# Gromov-Wasserstein Averaging in a Riemannian Framework

arXiv: https://arxiv.org/abs/1910.04308 · HTML: https://mlsanity.com/paper/1910.04308/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Oct 2019.

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

- [Unbalanced Optimal Transport, from Theory to Numerics](https://mlsanity.com/paper/2211.08775/) (2022)
- [Quantized Gromov-Wasserstein](https://mlsanity.com/paper/2104.02013/) (2021)
- [The Z-Gromov-Wasserstein Distance](https://mlsanity.com/paper/2408.08233/) (2024)
- [Sketching Merge Trees for Scientific Data Visualization](https://mlsanity.com/paper/2101.03196/) (2021)
- [Manifold Modeling in Quotient Space: Learning An Invariant Mapping with Decodability of Image Patches](https://mlsanity.com/paper/2203.05134/) (2022)
- [Metric properties of partial and robust Gromov-Wasserstein distances](https://mlsanity.com/paper/2411.02198/) (2024)
- [Fused Gromov-Wasserstein Variance Decomposition with Linear Optimal Transport](https://mlsanity.com/paper/2411.10204/) (2024)
- [Beyond Distance: Quantifying Point Cloud Dynamics with Persistent Homology and Dynamic Optimal Transport](https://mlsanity.com/paper/2603.15683/) (2026)
- [Gromov-Wasserstein Quantization and Clustering: Structure, Rates, and Algorithms](https://mlsanity.com/paper/2608.11016/) (2026)

## Abstract (arXiv)

We introduce a theoretical framework for performing statistical tasks---including, but not limited to, averaging and principal component analysis---on the space of (possibly asymmetric) matrices with arbitrary entries and sizes. This is carried out under the lens of the Gromov-Wasserstein (GW) distance, and our methods translate the Riemannian framework of GW distances developed by Sturm into practical, implementable tools for network data analysis. Our methods are illustrated on datasets of letter graphs, asymmetric stochastic blockmodel networks, and planar shapes viewed as metric spaces. On the theoretical front, we supplement the work of Sturm by producing additional results on the tangent structure of this "space of spaces", as well as on the gradient flow of the Fr'echet functional on this space.