# Stability of Entropic Wasserstein Barycenters and application to random geometric graphs

arXiv: https://arxiv.org/abs/2210.10535 · HTML: https://mlsanity.com/paper/2210.10535/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 33 references that are in the map.
- First posted on arXiv 19 Oct 2022.

## Builds on (most cited first, 15 of 15)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [Optimal Transport for structured data with application on graphs](https://mlsanity.com/paper/1805.09114/) (2018)
- [Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem](https://mlsanity.com/paper/1905.11882/) (2019)
- [Multilevel Clustering via Wasserstein Means](https://mlsanity.com/paper/1706.03883/) (2017)
- [Scalable Bayes via Barycenter in Wasserstein Space](https://mlsanity.com/paper/1508.05880/) (2015)
- [Wasserstein barycenters can be computed in polynomial time in fixed dimension](https://mlsanity.com/paper/2006.08012/) (2020)
- [Quantitative Stability of Regularized Optimal Transport and Convergence of Sinkhorn's Algorithm](https://mlsanity.com/paper/2110.06798/) (2021)
- [node2coords: Graph Representation Learning with Wasserstein Barycenters](https://mlsanity.com/paper/2007.16056/) (2020)
- [Entropic Optimal Transport in Random Graphs](https://mlsanity.com/paper/2201.03949/) (2022)
- [Quantitative Stability of Barycenters in the Wasserstein Space](https://mlsanity.com/paper/2209.10217/) (2022)
- [Graph Signal Representation with Wasserstein Barycenters](https://mlsanity.com/paper/1812.05517/) (2018)

## Built on by (most cited first, 1 of 1)

- [Doubly Regularized Entropic Wasserstein Barycenters](https://mlsanity.com/paper/2303.11844/) (2023)

## Abstract (arXiv)

As interest in graph data has grown in recent years, the computation of various geometric tools has become essential. In some area such as mesh processing, they often rely on the computation of geodesics and shortest paths in discretized manifolds. A recent example of such a tool is the computation of Wasserstein barycenters (WB), a very general notion of barycenters derived from the theory of Optimal Transport, and their entropic-regularized variant. In this paper, we examine how WBs on discretized meshes relate to the geometry of the underlying manifold. We first provide a generic stability result with respect to the input cost matrices. We then apply this result to random geometric graphs on manifolds, whose shortest paths converge to geodesics, hence proving the consistency of WBs computed on discretized shapes.