# Multi-Marginal Gromov-Wasserstein Transport and Barycenters

arXiv: https://arxiv.org/abs/2205.06725 · HTML: https://mlsanity.com/paper/2205.06725/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 May 2022.

## Built on by (most cited first, 9 of 9)

- [Hierarchical Multi-Marginal Optimal Transport for Network Alignment](https://mlsanity.com/paper/2310.04470/) (2023)
- [Contrasting Multiple Representations with the Multi-Marginal Matching Gap](https://mlsanity.com/paper/2405.19532/) (2024)
- [Joint Metric Space Embedding by Unbalanced OT with Gromov-Wasserstein Marginal Penalization](https://mlsanity.com/paper/2502.07510/) (2025)
- [Neural Estimation for Scaling Entropic Multimarginal Optimal Transport](https://mlsanity.com/paper/2506.00573/) (2025)
- [Optimal transport for automatic alignment of untargeted metabolomic data](https://mlsanity.com/paper/2306.03218/) (2023)
- [Binding Multiple Modalities via Multimodal Wasserstein Barycenter](https://mlsanity.com/paper/2609.33800/) (2026)
- [On sparsity, extremal structure, and monotonicity properties of Wasserstein and Gromov-Wasserstein optimal transport plans](https://mlsanity.com/paper/2602.16265/) (2026)
- [Gromov-Wasserstein Quantization and Clustering: Structure, Rates, and Algorithms](https://mlsanity.com/paper/2608.11016/) (2026)
- [A Unified Geometric Framework for Developmental Analysis of Spatial Transcriptomic Data](https://mlsanity.com/paper/2608.15306/) (2026)

## Abstract (arXiv)

Gromov-Wasserstein (GW) distances are combinations of Gromov-Hausdorff and Wasserstein distances that allow the comparison of two different metric measure spaces (mm-spaces). Due to their invariance under measure- and distance-preserving transformations, they are well suited for many applications in graph and shape analysis. In this paper, we introduce the concept of multi-marginal GW transport between a set of mm-spaces as well as its regularized and unbalanced versions. As a special case, we discuss multi-marginal fused variants, which combine the structure information of an mm-space with label information from an additional label space. To tackle the new formulations numerically, we consider the bi-convex relaxation of the multi-marginal GW problem, which is tight in the balanced case if the cost function is conditionally negative definite. The relaxed model can be solved by an alternating minimization, where each step can be performed by a multi-marginal Sinkhorn scheme. We show relations of our multi-marginal GW problem to (unbalanced, fused) GW barycenters and present various numerical results, which indicate the potential of the concept.