The Z-Gromov-Wasserstein Distance

2024-08-15

arXiv.org

AI Summary

Existing similarity measures for complex structured data—such as attributed graphs and higher-order relational data—lack a unified, mathematically rigorous, and computationally tractable framework. Method: We propose the Z-Gromov–Wasserstein (Z-GW) distance, a generalization of the Gromov–Wasserstein (GW) distance to measure spaces endowed with Z-valued kernels (Z-networks), where Z is an arbitrary metric space. Contribution/Results: We establish the first general GW theory parameterized by Z, proving that Z-GW is a well-defined metric inheriting separability, completeness, and geodesicity from Z. We derive a tight, computationally feasible lower bound and design efficient relaxation-based optimization and approximation algorithms. Compared to existing GW variants, Z-GW offers greater theoretical unification and practical scalability, providing a mathematically sound yet computationally viable similarity measure for attributed graphs and other complex structured data.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Complex Data Similarity
• Z-Gromov-Wasserstein Distance
• Graph Data Analysis

Innovation

• Z-Gromov-Wasserstein
• Complex and Continuous Data
• Computational Method