# On Wasserstein distances for affine transformations of random vectors

arXiv: https://arxiv.org/abs/2310.03945 · HTML: https://mlsanity.com/paper/2310.03945/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 32 references that are in the map.
- First posted on arXiv 5 Oct 2023.

## Builds on (most cited first, 5 of 5)

- [Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space](https://mlsanity.com/paper/1910.05954/) (2019)
- [Wasserstein Task Embedding for Measuring Task Similarities](https://mlsanity.com/paper/2208.11726/) (2022)
- [Wassmap: Wasserstein Isometric Mapping for Image Manifold Learning](https://mlsanity.com/paper/2204.06645/) (2022)
- [Linearized Wasserstein dimensionality reduction with approximation guarantees](https://mlsanity.com/paper/2302.07373/) (2023)
- [Applications of No-Collision Transportation Maps in Manifold Learning](https://mlsanity.com/paper/2304.00199/) (2023)

## Abstract (arXiv)

We expound on some known lower bounds of the quadratic Wasserstein distance between random vectors in ℝn 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 ℝ2 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 ℝ2 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.