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In the application of the Routh-Hurwitz criterion, two specific scenarios can arise that complicate stability analysis.
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On a Linear Gromov-Wasserstein Distance.

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    Summary
    This summary is machine-generated.

    This study introduces linear Gromov-Wasserstein distances, a novel approach inspired by linear optimal transport. This method offers a computationally efficient alternative for applications like shape classification.

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    Area of Science:

    • Optimal Transport Theory
    • Geometric Measure Theory
    • Machine Learning

    Background:

    • Gromov-Wasserstein (GW) distances generalize Wasserstein distances, offering invariance under isometries.
    • Existing linear optimal transport (LOT) methods are computationally efficient but lack a linear GW counterpart.
    • The absence of linear GW distances limits their application in complex geometric problems.

    Purpose of the Study:

    • To define and introduce the concept of linear Gromov-Wasserstein (LGW) distances.
    • To develop a generalized LOT model that motivates the LGW definition.
    • To demonstrate the practical utility of LGW distances in computational applications.

    Main Methods:

    • Definition of linear Gromov-Wasserstein distances.
    • Development of a generalized linear optimal transport model using barycentric projections.
    • Numerical implementation and evaluation of the proposed LGW distances.

    Main Results:

    • Successful formulation of linear Gromov-Wasserstein distances.
    • The proposed generalized LOT model provides a foundation for LGW.
    • Numerical examples show LGW distances can replace pairwise GW computations effectively.

    Conclusions:

    • Linear Gromov-Wasserstein distances provide a computationally tractable alternative to standard GW distances.
    • The proposed method enhances efficiency in applications such as shape classification.
    • This work opens new avenues for applying optimal transport in geometric data analysis.