  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) and propose
Earth Mover's Intersection (EMI), a positive definite analog to EMD for sets of
different sizes. We provide conditions under which EMD or certain
approximations to EMD are negative definite. We also present a
positive-definite-preserving transformation that can be applied to any kernel
and can also be used to derive positive definite EMD-based kernels and show
that the Jaccard index is simply the result of this transformation. Finally, we
evaluate kernels based on EMI and the proposed transformation versus EMD in
various computer vision tasks and show that EMD is generally inferior even with
indefinite kernel techniques.
