Mathematics, Computation and Geometry of Data

Volume 1 (2021)

Number 1

Comparison between variational optimal mass transportation and Lie advection

Pages: 99 – 130

DOI: https://dx.doi.org/10.4310/MCGD.2021.v1.n1.a4

Authors

Kehua Su (School of Computer Science, Wuhan University, Wuhan, Hubei, China)

Chenchen Li (School of Computer Science, Wuhan University, Wuhan, Hubei, China)

Shifan Zhao (School of Computer Science, Wuhan University, Wuhan, Hubei, China)

Na Lei (Dalian University of Technology, Dalian, China)

Lok Ming Lui (Department of Mathematics, Chinese University of Hong Kong)

Shikui Chen (State University of New York, Stony Brook, N.Y., U.S.A.)

Xianfeng Gu (State University of New York, Stony Brook, N.Y., U.S.A.)

Abstract

Optimal mass transportation plays a fundamental role in graphics, vision and machine learning. Conventional variational approach based on Brenier’s theorem gives accurate optimal transportation mapping and the Wasserstein distance but with high computational cost.

This work generalizes the Lie advection method to Riemannian manifolds with any dimensions, and compares the variational approach with Lie advection approach. Our experimental results show the efficiency and efficacy of the Lie advection method and demonstrate the Lie advection map can approximate the optimal transportation map with high accuracy.

Keywords

optimal mass transportation, Lie advection, Monge–Ampère, measure-preserving, Wasserstein distance

2010 Mathematics Subject Classification

Primary 68U05. Secondary 52B55.

The project is partially supported by NSFC 61772105, 61772379, 61328206, and 61720106005; and by NSF DMS-1418255, AFOSR FA9550-14-1-0193.

Received 20 September 2019

Published 15 July 2022