On Wasserstein distances for affine transformations of random vectors

TL;DR

The paper addresses the problem of quantifying the Wasserstein distance $W_2$ between a distribution and its affine transformations, with a focus on rotations and their role in rotational manifolds within Wasserstein space. It leverages the Dowson–Landau lower bound and Gelbrich’s equality framework to derive explicit, covariance-based lower bounds, and in 2D derives a closed-form rotation bound using the trace of square roots of covariance products; it also provides an upper bound for compositions of translations, rotations, and dilations. Key contributions include a closed-form expression for $\operatorname{tr}[(\Sigma_{R_\theta X}\Sigma_{R_\varphi X})^{1/2}]$ and sharp lower bounds that are attained for Gaussian or elliptically contoured distributions, together with practical upper bounds for affine compositions and concrete demonstrations on 1D submanifolds and handwriting-inspired data for manifold learning via Wassmap. The work advances understanding of Wasserstein geometry under affine deformations and offers practical tools for synthetic data generation and dimensionality reduction benchmarking in applications like handwriting analysis and image-based manifold learning, while highlighting open questions about centering, symmetry, and the influence of data geometry on the tightness of the bounds.

Abstract

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.

Figures (11)

• Figure 1: Rotation of $[0,1]^2$ from Example \ref{['EX:01Rotation']}. (Left) $W_2(X,R_\theta X)$ and the lower bound $\sqrt{1-\cos(\theta)}$ for $\theta\in[0,\frac{\pi}{2}]$. (Right) Relative error $(W_2(X,R_\theta X)-\sqrt{1-\cos(\theta)})/W_2(X,R_\theta X)$ for $\theta\in[0,\frac{\pi}{2}]$.
• Figure 2: Rotation of $[-\frac{1}{2},\frac{1}{2}]^2$ from Example \ref{['EX:CenteredCubeRotation']}. Plotted is $W_2(X,R_\theta X)$ for $\theta\in[0,\frac{\pi}{2}]$.
• Figure 3: Rotation of $[0,2]\times[0,1]$ from Example \ref{['EX:RectangleRotation']}. (Left) $W_2(X,R_\theta X)$ and the lower bound of \ref{['EQN:RectangleLowerBound']} for $\theta\in[0,\frac{\pi}{2}]$. (Right) Relative error $(W_2(X,R_\theta X)-$lower bound)$/W_2(X,R_\theta X)$ for $\theta\in[0,\frac{\pi}{2}]$.
• Figure 4: Figure illustrating Example \ref{['EX:GaussianComposition']}. (Top) Sample of $X\sim\mathcal{N}(0,\Sigma_1)$ under the map $T_\alpha\circ R_\theta\circ S_\lambda$ for $\theta\in[0,\frac{\pi}{2}]$ with $\alpha = 0$, $\lambda = \frac{1}{2}2$. (Bottom) Relative error (upper bound $- W_2(T_\alpha\circ R_\theta\circ S_\lambda X,X))/W_2(T_\alpha\circ R_\theta\circ S_\lambda X,X)$ for $X\sim(0,\Sigma_i)$, $i=1,2,3$ using the upper bound of Theorem \ref{['THM:CompositionW2']} (which is valid because Gaussians satisfy equality in Theorem \ref{['THM:RotationW2']}).
• Figure 5: (Left) $X$ drawn from the uniform distribution on the unit circle in $\mathbb{R}^2$ as in Example \ref{['EX:CircleComposition']}. Shown is the relative error between the Wasserstein distance and the upper bound. (Right) Relative error between Wasserstein distance and upper bound for $X$ drawn from the uniform distribution on $[-\frac{1}{2},\frac{1}{2}]$ in $\mathbb{R}^2$ as in Example \ref{['EX:LineComposition']}.

Theorems & Definitions (39)

• Proposition 1.1
• proof
• Theorem 1.2: dowson1982frechet
• Corollary 1.3
• Proposition 1.4
• proof
• Proposition 2.1
• proof
• Proposition 2.2
• proof