Quadratic Wasserstein distance between Gaussian laws revisited with correlation

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 ${\mathbb{R}}^d$ with respective covariance matrices ${\Sigma }_{\mu }$ and ${\Sigma }_{\nu }$. This derivation relies on the existence of an orthogonal matrix $O$ such that $O^{\ast }{\Sigma }_{\mu }O$ and $O^{\ast }{\Sigma }_{\nu }O$ 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.