On Wasserstein distances for affine transformations of random vectors

Figure 1.  Rotation of $ [0,1]^2 $ from Example 2.13. (Left) $ W_2(X,R_\theta X) $ and the lower bound $ \sqrt{1-\cos(\theta)} $ for $ \theta\in[0,\frac\pi2] $. (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 $ [-\frac12,\frac12]^2 $ from Example 2.14. 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 2.15. (Left) $ W_2(X,R_\theta X) $ and the lower bound of (8) for $ \theta\in[0,\frac\pi2] $. (Right) Relative error $ (W_2(X,R_\theta X)- $lower bound)$ /W_2(X,R_\theta X) $ for $ \theta\in[0,\frac\pi2] $

Figure 4.  Figure illustrating Example 2.20. (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 = \begin{bmatrix} \frac12 \\ 2 \end{bmatrix} $. (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 2.17 (which is valid because Gaussians satisfy equality in Theorem 2.10). Error bars show one standard deviation of the error over 20 repeated trials

Figure 5.  (Left) $ X $ drawn from the uniform distribution on the unit circle in $ \mathbb{R}^2 $ as in Example 3.1. 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 $ [-\frac12,\frac12] $ in $ \mathbb{R}^2 $ as in Example 3.2

Figure 6.  (Left) $ X_1 $ and $ X_2 $ vs. $ t $. (Right) $ X = \begin{bmatrix}X_1\\X_2\end{bmatrix} $ forming the letter C

Figure 7.  (Left) $ X_1 $ and $ X_2 $ vs. $ t $. (Right) $ X = \begin{bmatrix}X_1\\X_2\end{bmatrix} $ forming the letter A

Figure 8.  (Left) Letter A undergoing rotation. (Right) Relative error $ (W_2(R_\theta X,X)- $lower bound)$ /W_2(R_\theta X,X) $ for the lower bound of Theorem 2.10

Figure 9.  (Left) $ X_1 $ and $ X_2 $ vs. $ t $ for the parametrization (11) (top) and (12) (bottom). (Right) $ X = \begin{bmatrix}X_1\\X_2\end{bmatrix} $ forming the letter T

Figure 10.  (Left) Letter T undergoing rotation. (Center) Relative error $ (W_2(R_\theta X,X)- $lower bound)$ /W_2(R_\theta X,X) $ for the lower bound of Theorem 2.10 for the T parametrized by (11) (Right) Relative error for the T with nonzero mean parametrized by (12)

Figure 11.  (Left) Wassmap embedding of rotated copies of letter T parametrized by (11). (Center) Same for T parametrized by (12). (Right) MDS embedding using the lower bound of Theorem 2.10 for T parametrized by (11)