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 $\mathbb{R}^2$ 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.

翻译：本文详述了$\mathbb{R}^n$中随机向量之间的二次Wasserstein距离的已知下界，重点研究了Wasserstein空间流形学习中使用的仿射变换。特别地，我们通过计算协方差矩阵之间的Bures度量，给出了$\mathbb{R}^2$中随机向量旋转副本的具体下界。我们还推导了仿射映射复合的上界，这些映射为初始数据测度提供了丰富的微分同胚变换。我们将这些界应用于各种分布，包括位于$\mathbb{R}^2$一维流形上的分布，并展示了这些界的效果。最后，我们给出了一个可应用于流形学习框架的手写数字或字母数据集的模拟框架。