# Wasserstein Auto-Encoders of Merge Trees (and Persistence Diagrams)

arXiv: https://arxiv.org/abs/2307.02509 · HTML: https://mlsanity.com/paper/2307.02509/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 39 of its 105 references that are in the map.
- First posted on arXiv 5 Jul 2023.

## Builds on (most cited first, 30 of 39)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [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)
- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [Sliced Wasserstein Kernel for Persistence Diagrams](https://mlsanity.com/paper/1706.03358/) (2017)
- [A Stable Multi-Scale Kernel for Topological Machine Learning](https://mlsanity.com/paper/1412.6821/) (2014)
- [Topology-Aware Segmentation Using Discrete Morse Theory](https://mlsanity.com/paper/2103.09992/) (2021)
- [Geometry Score: A Method For Comparing Generative Adversarial Networks](https://mlsanity.com/paper/1802.02664/) (2018)
- [Topologically faithful image segmentation via induced matching of persistence barcodes](https://mlsanity.com/paper/2211.15272/) (2022)
- [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)
- [Exposition and Interpretation of the Topology of Neural Networks](https://mlsanity.com/paper/1810.03234/) (2018)
- [Evaluating the Disentanglement of Deep Generative Models through Manifold Topology](https://mlsanity.com/paper/2006.03680/) (2020)
- [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)
- [Scalar Field Comparison with Topological Descriptors: Properties and Applications for Scientific Visualization](https://mlsanity.com/paper/2106.00157/) (2021)
- [Ranking Viscous Finger Simulations to an Acquired Ground Truth with Topology-aware Matchings](https://mlsanity.com/paper/1908.07841/) (2019)
- [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)
- [Discrete Morse Sandwich: Fast Computation of Persistence Diagrams for Scalar Data -- An Algorithm and A Benchmark](https://mlsanity.com/paper/2206.13932/) (2022)
- [Topological Analysis of Ensembles of Hydrodynamic Turbulent Flows -- An Experimental Study](https://mlsanity.com/paper/2207.14080/) (2022)
- [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)
- [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)

## Built on by (most cited first, 3 of 3)

- [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)
- [Topology Aware Neural Interpolation of Scalar Fields](https://mlsanity.com/paper/2508.17995/) (2025)

## Abstract (arXiv)

This paper presents a computational framework for the Wasserstein auto-encoding of merge trees (MT-WAE), a novel extension of the classical auto-encoder neural network architecture to the Wasserstein metric space of merge trees. In contrast to traditional auto-encoders which operate on vectorized data, our formulation explicitly manipulates merge trees on their associated metric space at each layer of the network, resulting in superior accuracy and interpretability. Our novel neural network approach can be interpreted as a non-linear generalization of previous linear attempts [79] at merge tree encoding. It also trivially extends to persistence diagrams. Extensive experiments on public ensembles demonstrate the efficiency of our algorithms, with MT-WAE computations in the orders of minutes on average. We show the utility of our contributions in two applications adapted from previous work on merge tree encoding [79]. First, we apply MT-WAE to merge tree compression, by concisely representing them with their coordinates in the final layer of our auto-encoder. Second, we document an application to dimensionality reduction, by exploiting the latent space of our auto-encoder, for the visual analysis of ensemble data. We illustrate the versatility of our framework by introducing two penalty terms, to help preserve in the latent space both the Wasserstein distances between merge trees, as well as their clusters. In both applications, quantitative experiments assess the relevance of our framework. Finally, we provide a C++ implementation that can be used for reproducibility.