Title: Max-Min Sliced Gromov-Wasserstein
Primary area: other topics in machine learning (i.e., none of the above)
Keywords: Optimal Transport, Gromov-Wasserstein, Isometry

Abstract: The Gromov-Wasserstein (GW) distance is a powerful tool for comparing objects across different metric spaces, but its high computational complexity limits its applicability. Although the Sliced Gromov-Wasserstein (SGW) discrepancy addresses this issue by projecting onto 1D distributions, it sacrifices key isometric properties, such as reflection and rotation invariance. In this work, we introduce the max-min Sliced Gromov-Wasserstein (MSGW), a new variant that preserves the computational efficiency of SGW while ensuring essential isometric properties. This method can be viewed as an adversarial game and is closely tied to the Hausdorff distance. Empirical results demonstrate that MSGW achieves competitive performance with a limited number of projections and excels in scenarios with varying dimensions, making it a practical and robust alternative to existing
methods.