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

On Wasserstein distances for affine transformations of random vectors