Contents Online
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
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