  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.
