# Robust Barycenters of Persistence Diagrams

arXiv: https://arxiv.org/abs/2509.14904 · HTML: https://mlsanity.com/paper/2509.14904/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 88 references that are in the map.
- First posted on arXiv 18 Sep 2025.

## Builds on (most cited first, 16 of 16)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Sketching as a Tool for Numerical Linear Algebra](https://mlsanity.com/paper/1411.4357/) (2014)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [Geometry Helps to Compare Persistence Diagrams](https://mlsanity.com/paper/1606.03357/) (2016)
- [Large Scale computation of Means and Clusters for Persistence Diagrams using Optimal Transport](https://mlsanity.com/paper/1805.08331/) (2018)
- [Wasserstein Distances, Geodesics and Barycenters of Merge Trees](https://mlsanity.com/paper/2107.07789/) (2021)
- [Task-based Augmented Contour Trees with Fibonacci Heaps](https://mlsanity.com/paper/1902.04805/) (2019)
- [Progressive Wasserstein Barycenters of Persistence Diagrams](https://mlsanity.com/paper/1907.04565/) (2019)
- [Ranking Viscous Finger Simulations to an Acquired Ground Truth with Topology-aware Matchings](https://mlsanity.com/paper/1908.07841/) (2019)
- [Persistence Atlas for Critical Point Variability in Ensembles](https://mlsanity.com/paper/1807.11212/) (2018)
- [Uncertainty Visualization of 2D Morse Complex Ensembles Using Statistical Summary Maps](https://mlsanity.com/paper/1912.06341/) (2019)
- [Topological data analysis of vortices in the magnetically-induced current density in LiH molecule](https://mlsanity.com/paper/2212.08690/) (2022)
- [Principal Geodesic Analysis of Merge Trees (and Persistence Diagrams)](https://mlsanity.com/paper/2207.10960/) (2022)
- [Wasserstein Dictionaries of Persistence Diagrams](https://mlsanity.com/paper/2304.14852/) (2023)
- [Wasserstein medians: robustness, PDE characterization and numerics](https://mlsanity.com/paper/2307.01765/) (2023)

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

- [Sierpi'nski--Knopp Wasserstein Distance for Persistence Diagrams and Applications to 2-Wasserstein Approximation](https://mlsanity.com/paper/2609.01528/) (2026)

## Abstract (arXiv)

This short paper presents a general approach for computing robust Wasserstein barycenters of persistence diagrams. The classical method consists in computing assignment arithmetic means after finding the optimal transport plans between the barycenter and the persistence diagrams. However, this procedure only works for the transportation cost related to the q-Wasserstein distance Wq when q=2. We adapt an alternative fixed-point method to compute a barycenter diagram for generic transportation costs (q > 1), in particular those robust to outliers, q ∈ (1,2). We show the utility of our work in two applications: (i) the clustering of persistence diagrams on their metric space and (ii) the dictionary encoding of persistence diagrams. In both scenarios, we demonstrate the added robustness to outliers provided by our generalized framework. Our Python implementation is available at this address: https://github.com/Keanu-Sisouk/RobustBarycenter .