On the Definiteness of Earth Mover’s Distance and Its Relation to Set Intersection
本文提出推土机交集（EMI），一种正定版本的推土机距离（EMD），适用于不同大小的集合，并证明EMD在某些条件下是负定的，同时展示EMI在计算机视觉任务中优于EMD。
Positive definite (PD) 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 PD 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 PD-preserving transformation that can be applied to any kernel and can also be used to derive PD 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.