Earth Mover’s Distance (EMD) is a robust metric for comparing probability distributions. Unlike many conventional distances, EMD exploits the geometry of the space on which the distributions are defined. However, evaluating EMD directly is computationally expensive. In this talk, I will present metrics equivalent to EMD that can be computed rapidly. I will explain how these metrics emerge from the theory of Hölder spaces and their duals. I will also show that on any space equipped with a one-parameter family of operators of the kind widely used in machine learning, there is a natural metric with respect to which the classical characterizations of Hölder spaces and their duals generalize. Finally, I will extend these results to multi-parameter operators.