Distance-Matrix Wasserstein Statistics for Scalable Gromov--Wasserstein Learning

TL;DR

This paper introduces Distance-Matrix Wasserstein (DMW), a scalable and theoretically grounded method for approximating Gromov--Wasserstein distances using sampled distance matrices, with applications in graph and shape analysis.

Contribution

The authors propose DMW, a hierarchy of Wasserstein statistics that approximates GW distances efficiently, with proven bounds and scalable computation methods.

Findings

DMW provides a lower bound and relaxation of GW distances.

Finite-sample bounds depend on data manifold, not ambient dimension.

Experiments demonstrate DMW's effectiveness in graph classification and two-sample testing.

Abstract

Gromov--Wasserstein (GW) distances compare graphs, shapes, and point clouds through internal distances, without requiring a common coordinate system. This invariance is powerful, but discrete GW is a nonconvex quadratic optimal transport problem and is difficult to estimate at scale. We propose \emph{Distance-Matrix Wasserstein} (DMW), a hierarchy of Wasserstein statistics comparing laws of random finite distance matrices. Rather than optimizing a global point-level alignment, DMW samples $n$ points from each space, records their pairwise distances, and transports the resulting matrix laws. We prove that DMW is a relaxation and lower bound of GW, and establish a reverse approximation inequality: the GW--DMW gap is controlled by the Wasserstein error of approximating each original measure with $n$ samples. Hence population DMW converges to GW as sampled subspaces become dense. We further give finite-sample bounds, including intrinsic-dimensional rates that depend on the data manifold rather than the ambient matrix dimension $\binom n2$. For scalable computation, we introduce sliced and multi-scale DMW; for $p=1$, the sliced multi-scale dissimilarity yields positive-definite exponential kernels. Experiments on synthetic metric spaces, scalability benchmarks, graph classification, and two-sample testing validate the theory and demonstrate an interpretable GW-style proxy for structural comparison.