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Optimal transport mapping via input convex neural networks
Ashok Vardhan Makkuva · Amirhossein Taghvaei · Sewoong Oh · Jason Lee

Wed Jul 15 08:00 AM -- 08:45 AM & Wed Jul 15 08:00 PM -- 08:45 PM (PDT) @ None #None

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.

Author Information

Ashok Makkuva (UIUC)
Amir Taghvaei (University of Illinois at Urbana-Champaign)

Amirhossein Taghvaei was born and raised in southern beach of Caspian sea, Mazandaran, Iran. After obtaining two B.S degrees in Mechanical engineering and Physics from Sharif Univ. of Technology, Tehran, Iran, he joined University of Illinois at Urbana-Champaign as a PhD student in Mechanical Science and Engineering department. While pursuing PhD, he obtained his Masters degree in Mathematics, in May 2017. He is now in the last stage of his PhD program looking for opportunities to continue an academic path. He is a member of the Decision and Control group in Coordinated Science Laboratory advised by Prof. Prashant Mehta. His research interest lies mostly in the intersection of control theory and machine learning.

Sewoong Oh (University of Washington)
Jason Lee (Princeton)

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