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Related Experiment Videos

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

Andrew Gardner, Christian A Duncan, Jinko Kanno

    IEEE Transactions on Cybernetics
    |July 11, 2018
    PubMed
    Summary

    This study introduces Earth Mover's Intersection (EMI), a positive definite kernel for comparing sets of different sizes. EMI and a novel transformation outperform Earth Mover's Distance (EMD) in computer vision tasks.

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    Area of Science:

    • Machine Learning
    • Computer Vision
    • Kernel Methods

    Background:

    • Positive definite (PD) kernels are crucial for simplifying complex machine learning problems by linearizing geometry.
    • Earth Mover's Distance (EMD) is a valuable metric but can be negative definite and challenging for sets of varying sizes.

    Purpose of the Study:

    • To develop a set-theoretic interpretation of EMD and introduce Earth Mover's Intersection (EMI) as a PD analog.
    • To propose a PD-preserving transformation applicable to any kernel, including deriving PD EMD-based kernels.
    • To evaluate EMI and the transformation against EMD in computer vision applications.

    Main Methods:

    • Developed a set-theoretic framework for EMD and introduced EMI for sets of differing sizes.
    • Identified conditions for EMD and its approximations to be negative definite.

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  • Proposed and applied a PD-preserving transformation, demonstrating its connection to the Jaccard index.

    • Main Results:

    • EMI is a positive definite kernel suitable for comparing sets of unequal sizes.
    • The proposed transformation yields PD kernels, with the Jaccard index as a specific instance.
    • EMI and the transformation-based kernels demonstrated superior performance over EMD in computer vision tasks.

    Conclusions:

    • EMI offers a robust, positive definite alternative to EMD for set comparison.
    • The PD-preserving transformation provides a versatile tool for kernel development.
    • The proposed methods represent a significant advancement over traditional EMD in practical applications.