Quadratic Wasserstein distance between Gaussian laws revisited with correlation

TL;DR

This paper revisits the formula for the quadratic Wasserstein distance between Gaussian distributions, providing a simple derivation based on correlation matrices and optimal couplings, enhancing understanding of Gaussian law distances.

Contribution

It offers a straightforward derivation of the Wasserstein distance formula for Gaussian laws using correlation matrices and orthogonal transformations, clarifying previous complex proofs.

Findings

Simplified derivation of Wasserstein distance formula for Gaussian distributions.

Utilizes orthogonal matrices to relate covariance matrices via correlation.

Highlights the role of copulas and optimal couplings in the derivation.

Abstract

In this note, we give a simple derivation of the formula obtained in Dowson and Landau (1982), Olkin and Pukelsheim (1982) and Givens and Shortt (1984) for the quadratic Wasserstein distance between two Gaussian distributions on $\R^d$ with respective covariance matrices $\Sigma_\mu$ and $\Sigma_\nu$. This derivation relies on the existence of an orthogonal matrix $O$ such that $O^*\Sigma_\mu O$ and $O^*\Sigma_\nu O$ share the same correlation matrix and on the simplicity of optimal couplings in the case with the same correlation matrix and therefore the same copula.