TY - GEN
T1 - The Singularity Set of Optimal Transportation Maps
AU - Luo, Zhongxuan
AU - Chen, Wei
AU - Lei, Na
AU - Guo, Yang
AU - Zhao, Tong
AU - Gu, Xianfeng
N1 - Publisher Copyright:
© 2021, Springer Nature Switzerland AG.
PY - 2021
Y1 - 2021
N2 - Optimal transportation plays an important role in many engineering fields, especially in deep learning. By Brenier theorem, computating optimal transportation maps is reduced to solving Monge–Ampère equations, which in turn is equivalent to construct 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 homotopic equivalent as well.
AB - Optimal transportation plays an important role in many engineering fields, especially in deep learning. By Brenier theorem, computating optimal transportation maps is reduced to solving Monge–Ampère equations, which in turn is equivalent to construct 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 homotopic equivalent as well.
KW - Convex hull
KW - Power Diagram
KW - Secondary polytope
KW - Upper Envelope
KW - Weighted Delaunay triangulation
UR - https://www.scopus.com/pages/publications/85116391728
U2 - 10.1007/978-3-030-76798-3_4
DO - 10.1007/978-3-030-76798-3_4
M3 - Conference contribution
AN - SCOPUS:85116391728
SN - 9783030767976
T3 - Lecture Notes in Computational Science and Engineering
SP - 61
EP - 80
BT - Numerical Geometry, Grid Generation and Scientific Computing - Proceedings of the 10th International Conference, NUMGRID 2020 / Delaunay 130, Celebrating the 130th Anniversary of Boris Delaunay, 2020
A2 - Garanzha, Vladimir A.
A2 - Si, Hang
A2 - Kamenski, Lennard
PB - Springer Science and Business Media Deutschland GmbH
T2 - 10th International Conference on Numerical Geometry, Grid Generation, and Scientific Computing celebrating the 130th anniversary of B. N. Delaunay, NUMGRID 2020
Y2 - 25 November 2020 through 27 November 2020
ER -