A remark on the optimal transport between two probability measures sharing the same copula

2013/07/16 by Aurélien Alfonsi, Benjamin Jourdain, Alfonsi, Aurélien +1 · 1 citation
Mathematics · Physics and Astronomy · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Statistical Mechanics and Entropy #math.PR

paper · pdf · doi:10.48550/arxiv.1307.4249

arxiv created 2013/07/16 · openalex publication_date 2013/07/16 · arxiv updated 2013/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We are interested in the Wasserstein distance between two probability measures on \Rⁿ sharing the same copula C. The image of the probability measure dC by the vectors of pseudo-inverses of marginal distributions is a natural generalization of the coupling known to be optimal in dimension n=1. It turns out that for cost functions c(x,y) equal to the p-th power of the L^q norm of x-y in \Rⁿ, this coupling is optimal only when p=q i.e. when c(x,y) may be decomposed as the sum of coordinate-wise costs.

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