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The Z-Gromov-Wasserstein Distance

Martin Bauer, Facundo Mémoli, Tom Needham, Mao Nishino

TL;DR

This work introduces the Z-Gromov-Wasserstein (Z-GW) distance, a unifying framework that extends the classical Gromov-Wasserstein distance to measure networks whose kernel takes values in a fixed metric space Z. By defining Z-networks as measure spaces equipped with a Z-valued kernel and the GW_p^Z objective, the authors establish that GW_p^Z is a metric on Z-networks up to weak isomorphism when Z is separable, and they prove fundamental properties including existence of optimal couplings, separability, completeness (iff Z is complete), path-connectedness, and geodesicity when Z is geodesic. They show that many known GW variants (e.g., Wasserstein, GW, ultrametric GW, fused GW, spectral GW, dynamic metric-space GW) arise as special cases under appropriate choices of Z, providing a unified theory and enabling new analyses of complex data types (attributed graphs, shape graphs, probabilistic metric spaces, etc.). The work also develops lower bounds and practical approximations (e.g., via R^n embeddings) and outlines a numerical algorithm, making the theory actionable for applications in machine learning and data analysis. Overall, the Z-GW framework offers a scalable, theoretically grounded path to comparing highly structured objects beyond traditional metric spaces, with broad implications for theory, computation, and applications in data science.

Abstract

The Gromov-Wasserstein (GW) distance is a powerful tool for comparing metric measure spaces which has found broad applications in data science and machine learning. Driven by the need to analyze datasets whose objects have increasingly complex structure (such as node and edge-attributed graphs), several variants of GW distance have been introduced in the recent literature. With a view toward establishing a general framework for the theory of GW-like distances, this paper considers a vast generalization of the notion of a metric measure space: for an arbitrary metric space $Z$, we define a $Z$-network to be a measure space endowed with a kernel valued in $Z$. We introduce a method for comparing $Z$-networks by defining a generalization of GW distance, which we refer to as $Z$-Gromov-Wasserstein ($Z$-GW) distance. This construction subsumes many previously known metrics and offers a unified approach to understanding their shared properties. This paper demonstrates that the $Z$-GW distance defines a metric on the space of $Z$-networks which retains desirable properties of $Z$, such as separability, completeness, and geodesicity. Many of these properties were unknown for existing variants of GW distance that fall under our framework. Our focus is on foundational theory, but our results also include computable lower bounds and approximations of the distance which will be useful for practical applications.

The Z-Gromov-Wasserstein Distance

TL;DR

This work introduces the Z-Gromov-Wasserstein (Z-GW) distance, a unifying framework that extends the classical Gromov-Wasserstein distance to measure networks whose kernel takes values in a fixed metric space Z. By defining Z-networks as measure spaces equipped with a Z-valued kernel and the GW_p^Z objective, the authors establish that GW_p^Z is a metric on Z-networks up to weak isomorphism when Z is separable, and they prove fundamental properties including existence of optimal couplings, separability, completeness (iff Z is complete), path-connectedness, and geodesicity when Z is geodesic. They show that many known GW variants (e.g., Wasserstein, GW, ultrametric GW, fused GW, spectral GW, dynamic metric-space GW) arise as special cases under appropriate choices of Z, providing a unified theory and enabling new analyses of complex data types (attributed graphs, shape graphs, probabilistic metric spaces, etc.). The work also develops lower bounds and practical approximations (e.g., via R^n embeddings) and outlines a numerical algorithm, making the theory actionable for applications in machine learning and data analysis. Overall, the Z-GW framework offers a scalable, theoretically grounded path to comparing highly structured objects beyond traditional metric spaces, with broad implications for theory, computation, and applications in data science.

Abstract

The Gromov-Wasserstein (GW) distance is a powerful tool for comparing metric measure spaces which has found broad applications in data science and machine learning. Driven by the need to analyze datasets whose objects have increasingly complex structure (such as node and edge-attributed graphs), several variants of GW distance have been introduced in the recent literature. With a view toward establishing a general framework for the theory of GW-like distances, this paper considers a vast generalization of the notion of a metric measure space: for an arbitrary metric space , we define a -network to be a measure space endowed with a kernel valued in . We introduce a method for comparing -networks by defining a generalization of GW distance, which we refer to as -Gromov-Wasserstein (-GW) distance. This construction subsumes many previously known metrics and offers a unified approach to understanding their shared properties. This paper demonstrates that the -GW distance defines a metric on the space of -networks which retains desirable properties of , such as separability, completeness, and geodesicity. Many of these properties were unknown for existing variants of GW distance that fall under our framework. Our focus is on foundational theory, but our results also include computable lower bounds and approximations of the distance which will be useful for practical applications.
Paper Structure (44 sections, 29 theorems, 143 equations, 2 figures, 1 table)

This paper contains 44 sections, 29 theorems, 143 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main Contributions and Outline
  3. Related Work
  4. Z-Gromov-Wasserstein Distances
  5. Background
  6. Basic Terminology and Notation
  7. Optimal Transport Distances
  8. Metric Space-Valued Lp-Spaces
  9. Z-Valued Measure Networks
  10. Main Definitions
  11. Examples of Z-Networks and Z-Gromov-Wasserstein Distances
  12. Examples of Existing Z-Gromov-Wasserstein Distances in the Literature
  13. Wasserstein Distance and Gromov-Wasserstein Distances
  14. Attributed Graphs
  15. Diffusions and Stochastic Processes
  16. ...and 29 more sections

Key Result

Proposition 8

Let $X$ be a measure space equipped with a measure $\mu_X$, and suppose $(Y,d_Y)$ is a separable metric space. Then:

Figures (2)

  • Figure 1: Schematic illustration of types of networks. (a) A graph with edge weights and node weights (each visualized by size variations). This structure is encoded as a measure network, and two such structures can be compared through the Gromov-Wasserstein distance memoli2007chowdhury2019gromov. (b) A weighted graph with additional node features, consisting of an assignment of a vector in $\mathbb{R}^n$ to each node (visualized as a column vector). These objects can be compared via the Fused Gromov-Wasserstein distance titouan2019optimalvayer2020fused. (c) Additionally, a graph can be endowed with edge features, assigning a point in some fixed metric space $Z$ to each edge (here, we visualize a 1-dimensional probability distribution attached to each edge). These complex objects are modeled as $Z$-networks, in the language of this paper, and two such objects can be compared through our proposed framework. By choosing an appropriate target space $Z$, one recovers many notions of distance between structured objects that have appeared previously in the literature---see \ref{['tab:metrics']}.
  • Figure 2: Top: Simple example of a shape graph in $\mathbb{R}^2$. Bottom: Representation as an attributed network. The intersection points $x$ and $x'$ (circles) are nodes in $X$, $\omega_X(x,x')$ (dashed) is an element of the space of curves $Z$.

Theorems & Definitions (51)

  • Definition 1: Coupling
  • Definition 2: Wasserstein Distance
  • Definition 3: Measure Network
  • Definition 5: Gromov-Wasserstein $p$-Distance
  • Definition 6: Metric Space-Valued $L^p$ Spaces
  • Remark 7
  • Proposition 8
  • Remark 9
  • Definition 10: $Z$-Network
  • Definition 11: Gromov-Wasserstein Distance for $(Z,p)$-Networks
  • ...and 41 more