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The Wasserstein distance is a way of quantifying the distance between probability distributions on a metric space. Suppose \(P(x), Q(x)\) are two cumulative distributions functions of the real random variable \(X\). The Wasserstein metric is defined as

\[W_p(P,Q) = \Big(\int_0^1 du \lvert P^{-1}(u) - Q^{-1}(u)\lvert^p \Big)^{1/p}\]Visually