Recent findings for optimal transport maps between distribution functions sharing the same copula show that componentwise the solution is the optimal map between marginal distributions. This is an important discovery since in the multivariate setting optimal maps are difficult to find and only known in a few special cases. In this paper, we extend the result on common copulas by showing that orthonormal transformations of variables sharing a common copula also have a known optimal map. We illustrate this by establishing optimal maps between members of a class of scale mixture of normal distributions.

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Parseval 的恒等式和最优运输图

共享相同 copula 的分布函数之间的最优传输图的最新发现表明，组件方面的解决方案是边缘分布之间的最优图。这是一个重要的发现，因为在多变量设置中很难找到最佳映射，并且仅在少数特殊情况下才知道。在本文中，我们通过展示共享公共 copula 的变量的正交变换也具有已知的最优映射来扩展公共 copula 的结果。我们通过在一类正态分布混合尺度的成员之间建立最佳映射来说明这一点。