A geometric perspective on regularized optimal transport

TL;DR

This paper offers a geometric framework for understanding regularized optimal transport, introducing variational problems on Riemannian manifolds and analogues of classical problems like Schrödinger bridge and Yasue, along with a Hopf--Cole transformation.

Contribution

It introduces a novel geometric perspective on regularized optimal transport, including new variational problems and transformations on Riemannian manifolds.

Findings

New geometric intuition for regularized optimal transport

Variational problems on Riemannian manifolds related to Schrödinger bridge and Yasue

Geometric analogue of the Hopf--Cole transformation

Abstract

We present new geometric intuition on dynamical versions of regularized optimal transport. We introduce two families of variational problems on Riemannian manifolds which contain analogues of the Schr\"odinger bridge problem and the Yasue problem. We also propose an analogue of the Hopf--Cole transformation in the geometric setting.