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Estimating Wasserstein distances between two high-dimensional densities suffers from the curse of dimensionality: one needs an exponential (wrt dimension) number of samples to ensure that the distance between two empirical measures is comparable to the distance between the original densities.
Computational optimal transport
Peyré, G. and Cuturi, M · 1935
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The speed of mean Glivenko-Cantelli convergence
Dudley, R. M · 1969
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