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Wasserstein Measure If P and Q are two probability distributions, then Wasserstein distance is defined as W(PllQ) = __inf 'Eq .y llz—yll] J_GH(P,@ P('j): r('x, )J)\. « II(P, Q) denotes the set of all joint distributions y (x, y) whose marginals are P and Q respectively * y(x,y) represents the "mass" transported from x (in distribution P) to y (in distribution Q) * The Wasserstein distance (Earth mover distance) is minimum cost of undertaking such a transport The Wasserstein distance has a dual form (Kantorovich-Rubinstein): W (Pr, Pg) = SUP< ]EmNPr@ Eqnpy [ ()] Supremum over W P/ 8 ) HlfllL-i o K-Lipschitz =— sup E,op [f(2)] —Egnp,[f(z)] ([ Tunctions fllL<K M M 7Y R ,f IR = where, f(.) is K-Lipschitz if"f(x) —fI <K|x—y]| N N=4 Arjovsky et al., Wasserstein GAN, 2017
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