Parseval's Identity and Optimal Transport Maps

TL;DR

This paper extends the understanding of optimal transport maps by showing that orthonormal transformations of variables with a shared copula also have known optimal maps, with applications to scale mixture of normal distributions.

Contribution

It generalizes previous results on copula-based optimal maps to include orthonormal transformations, providing explicit solutions for a broader class of distributions.

Findings

Optimal maps exist for orthonormal transformations sharing a common copula.

Explicit optimal transport maps are derived for scale mixture of normal distributions.

The results simplify the computation of optimal transport in multivariate settings.

Abstract

Recent findings for optimal transport maps between distribution functions sharing the same copula show that componentwise the solution is the optimal map between marginal distributions. This is an important discovery since in the multivariate setting optimal maps are difficult to find and only known in a few special cases. In this paper, we extend the result on common copulas by showing that orthonormal transformations of variables sharing a common copula also have a known optimal map. We illustrate this by establishing optimal maps between members of a class of scale mixture of normal distributions.