Orthogonal Gromov-Wasserstein Discrepancy with Efficient Lower Bound

TL;DR

This paper introduces the orthogonal Gromov-Wasserstein (OGW) discrepancy, a new surrogate for GW that provides a tight, efficient lower bound and extends to fused GW, improving graph comparison tasks.

Contribution

The paper proposes OGW as a novel surrogate for GW with a closed-form lower bound, extending to fused GW and enabling efficient, accurate graph data analysis.

Findings

OGW lower bound is tight and computationally efficient.

OGW extends naturally to fused Gromov-Wasserstein incorporating node features.

Experiments demonstrate accurate predictions and effective barycenters for graph sets.

Abstract

Comparing structured data from possibly different metric-measure spaces is a fundamental task in machine learning, with applications in, e.g., graph classification. The Gromov-Wasserstein (GW) discrepancy formulates a coupling between the structured data based on optimal transportation, tackling the incomparability between different structures by aligning the intra-relational geometries. Although efficient \emph{local} solvers such as conditional gradient and Sinkhorn are available, the inherent non-convexity still prevents a tractable evaluation, and the existing lower bounds are not tight enough for practical use. To address this issue, we take inspiration from the connection with the quadratic assignment problem, and propose the orthogonal Gromov-Wasserstein (OGW) discrepancy as a surrogate of GW. It admits an efficient and \emph{closed-form} lower bound with $\mathcal{O}(n^3)$ complexity, and directly extends to the fused Gromov-Wasserstein (FGW) distance, incorporating node features into the coupling. Extensive experiments on both the synthetic and real-world datasets show the tightness of our lower bounds, and both OGW and its lower bounds efficiently deliver accurate predictions and satisfactory barycenters for graph sets.