Structure preservation via the Wasserstein distance

Abstract

We show that under minimal assumptions on a random vector X ∈ R d and with high probability, given m independent copies of X , the coordinate distribution of each vector ( 〈 X i , θ 〉 ) i = 1 m is dictated by the distribution of the true marginal 〈 X , θ 〉 . Specifically, we show that with high probability, sup θ ∈ S d − 1 ⁡ ( 1 m ∑ i = 1 m | 〈 X i , θ 〉 ♯ − λ i θ | 2 ) 1 / 2 ≤ c ( d m ) 1 / 4 , where λ i θ = m ∫ ( i − 1 m , i m ] F 〈 X , θ 〉 − 1 ( u ) d u 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, 1 m ∑ i = 1 m δ 〈 X i , θ 〉 .

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