# Principal Geodesic Analysis of Merge Trees (and Persistence Diagrams)

arXiv: https://arxiv.org/abs/2207.10960 · HTML: https://mlsanity.com/paper/2207.10960/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 125 references that are in the map.
- First posted on arXiv 22 Jul 2022.

## Builds on (most cited first, 23 of 23)

- [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)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [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)
- [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)
- [Wasserstein Distances, Geodesics and Barycenters of Merge Trees](https://mlsanity.com/paper/2107.07789/) (2021)
- [Principal Geodesic Analysis for Probability Measures under the Optimal Transport Metric](https://mlsanity.com/paper/1506.07944/) (2015)
- [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)
- [A Structural Average of Labeled Merge Trees for Uncertainty Visualization](https://mlsanity.com/paper/1908.00113/) (2019)
- [Edit Distance between Merge Trees](https://mlsanity.com/paper/2207.08511/) (2022)
- [A Riemannian Framework for Statistical Analysis of Topological Persistence Diagrams](https://mlsanity.com/paper/1605.08912/) (2016)
- [Measuring Distance between Reeb Graphs](https://mlsanity.com/paper/1307.2839/) (2013)
- [Intrinsic Interleaving Distance for Merge Trees](https://mlsanity.com/paper/1908.00063/) (2019)
- [Utilising urgent computing to tackle the spread of mosquito-borne diseases](https://mlsanity.com/paper/2111.05654/) (2021)
- [Reeb Graph Metrics from the Ground Up](https://mlsanity.com/paper/2110.05631/) (2021)
- [Sketching Merge Trees for Scientific Data Visualization](https://mlsanity.com/paper/2101.03196/) (2021)

## Built on by (most cited first, 8 of 8)

- [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)
- [Rapid and Precise Topological Comparison with Merge Tree Neural Networks](https://mlsanity.com/paper/2404.05879/) (2024)
- [Robust Barycenters of Persistence Diagrams](https://mlsanity.com/paper/2509.14904/) (2025)
- [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)
- [Sierpi'nski--Knopp Wasserstein Distance for Persistence Diagrams and Applications to 2-Wasserstein Approximation](https://mlsanity.com/paper/2609.01528/) (2026)

## Abstract (arXiv)

This paper presents a computational framework for the Principal Geodesic Analysis of merge trees (MT-PGA), a novel adaptation of the celebrated Principal Component Analysis (PCA) framework [87] to the Wasserstein metric space of merge trees [92]. We formulate MT-PGA computation as a constrained optimization problem, aiming at adjusting a basis of orthogonal geodesic axes, while minimizing a fitting energy. We introduce an efficient, iterative algorithm which exploits shared-memory parallelism, as well as an analytic expression of the fitting energy gradient, to ensure fast iterations. Our approach also trivially extends to extremum persistence diagrams. Extensive experiments on public ensembles demonstrate the efficiency of our approach - with MT-PGA computations in the orders of minutes for the largest examples. We show the utility of our contributions by extending to merge trees two typical PCA applications. First, we apply MT-PGA to data reduction and reliably compress merge trees by concisely representing them by their first coordinates in the MT-PGA basis. Second, we present a dimensionality reduction framework exploiting the first two directions of the MT-PGA basis to generate two-dimensional layouts of the ensemble. We augment these layouts with persistence correlation views, enabling global and local visual inspections of the feature variability in the ensemble. In both applications, quantitative experiments assess the relevance of our framework. Finally, we provide a C++ implementation that can be used to reproduce our results.