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Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 2022, Volume 62, Number 8, Pages 1341–1359
DOI: https://doi.org/10.31857/S0044466922080099
(Mi zvmmf11439)
 

10th International Conference "Numerical Geometry, Meshing and High Performance Computing (NUMGRID 2020/Delaunay 130)"
Optimal control

The singularity set of optimal transportation maps

Zh. Luoa, W. Chenb, N. Leic, Ya. Guod, T. Zhaoe, J. Liuf, X. D. Gud

a Liaoning Province Ubiquitous Networking and Service Software Key Laboratory
b School of Software Technology, Dalian University of Technology
c International School of Information Science & Engineering, Dalian University of Technology and Ritsumeikan University
d Department of Computer Science, Stony Brook University
e Institut National de Recherche en Informatique et en Automatique, Sophia Antipolis – Méditerranée
f University of Wollongong
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.
Key words: upper envelope, convex hull, power diagram, weighted Delaunay triangulation, secondary polytope, normal Fréсhet distance, obliqueness, curvature.
Funding agency Grant number
National Natural Science Foundation of China 61720106005
61772105
61936002
CMMI-1762287
Fundamental Research Funds for the Central Universities of China DUT20TD107
DUT20JC32
URP 2017-9198R
National Institutes of Health R21EBO29733
R01LM012434
Australian Research Council DP170100929
DP200101084
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.
Received: 09.10.2021
Revised: 21.01.2022
Accepted: 11.03.2022
English version:
Computational Mathematics and Mathematical Physics, 2022, Volume 62, Issue 8, Pages 1313–1330
DOI: https://doi.org/10.1134/S0965542522080097
Bibliographic databases:
Document Type: Article
UDC: 519.85
Language: Russian
Citation: Zh. Luo, W. Chen, N. Lei, Ya. Guo, T. Zhao, J. Liu, X. D. Gu, “The singularity set of optimal transportation maps”, Zh. Vychisl. Mat. Mat. Fiz., 62:8 (2022), 1341–1359; Comput. Math. Math. Phys., 62:8 (2022), 1313–1330
Citation in format AMSBIB
\Bibitem{LuoCheLei22}
\by Zh.~Luo, W.~Chen, N.~Lei, Ya.~Guo, T.~Zhao, J.~Liu, X.~D.~Gu
\paper The singularity set of optimal transportation maps
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2022
\vol 62
\issue 8
\pages 1341--1359
\mathnet{http://mi.mathnet.ru/zvmmf11439}
\crossref{https://doi.org/10.31857/S0044466922080099}
\elib{https://elibrary.ru/item.asp?id=49273509}
\transl
\jour Comput. Math. Math. Phys.
\yr 2022
\vol 62
\issue 8
\pages 1313--1330
\crossref{https://doi.org/10.1134/S0965542522080097}
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    Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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