Structure preservation via the Wasserstein distance

Abstract

We show that under minimal assumptions on a random vector $X\in R^d$ and with high probability, given m independent copies of X, the coordinate distribution of each vector ${(〈X_i,\theta 〉)}_{i=1}^m$ is dictated by the distribution of the true marginal $〈X,\theta 〉$. Specifically, we show that with high probability,$\underset{\theta \in S^{d-1}}{\sup }{\left(\frac{1}{m}\sum _{i=1}^m{|{〈X_i,\theta 〉}^{♯}-{\lambda }_i^{\theta }|}^2\right)}^{1/2}\le c{\left(\frac{d}{m}\right)}^{1/4},$ where ${\lambda }_i^{\theta }=m{\int }_{(\frac{i-1}{m},\frac{i}{m}]}{F}_{〈X,\theta 〉}^{-1}\left(u\right)du$ and $a^{♯}$ denotes the monotone non-decreasing rearrangement of a. Moreover, this estimate is optimal.

The proof follows from a sharp estimate on the worst Wasserstein distance between a marginal of X and its empirical counterpart, $\frac{1}{m}{\sum }_{i=1}^m{\delta }_{〈X_i,\theta 〉}$.