Comparison of transport map generated by heat flow interpolation and the optimal transport Brenier map

TL;DR

This paper compares two different methods for generating transport maps: heat flow interpolation and the optimal transport Brenier map, highlighting their differences.

Contribution

It demonstrates that the heat flow interpolation map generally differs from the Brenier optimal transport map, clarifying their relationship.

Findings

Heat flow interpolation map is not generally the Brenier optimal transport map.

The two transport maps can differ significantly in certain cases.

The note clarifies the distinction between these two approaches.

Abstract

This note shows that the non-expansive transport map constructed by Y.-H. Kim and E. Milman using heat flow interpolation is in general different from the optimal transport Brenier map.