TY - GEN
T1 - Optimal transport mapping via input convex neural networks
AU - Makkuva, Ashok Vardhan
AU - Taghvaei, Amirhossein
AU - Lee, Jason D.
AU - Oh, Sewoong
N1 - Publisher Copyright:
© 2020 37th International Conference on Machine Learning, ICML 2020. All rights reserved.
PY - 2020
Y1 - 2020
N2 - 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 to estimate the optimal transport mapping as the gradient of a convex function that is trained via minimax optimization. Numerical experiments confirm the accuracy of the learned transport map. Our approach can be readily used to train a deep generative model. When trained between a simple distribution in the latent space and a target distribution, the learned optimal transport map acts as a deep generative model. Although scaling this to a large dataset is challenging, we demonstrate two important strengths over standard adversarial training: robustness and discontinuity. As we seek the optimal transport, the learned generative model provides the same mapping regardless of how we initialize the neural networks. Further, a gradient of a neural network can easily represent discontinuous mappings, unlike standard neural networks that are constrained to be continuous. This allows the learned transport map to match any target distribution with many discontinuous supports and achieve sharp boundaries.
AB - 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 to estimate the optimal transport mapping as the gradient of a convex function that is trained via minimax optimization. Numerical experiments confirm the accuracy of the learned transport map. Our approach can be readily used to train a deep generative model. When trained between a simple distribution in the latent space and a target distribution, the learned optimal transport map acts as a deep generative model. Although scaling this to a large dataset is challenging, we demonstrate two important strengths over standard adversarial training: robustness and discontinuity. As we seek the optimal transport, the learned generative model provides the same mapping regardless of how we initialize the neural networks. Further, a gradient of a neural network can easily represent discontinuous mappings, unlike standard neural networks that are constrained to be continuous. This allows the learned transport map to match any target distribution with many discontinuous supports and achieve sharp boundaries.
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M3 - Conference contribution
AN - SCOPUS:85105203547
T3 - 37th International Conference on Machine Learning, ICML 2020
SP - 6628
EP - 6637
BT - 37th International Conference on Machine Learning, ICML 2020
A2 - Daume, Hal
A2 - Singh, Aarti
PB - International Machine Learning Society (IMLS)
T2 - 37th International Conference on Machine Learning, ICML 2020
Y2 - 13 July 2020 through 18 July 2020
ER -