On the Definiteness of Earth Mover's Distance and Its Relation to Set Intersection

TL;DR

This paper introduces Earth Mover's Intersection (EMI), a positive definite analog to Earth Mover's Distance (EMD), enabling more effective kernel methods in machine learning and demonstrating its advantages in computer vision tasks.

Contribution

The paper develops EMI as a positive definite set-based measure related to EMD, and proposes a transformation to derive positive definite EMD kernels, improving performance in vision applications.

Findings

EMI is positive definite and effective for set comparisons.

Transformations can convert EMD into positive definite kernels.

EMI-based kernels outperform EMD in vision tasks.

Abstract

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.