Quadratic Wasserstein distance between Gaussian laws revisited with correlation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912457687040000 |
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| author | Alfonsi, Aurélien Jourdain, Benjamin |
| author_facet | Alfonsi, Aurélien Jourdain, Benjamin |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_23742 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quadratic Wasserstein distance between Gaussian laws revisited with correlation Alfonsi, Aurélien Jourdain, Benjamin Probability 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. |
| title | Quadratic Wasserstein distance between Gaussian laws revisited with correlation |
| topic | Probability |
| url | https://arxiv.org/abs/2506.23742 |