A Novel Sliced Fused Gromov-Wasserstein Distance
This work addresses computational bottlenecks in comparing heterogeneous data for researchers in machine learning and data analysis, offering an incremental improvement over existing sliced methods.
The authors tackled the computational challenge of Gromov-Wasserstein and fused Gromov-Wasserstein distances by proposing a novel slicing technique that reduces numerical effort while preserving invariance to isometric transformations and allowing comparison of arbitrary geometries, resulting in a more robust and reliable distance for applications like shape retrieval and graph isomorphism testing.
The Gromov--Wasserstein (GW) distance and its fused extension (FGW) are powerful tools for comparing heterogeneous data. Their computation is, however, challenging since both distances are based on non-convex, quadratic optimal transport (OT) problems. Leveraging 1D OT, a sliced version of GW has been proposed to lower the computational burden. Unfortunately, this sliced version is restricted to Euclidean geometry and loses invariance to isometries, strongly limiting its application in practice. To overcome these issues, we propose a novel slicing technique for GW as well as for FGW that is based on an appropriate lower bound, hierarchical OT, and suitable quadrature rules for the underlying 1D OT problems. Our novel sliced FGW significantly reduces the numerical effort while remaining invariant to isometric transformations and allowing the comparison of arbitrary geometries. We show that our new distance actually defines a pseudo-metric for structured spaces that bounds FGW from below and study its interpolation properties between sliced Wasserstein and GW. Since we avoid the underlying quadratic program, our sliced distance is numerically more robust and reliable than the original GW and FGW distance; especially in the context of shape retrieval and graph isomorphism testing.