# The Z-Gromov-Wasserstein Distance arXiv: https://arxiv.org/abs/2408.08233 ยท HTML: https://mlsanity.com/paper/2408.08233/ ## Key facts - Cited by 12 papers in the map (data through 14 Oct 2026). - Builds on 26 of its 91 references that are in the map. - First posted on arXiv 15 Aug 2024. ## Builds on (most cited first, 26 of 26) - [Optimal Transport for structured data with application on graphs](https://mlsanity.com/paper/1805.09114/) (2018) - [Gromov-Wasserstein Alignment of Word Embedding Spaces](https://mlsanity.com/paper/1809.00013/) (2018) - [Scalable Gromov-Wasserstein Learning for Graph Partitioning and Matching](https://mlsanity.com/paper/1905.07645/) (2019) - [Fused Gromov-Wasserstein distance for structured objects: theoretical foundations and mathematical properties](https://mlsanity.com/paper/1811.02834/) (2018) - [Linear-Time Gromov Wasserstein Distances using Low Rank Couplings and Costs](https://mlsanity.com/paper/2106.01128/) (2021) - [CO-Optimal Transport](https://mlsanity.com/paper/2002.03731/) (2020) - [Generalized Spectral Clustering via Gromov-Wasserstein Learning](https://mlsanity.com/paper/2006.04163/) (2020) - [Gromov-Wasserstein Factorization Models for Graph Clustering](https://mlsanity.com/paper/1911.08530/) (2019) - [Clustering attributed graphs: models, measures and methods](https://mlsanity.com/paper/1501.01676/) (2015) - [Quantized Gromov-Wasserstein](https://mlsanity.com/paper/2104.02013/) (2021) - [On the existence of Monge maps for the Gromov-Wasserstein problem](https://mlsanity.com/paper/2210.11945/) (2022) - [Generalized Dimension Reduction Using Semi-Relaxed Gromov-Wasserstein Distance](https://mlsanity.com/paper/2405.15959/) (2024) - [Efficient Approximation of Gromov-Wasserstein Distance Using Importance Sparsification](https://mlsanity.com/paper/2205.13573/) (2022) - [Hypergraph Co-Optimal Transport: Metric and Categorical Properties](https://mlsanity.com/paper/2112.03904/) (2021) - [Gromov-Wasserstein Averaging in a Riemannian Framework](https://mlsanity.com/paper/1910.04308/) (2019) - [The Gromov-Wasserstein distance between spheres](https://mlsanity.com/paper/2306.10586/) (2023) - [Spatio-temporal Persistent Homology for Dynamic Metric Spaces](https://mlsanity.com/paper/1812.00949/) (2018) - [Statistical Shape Analysis of Brain Arterial Networks (BAN)](https://mlsanity.com/paper/2007.04793/) (2020) - [A new variational model for shape graph registration with partial matching constraints](https://mlsanity.com/paper/2105.00678/) (2021) - [On a Generalization of Wasserstein Distance and the Beckmann Problem to Connection Graphs](https://mlsanity.com/paper/2312.10295/) (2023) - [Partial Optimal Transport with Applications on Positive-Unlabeled Learning](https://mlsanity.com/paper/2002.08276/) (2020) - [Orthogonal Matrix Retrieval in Cryo-Electron Microscopy](https://mlsanity.com/paper/1412.0494/) (2014) - [Shape-Graph Matching Network (SGM-net): Registration for Statistical Shape Analysis](https://mlsanity.com/paper/2308.06869/) (2023) - [Scalable unsupervised alignment of general metric and non-metric structures](https://mlsanity.com/paper/2406.13507/) (2024) - [Statistical Shape Analysis of Shape Graphs with Applications to Retinal Blood-Vessel Networks](https://mlsanity.com/paper/2211.15514/) (2022) - [Exploiting Edge Features in Graphs with Fused Network Gromov-Wasserstein Distance](https://mlsanity.com/paper/2309.16604/) (2023) ## Built on by (most cited first, 12 of 12) - [Metric properties of partial and robust Gromov-Wasserstein distances](https://mlsanity.com/paper/2411.02198/) (2024) - [Conformal Graph Prediction with Z-Gromov-Wasserstein Distances](https://mlsanity.com/paper/2603.02460/) (2026) - [Metrics for Parametric Families of Networks](https://mlsanity.com/paper/2509.22549/) (2025) - [Neural Correlates of Language Models Are Specific to Human Language](https://mlsanity.com/paper/2510.03156/) (2025) - [SPRig: Self-Supervised Pose-Invariant Rigging from Mesh Sequences](https://mlsanity.com/paper/2602.12740/) (2026) - [On Robust Cross Domain Alignment](https://mlsanity.com/paper/2412.15861/) (2024) - [Illuminant and light direction estimation using Wasserstein distance method](https://mlsanity.com/paper/2503.05802/) (2025) - [The Joint Gromov Wasserstein Objective for Multiple Object Matching](https://mlsanity.com/paper/2511.16868/) (2025) - [Network Learning with Semi-relaxed Gromov-Wasserstein](https://mlsanity.com/paper/2606.02223/) (2026) - [k-Nearest Neighbors in Gromov--Wasserstein Space](https://mlsanity.com/paper/2606.10295/) (2026) - [Gromov-Monge Flow Matching for Equivariant Graph Generation](https://mlsanity.com/paper/2608.26961/) (2026) - [Unsupervised Multi-Scale Gromov-Wasserstein Hypergraph Alignment](https://mlsanity.com/paper/2608.29635/) (2026) ## Abstract (arXiv) 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.