Structure preservation via the Wasserstein distance

We show that under minimal assumptions on a random vector $X\in {\mathbb{R}}^d$ 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, $$\underset{\theta \in S^{d-1}}{\sup }{\left(\frac{1}{m}\sum _{i=1}^m{\mid ⟨X_i,\theta {⟩}^{\mathrm{♯}}-{\lambda }_i^{\theta }\mid }^2\right)}^{1\mathrm{/}2}\le c{\left(\frac{d}{m}\right)}^{1\mathrm{/}4},$$ where ${\lambda }_i^{\theta }=m{\int }_{(\frac{i-1}{m},\frac{i}{m}]}{F}_{⟨X,\theta ⟩}^{-1}(u)du$ and $a^{\mathrm{♯}}$ 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, $\frac{1}{m}{\sum }_{i=1}^m{\delta }_{⟨X_i,\theta ⟩}$.