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.
Bonnotte.Unidimensional and evolution methods for optimal transportation
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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.