Calculus of Variations and Geometric Measure Theory

H. Lavenant - F. Santambrogio

The flow map of the Fokker-Planck equation does not provide optimal transport

created by lavenant on 08 Mar 2022
modified on 31 May 2022


Accepted Paper

Inserted: 8 mar 2022
Last Updated: 31 may 2022

Journal: Applied Mathematics Letters
Year: 2022


In Khrulkov and Oseledets. Understanding DDPM Latent Codes Through Optimal Transport. \emph{arXiv preprint arXiv:2202.07477} the authors conjecture that, by integrating the flow of the ODE given by the Wasserstein velocity in a Fokker-Planck equation, one obtains an optimal transport map. On the other hand this result was thought to be false in Kim and Milman. A generalization of Caffarelli’s contraction theorem via (reverse) heat flow. \emph{Mathematische Annalen 354.3 (2012): 827-862} but no proof was provided. In this note we show that the result claimed by Khrulkov and Oseledets cannot hold. This strengthens a counterexample which was built in Tanana. Comparison of transport map generated by heat flow interpolation and the optimal transport Brenier map. \emph{Communications in Contemporary Mathematics, 23(6)}, 2021.