# Fused Gromov-Wasserstein distance for structured objects: theoretical foundations and mathematical properties

arXiv: https://arxiv.org/abs/1811.02834 · HTML: https://mlsanity.com/paper/1811.02834/

## Key facts

- Cited by 93 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Nov 2018.

## Built on by (most cited first, 30 of 93)

- [Linear-Time Gromov Wasserstein Distances using Low Rank Couplings and Costs](https://mlsanity.com/paper/2106.01128/) (2021)
- [Unbalanced Optimal Transport, from Theory to Numerics](https://mlsanity.com/paper/2211.08775/) (2022)
- [Temporally Consistent Unbalanced Optimal Transport for Unsupervised Action Segmentation](https://mlsanity.com/paper/2404.01518/) (2024)
- [Online Graph Dictionary Learning](https://mlsanity.com/paper/2102.06555/) (2021)
- [Aligning individual brains with Fused Unbalanced Gromov-Wasserstein](https://mlsanity.com/paper/2206.09398/) (2022)
- [Learning to Predict Graphs with Fused Gromov-Wasserstein Barycenters](https://mlsanity.com/paper/2202.03813/) (2022)
- [Semi-relaxed Gromov-Wasserstein divergence with applications on graphs](https://mlsanity.com/paper/2110.02753/) (2021)
- [Template based Graph Neural Network with Optimal Transport Distances](https://mlsanity.com/paper/2205.15733/) (2022)
- [CSOT: Curriculum and Structure-Aware Optimal Transport for Learning with Noisy Labels](https://mlsanity.com/paper/2312.06221/) (2023)
- [Pave Your Own Path: Graph Gradual Domain Adaptation on Fused Gromov-Wasserstein Geodesics](https://mlsanity.com/paper/2505.12709/) (2025)
- [Robust Attributed Graph Alignment via Joint Structure Learning and Optimal Transport](https://mlsanity.com/paper/2301.12721/) (2023)
- [Weisfeiler-Lehman meets Gromov-Wasserstein](https://mlsanity.com/paper/2202.02495/) (2022)
- [The Z-Gromov-Wasserstein Distance](https://mlsanity.com/paper/2408.08233/) (2024)
- [InfiGFusion: Graph-on-Logits Distillation via Efficient Gromov-Wasserstein for Model Fusion](https://mlsanity.com/paper/2505.13893/) (2025)
- [Efficient Approximation of Gromov-Wasserstein Distance Using Importance Sparsification](https://mlsanity.com/paper/2205.13573/) (2022)
- [A Convergent Single-Loop Algorithm for Relaxation of Gromov-Wasserstein in Graph Data](https://mlsanity.com/paper/2303.06595/) (2023)
- [Graph Classification via Reference Distribution Learning: Theory and Practice](https://mlsanity.com/paper/2408.11370/) (2024)
- [Joint Self-Supervised Video Alignment and Action Segmentation](https://mlsanity.com/paper/2503.16832/) (2025)
- [InfoOT: Information Maximizing Optimal Transport](https://mlsanity.com/paper/2210.03164/) (2022)
- [A Novel Sliced Fused Gromov-Wasserstein Distance](https://mlsanity.com/paper/2508.02364/) (2025)
- [Structured Transforms Across Spaces with Cost-Regularized Optimal Transport](https://mlsanity.com/paper/2311.05788/) (2023)
- [Efficient Graph Field Integrators Meet Point Clouds](https://mlsanity.com/paper/2302.00942/) (2023)
- [Unbalanced Low-rank Optimal Transport Solvers](https://mlsanity.com/paper/2305.19727/) (2023)
- [Any2Graph: Deep End-To-End Supervised Graph Prediction With An Optimal Transport Loss](https://mlsanity.com/paper/2402.12269/) (2024)
- [Proximity Matters: Local Proximity Enhanced Balancing for Treatment Effect Estimation](https://mlsanity.com/paper/2407.01111/) (2024)
- [Graph Data Selection for Domain Adaptation: A Model-Free Approach](https://mlsanity.com/paper/2505.17293/) (2025)
- [Modeling Microenvironment Trajectories on Spatial Transcriptomics with NicheFlow](https://mlsanity.com/paper/2511.00977/) (2025)
- [Computing Approximate Graph Edit Distance via Optimal Transport](https://mlsanity.com/paper/2412.18857/) (2024)
- [Learning Graphon Autoencoders for Generative Graph Modeling](https://mlsanity.com/paper/2105.14244/) (2021)
- [Wasserstein Graph Distance Based on L1-Approximated Tree Edit Distance between Weisfeiler-Lehman Subtrees](https://mlsanity.com/paper/2207.04216/) (2022)

## Abstract (arXiv)

Optimal transport theory has recently found many applications in machine learning thanks to its capacity for comparing various machine learning objects considered as distributions. The Kantorovitch formulation, leading to the Wasserstein distance, focuses on the features of the elements of the objects but treat them independently, whereas the Gromov-Wasserstein distance focuses only on the relations between the elements, depicting the structure of the object, yet discarding its features. In this paper we propose to extend these distances in order to encode simultaneously both the feature and structure informations, resulting in the Fused Gromov-Wasserstein distance. We develop the mathematical framework for this novel distance, prove its metric and interpolation properties and provide a concentration result for the convergence of finite samples. We also illustrate and interpret its use in various contexts where structured objects are involved.