We expound on some known lower bounds of the quadratic Wasserstein distance between random vectors in $\mathbb{R}^n$ with an emphasis on affine transformations that have been used in manifold learning of data in Wasserstein space. In particular, we give concrete lower bounds for rotated copies of random vectors in $\mathbb{R}^2$ with uncorrelated components by computing the Bures metric between the covariance matrices. We also derive upper bounds for compositions of affine maps which yield a fruitful variety of diffeomorphisms applied to an initial data measure. We apply these bounds to various distributions including those lying on a 1-dimensional manifold in $\mathbb{R}^2$ and illustrate the quality of the bounds. Finally, we give a framework for mimicking handwritten digit or alphabet datasets that can be applied in a manifold learning framework.

论文阐述了位于Wasserstein空间的数据流形学习中的关于随机向量在$\mathbb{R}^n$中的二次Wasserstein距离的一些已知下界，重点考虑应用于数据的仿射变换。具体而言，通过计算协方差矩阵之间的Bures度量，给出了关于在$\mathbb{R}^2$中具有不相关分量的随机向量的旋转副本的具体下界。我们还推导了由仿射映射组成的上界，从而产生了多样的微分同胚，应用于初始数据度量。我们将这些界限应用于各种分布，包括位于$\mathbb{R}^2$中的1维流形上的分布，并展示了界限的质量。最后，我们提出了一个可以应用于流形学习框架中的模仿手写数字或字母数据集的框架。

关于随机向量的仿射变换的Wasserstein距离