A Novel Sliced Fused Gromov-Wasserstein Distance

TL;DR

This paper introduces a new sliced Gromov-Wasserstein distance that is computationally efficient, invariant to isometries, and applicable to arbitrary geometries, improving shape retrieval and graph comparison tasks.

Contribution

A novel slicing technique for GW and FGW based on hierarchical OT and quadrature rules, enhancing efficiency and invariance properties.

Findings

Reduces computational effort significantly.

Maintains invariance to isometric transformations.

Improves robustness in shape retrieval and graph testing.

Abstract

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.