On the Pythagorean Structure of the Optimal Transport for Separable Cost Functions

TL;DR

This paper explores the structure of optimal transport problems with separable cost functions, revealing a Pythagorean framework and conditional independence properties, with explicit solutions in Euclidean spaces for certain cases.

Contribution

It introduces a Pythagorean structure for optimal transport with separable costs and derives explicit formulas in Euclidean spaces when one measure is supported on a line.

Findings

Optimal transport can be decomposed into lower-dimensional movements.

Conditional independence property of the transportation plan.

Explicit formula for transport plans when one measure is supported on a line.

Abstract

In this paper, we study the optimal transport problem induced by separable cost functions. In this framework, transportation can be expressed as the composition of two lower-dimensional movements. Through this reformulation, we prove that the random variable inducing the optimal transportation plan enjoys a conditional independence property. We conclude the paper by focusing on some significant settings. In particular, we study the problem in the Euclidean space endowed with the squared Euclidean distance. In this instance, we retrieve an explicit formula for the optimal transportation plan between any couple of measures as long as one of them is supported on a straight line.