A remark on the optimal transport between two probability measures sharing the same copula

TL;DR

This paper investigates the Wasserstein distance between two probability measures sharing the same copula, revealing that the natural coupling is optimal only under specific cost functions where the cost decomposes coordinate-wise.

Contribution

It characterizes when the pseudo-inverse based coupling is optimal for measures sharing the same copula, highlighting the special case when costs decompose coordinate-wise.

Findings

Coupling based on pseudo-inverses is optimal only when p=q in the cost function.

Optimality of the coupling depends on the cost function's structure.

The result generalizes the known optimal coupling in one dimension to higher dimensions.

Abstract

We are interested in the Wasserstein distance between two probability measures on $\R^n$ sharing the same copula $C$. The image of the probability measure $dC$ by the vectors of pseudo-inverses of marginal distributions is a natural generalization of the coupling known to be optimal in dimension $n=1$. It turns out that for cost functions $c(x,y)$ equal to the $p$-th power of the $L^q$ norm of $x-y$ in $\R^n$, this coupling is optimal only when $p=q$ i.e. when $c(x,y)$ may be decomposed as the sum of coordinate-wise costs.