On Wasserstein distances for affine transformations of random vectors
This provides incremental theoretical tools for manifold learning applications in Wasserstein space.
The paper derived concrete lower and upper bounds for Wasserstein distances between affine transformations of random vectors, specifically computing Bures metric bounds for rotated copies in ℝ² and applying them to distributions on 1D manifolds.
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.