Earth Mover's Distance Yields Positive Definite Kernels For Certain Ground Distances

Positive definite kernels are an important tool in machine learning that enable efficient solutions to otherwise difficult or intractable problems by implicitly linearizing the problem geometry. In this paper we develop a set-theoretic interpretation of the Earth Mover’s Distance (EMD) that naturally yields metric and kernel forms of EMD as generalizations of elementary set operations. In particular, EMD is generalized to sets of unequal size. We also offer the first proof of positive definite kernels based directly on EMD, and provide propositions and conjectures concerning what properties are necessary and sufficient for EMD to be conditionally negative definite. In particular, we show that three distinct positive definite kernels – intersection, minimum, and Jaccard index – can be derived from EMD with various ground distances. In the process we show that the Jaccard index is simply the result of a positive definite preserving transformation that can be applied to any kernel. Finally, we evaluate the proposed kernels in various computer vision tasks.