A geometric view of optimal transportation and generative model

Na Lei Dalian University of Technology Kehua Su Wuhan University Li Cui Beijing Normal University Shing-Tung Yau Harvard University Xianfeng Gu Stony Brook University

Computational Geometry Convex and Discrete Geometry Machine Learning TBD mathscidoc:1905.09001

Computer Aided Geometric Design, 68, 1-21, 2019.1
In this work, we give a geometric interpretation to the Generative Adversarial Networks (GANs). The geometric view is based on the intrinsic relation between Optimal Mass Transportation (OMT) theory and convex geometry, and leads to a variational approach to solve the Alexandrov problem: constructing a convex polytope with prescribed face normals and volumes. By using the optimal transportation view of GAN model, we show that the discriminator computes the Wasserstein distance via the Kantorovich potential, the generator calculates the transportation map. For a large class of transportation costs, the Kantorovich potential can give the optimal transportation map by a close-form formula. Therefore, it is sufficient to solely optimize the discriminator. This shows the adversarial competition can be avoided, and the computational architecture can be simplified. Preliminary experimental results show the geometric method outperforms the traditional Wasserstein GAN for approximating probability measures with multiple clusters in low dimensional space.
Optimal Transport; Monge-Ampere; GAN; Wasserstein distance
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  title={A geometric view of optimal transportation and generative model},
  author={Na Lei, Kehua Su, Li Cui, Shing-Tung Yau, and Xianfeng Gu},
  booktitle={Computer Aided Geometric Design},
Na Lei, Kehua Su, Li Cui, Shing-Tung Yau, and Xianfeng Gu. A geometric view of optimal transportation and generative model. 2019. Vol. 68. In Computer Aided Geometric Design. pp.1-21. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190504221807096046311.
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