# Wasserstein Distances, Geodesics and Barycenters of Merge Trees

arXiv: https://arxiv.org/abs/2107.07789 · HTML: https://mlsanity.com/paper/2107.07789/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 115 references that are in the map.
- First posted on arXiv 16 Jul 2021.

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

- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [Geometry Helps to Compare Persistence Diagrams](https://mlsanity.com/paper/1606.03357/) (2016)
- [The Topology ToolKit](https://mlsanity.com/paper/1805.09110/) (2018)
- [Large Scale computation of Means and Clusters for Persistence Diagrams using Optimal Transport](https://mlsanity.com/paper/1805.08331/) (2018)
- [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)
- [Scalar Field Comparison with Topological Descriptors: Properties and Applications for Scientific Visualization](https://mlsanity.com/paper/2106.00157/) (2021)
- [Distributed computation of persistent homology](https://mlsanity.com/paper/1310.0710/) (2013)
- [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)
- [Lifted Wasserstein Matcher for Fast and Robust Topology Tracking](https://mlsanity.com/paper/1808.05870/) (2018)
- [A Structural Average of Labeled Merge Trees for Uncertainty Visualization](https://mlsanity.com/paper/1908.00113/) (2019)
- [Measuring Distance between Reeb Graphs](https://mlsanity.com/paper/1307.2839/) (2013)

## Built on by (most cited first, 16 of 16)

- [Topological Analysis of Ensembles of Hydrodynamic Turbulent Flows -- An Experimental Study](https://mlsanity.com/paper/2207.14080/) (2022)
- [Principal Geodesic Analysis of Merge Trees (and Persistence Diagrams)](https://mlsanity.com/paper/2207.10960/) (2022)
- [A User's Guide to Sampling Strategies for Sliced Optimal Transport](https://mlsanity.com/paper/2502.02275/) (2025)
- [Wasserstein Dictionaries of Persistence Diagrams](https://mlsanity.com/paper/2304.14852/) (2023)
- [Wasserstein Auto-Encoders of Merge Trees (and Persistence Diagrams)](https://mlsanity.com/paper/2307.02509/) (2023)
- [TTK is Getting MPI-Ready](https://mlsanity.com/paper/2310.08339/) (2023)
- [Robust Barycenters of Persistence Diagrams](https://mlsanity.com/paper/2509.14904/) (2025)
- [The Universal ℓp-Metric on Merge Trees](https://mlsanity.com/paper/2112.12165/) (2021)
- [Multi-scale Cycle Tracking in Dynamic Planar Graphs](https://mlsanity.com/paper/2409.06476/) (2024)
- [Topology Aware Neural Interpolation of Scalar Fields](https://mlsanity.com/paper/2508.17995/) (2025)
- [Continuous Edit Distance, Geodesics and Barycenters of Time-varying Persistence Diagrams](https://mlsanity.com/paper/2512.12939/) (2025)
- [Cycles Communities from the Perspective of Dendrograms and Gradient Sampling](https://mlsanity.com/paper/2512.12974/) (2025)
- [MS-COOT: Comparing Morse-Smale Complexes with Co-Optimal Transport](https://mlsanity.com/paper/2606.08258/) (2026)
- [DebrisTracer: Reliable Tracking in Hypervelocity Impact Fast Imaging](https://mlsanity.com/paper/2607.15986/) (2026)
- [Sierpi'nski--Knopp Wasserstein Distance for Persistence Diagrams and Applications to 2-Wasserstein Approximation](https://mlsanity.com/paper/2609.01528/) (2026)
- [Direct Topology Tracking in Continuous Implicit Models](https://mlsanity.com/paper/2609.12157/) (2026)

## Abstract (arXiv)

This paper presents a unified computational framework for the estimation of distances, geodesics and barycenters of merge trees. We extend recent work on the edit distance [106] and introduce a new metric, called the Wasserstein distance between merge trees, which is purposely designed to enable efficient computations of geodesics and barycenters. Specifically, our new distance is strictly equivalent to the L2-Wasserstein distance between extremum persistence diagrams, but it is restricted to a smaller solution space, namely, the space of rooted partial isomorphisms between branch decomposition trees. This enables a simple extension of existing optimization frameworks [112] for geodesics and barycenters from persistence diagrams to merge trees. We introduce a task-based algorithm which can be generically applied to distance, geodesic, barycenter or cluster computation. The task-based nature of our approach enables further accelerations with shared-memory parallelism. Extensive experiments on public ensembles and SciVis contest benchmarks demonstrate the efficiency of our approach -- with barycenter computations in the orders of minutes for the largest examples -- as well as its qualitative ability to generate representative barycenter merge trees, visually summarizing the features of interest found in the ensemble. We show the utility of our contributions with dedicated visualization applications: feature tracking, temporal reduction and ensemble clustering. We provide a lightweight C++ implementation that can be used to reproduce our results.