Quadratic Wasserstein distance between Gaussian laws revisited with correlation

2025/06/30 by Alfonsi, Aurélien, Jourdain, Benjamin · 1 citation
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2506.23742

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 Σ_μ and Σ_ν. This derivation relies on the existence of an orthogonal matrix O such that O^*Σ_μO and O^*Σ_νO 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.

Citations

• A remark on the optimal transport between two probability measures sharing the same copula
• A class of Wasserstein metrics for probability distributions.

Cited by

• Wasserstein projections in the convex order: regularity and characterization in the quadratic Gaussian case

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