The observable Wasserstein distance is a hierarchy of lower bounds on Wasserstein distance via 1-Lipschitz projections to the line, with an injectivity theorem that recovers the full distance when the order exceeds the metric covering dimension of the support.
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The Observable Wasserstein Distance
The observable Wasserstein distance is a hierarchy of lower bounds on Wasserstein distance via 1-Lipschitz projections to the line, with an injectivity theorem that recovers the full distance when the order exceeds the metric covering dimension of the support.