Structure preservation via the Wasserstein distance

Abstract

We show that under minimal assumptions on a random vector X∈Rd and with high probability, given m independent copies of X, the coordinate distribution of each vector (〈Xi,θ〉)i=1m is dictated by the distribution of the true marginal 〈X,θ〉. Specifically, we show that with high probability, supθ∈Sd−1⁡([Formula presented])1/4, where λiθ=m∫([Formula presented]F〈X,θ〉−1(u)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, [Formula presented]∑i=1mδ〈Xi,θ〉. Show more