# Conic Formulations of Transport Metrics for Unbalanced Measure Networks and Hypernetworks

arXiv: https://arxiv.org/abs/2508.10888 · HTML: https://mlsanity.com/paper/2508.10888/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 41 references that are in the map.
- First posted on arXiv 14 Aug 2025.

## Builds on (most cited first, 13 of 13)

- [Unbalanced minibatch Optimal Transport; applications to Domain Adaptation](https://mlsanity.com/paper/2103.03606/) (2021)
- [The Unbalanced Gromov Wasserstein Distance: Conic Formulation and Relaxation](https://mlsanity.com/paper/2009.04266/) (2020)
- [CO-Optimal Transport](https://mlsanity.com/paper/2002.03731/) (2020)
- [Your diffusion model secretly knows the dimension of the data manifold](https://mlsanity.com/paper/2212.12611/) (2022)
- [Rates of Convergence for Laplacian Semi-Supervised Learning with Low Labeling Rates](https://mlsanity.com/paper/2006.02765/) (2020)
- [Hypergraph Co-Optimal Transport: Metric and Categorical Properties](https://mlsanity.com/paper/2112.03904/) (2021)
- [A New Robust Partial p-Wasserstein-Based Metric for Comparing Distributions](https://mlsanity.com/paper/2405.03664/) (2024)
- [Unbalanced CO-Optimal Transport](https://mlsanity.com/paper/2205.14923/) (2022)
- [Outlier-Robust Gromov-Wasserstein for Graph Data](https://mlsanity.com/paper/2302.04610/) (2023)
- [Partial Gromov-Wasserstein Metric](https://mlsanity.com/paper/2402.03664/) (2024)
- [Barycenters for the Hellinger--Kantorovich distance over ℝd](https://mlsanity.com/paper/1910.14572/) (2019)
- [Metric properties of partial and robust Gromov-Wasserstein distances](https://mlsanity.com/paper/2411.02198/) (2024)
- [Linear Partial Gromov-Wasserstein Embedding](https://mlsanity.com/paper/2410.16669/) (2024)

## Built on by (most cited first, 2 of 2)

- [Learning to Orchestrate Agents under Uncertainty](https://mlsanity.com/paper/2605.27073/) (2026)
- [A Unified Geometric Framework for Developmental Analysis of Spatial Transcriptomic Data](https://mlsanity.com/paper/2608.15306/) (2026)

## Abstract (arXiv)

The Gromov-Wasserstein (GW) variant of optimal transport, designed to compare probability densities defined over distinct metric spaces, has emerged as an important tool for the analysis of data with complex structure, such as ensembles of point clouds or networks. To overcome certain limitations, such as the restriction to comparisons of measures of equal mass and sensitivity to outliers, several unbalanced or partial transport relaxations of the GW distance have been introduced in the recent literature. This paper is concerned with the Conic Gromov-Wasserstein (CGW) distance introduced by S'ejourn'e, Vialard, and Peyr'e. We provide a novel formulation in terms of semi-couplings, and extend the framework beyond the metric measure space setting, to compare more general network and hypernetwork structures. With this new formulation, we establish several fundamental properties of the CGW metric, including its scaling behavior under dilation, variational convergence in the limit of volume growth constraints, and comparison bounds with established optimal transport metrics. We further derive quantitative bounds that characterize the robustness of the CGW metric to perturbations in the underlying measures. The hypernetwork formulation of CGW admits a simple and provably convergent block coordinate ascent algorithm for its estimation, and we demonstrate the computational tractability and scalability of our approach through experiments on synthetic and real-world high-dimensional and structured datasets.