On a Linear Gromov-Wasserstein Distance

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.