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The Singularity Set of Optimal Transportation Maps

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Abstract

Optimal transportation plays an important role in many engineering fields, especially in deep learning. By the Brenier theorem, computing optimal transportation maps is reduced to solving Monge–Ampère equations, which in turn is equivalent to constructing Alexandrov polytopes. Furthermore, the regularity theory of Monge–Ampère equation explains mode collapsing issue in deep learning. Hence, computing and studying the singularity sets of OT maps become important. In this work, we generalize the concept of medial axis to power medial axis, which describes the singularity sets of optimal transportation maps. Then we propose a computational algorithm based on variational approach using power diagrams. Furthermore, we prove that when the measures are changed homotopically, the corresponding singularity sets of the optimal transportation maps are homotopy equivalent as well. Furthermore, we generalize the Fréchet distance concept and utilize the obliqueness condition to give a sufficient condition for the existence of singularities of optimal transportation maps between planar domains. The condition is formulated using the boundary curvature.

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ACKNOWLEDGMENTS

This research of Luo, Chen, and Lei was partially supported by the National Natural Science Foundation of China under Grant nos. 61720106005, 61772105, 61936002, the Fundamental Research Funds for the Central Universities (DUT20TD107, DUT20JC32). The work of Guo and Gu was supported by the National Science Foundation (CMMI-1762287), the Ford University Research Program (URP no. 2017-9198R), the National Institute of Health (R21EB029733, R01LM012434). The research of Liu was supported by the Australian Research Council DP170100929 and DP200101084.

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Authors and Affiliations

  1. Key Laboratory for Ubiquitous Network and Service Software of Liaoning Province, 116620, Dalian, China

    Zhongxuan Luo

  2. School of Software Technology, Dalian University of Technology, 116620, Dalian, China

    Wei Chen

  3. DUT-RU ISE, Dalian University of Technology, 116620, Dalian, China

    Na Lei

  4. Department of Computer Science, Stony Brook University, 11794, Stony Brook, NY, USA

    Yang Guo & Xianfeng Gu

  5. INRIA Sophia-Antipolis & Telecom Paris, Paris, France

    Tong Zhao

  6. School of Mathematics and Applied Statistics, University of Wollongong, 2522, Wollongong, NSW, Australia

    Jiakun Liu

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  1. Zhongxuan Luo
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  2. Wei Chen
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  3. Na Lei
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  4. Yang Guo
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  5. Tong Zhao
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  6. Jiakun Liu
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  7. Xianfeng Gu
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Correspondence to Na Lei.

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Luo, Z., Chen, W., Lei, N. et al. The Singularity Set of Optimal Transportation Maps. Comput. Math. and Math. Phys. 62, 1313–1330 (2022). https://doi.org/10.1134/S0965542522080097

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  • DOI: https://doi.org/10.1134/S0965542522080097

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