a novel two-step approach for the fundamental problem of learning an optimal map from one distribution to another. First, we learn an optimal transport (OT) plan, which can be thought as a one-to-many map between the two distributions. To that end, we propose a stochastic dual approach of regularized OT, and show empirically that it scales better than a recent related approach when the amount of samples is very large. Second, we estimate a \textit{Monge map} as a deep neural network learned by approximating the barycentric projection of the previously-obtained OT plan. This parameterization allows generalization of the mapping outside the support of the input measure. We prove two theoretical stability results of regularized OT which show that our estimations converge to the OT plan and Monge map between the underlying continuous measures. We showcase our proposed approach on two applications: domain adaptation and generative modeling.

arXiv comment15 pages, 4 figures. To appear in the Proceedings of the International Conference on Learning Representations (ICLR) 2018

References in corpus (9)

• Generative Moment Matching Networks
• Learning with a Wasserstein Loss
• Bayesian Inference with Optimal Maps
• Stochastic Optimization for Large-scale Optimal Transport
• Smooth and Sparse Optimal Transport
• Joint Distribution Optimal Transportation for Domain Adaptation
• Regularized Optimal Transport and the Rot Mover's Distance
• Blind Source Separation with Optimal Transport Non-negative Matrix Factorization
• Wasserstein Training of Boltzmann Machines

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• Scalable Computations of Wasserstein Barycenter via Input Convex Neural Networks
• Continuous Regularized Wasserstein Barycenters
• Blind Source Separation with Optimal Transport Non-negative Matrix Factorization
• (Martingale) Optimal Transport And Anomaly Detection With Neural Networks: A Primal-dual Algorithm
• Scalable Unbalanced Optimal Transport using Generative Adversarial Networks
• Outlier-Robust Optimal Transport
• Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark
• Large-Scale Wasserstein Gradient Flows
• Continuous Wasserstein-2 Barycenter Estimation without Minimax Optimization
• Understanding the Effectiveness of Lipschitz-Continuity in Generative Adversarial Nets
• Generative Modeling with Optimal Transport Maps
• Neural Monge Map estimation and its applications
• Dataset Dynamics via Gradient Flows in Probability Space
• Learning transport cost from subset correspondence
• Large-Scale Optimal Transport via Adversarial Training with Cycle-Consistency
• A novel notion of barycenter for probability distributions based on optimal weak mass transport
• Affine Transport for Sim-to-Real Domain Adaptation
• Guiding the One-to-one Mapping in CycleGAN via Optimal Transport
• Physics Informed Convex Artificial Neural Networks (PICANNs) for Optimal Transport based Density Estimation
• Approximating the Optimal Transport Plan via Particle-Evolving Method