# Rapid and Precise Topological Comparison with Merge Tree Neural Networks

arXiv: https://arxiv.org/abs/2404.05879 · HTML: https://mlsanity.com/paper/2404.05879/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 57 references that are in the map.
- First posted on arXiv 8 Apr 2024.

## Builds on (most cited first, 27 of 27)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Graph Matching Networks for Learning the Similarity of Graph Structured Objects](https://mlsanity.com/paper/1904.12787/) (2019)
- [From Local Structures to Size Generalization in Graph Neural Networks](https://mlsanity.com/paper/2010.08853/) (2020)
- [SimGNN: A Neural Network Approach to Fast Graph Similarity Computation](https://mlsanity.com/paper/1808.05689/) (2018)
- [Position: Topological Deep Learning is the New Frontier for Relational Learning](https://mlsanity.com/paper/2402.08871/) (2024)
- [Learning-based Efficient Graph Similarity Computation via Multi-Scale Convolutional Set Matching](https://mlsanity.com/paper/1809.04440/) (2018)
- [The Topology ToolKit](https://mlsanity.com/paper/1805.09110/) (2018)
- [Topological Deep Learning: A Review of an Emerging Paradigm](https://mlsanity.com/paper/2302.03836/) (2023)
- [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)
- [Principal Geodesic Analysis of Merge Trees (and Persistence Diagrams)](https://mlsanity.com/paper/2207.10960/) (2022)
- [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)
- [Computing the Gromov-Hausdorff Distance for Metric Trees](https://mlsanity.com/paper/1509.05751/) (2015)
- [Geometry-Aware Merge Tree Comparisons for Time-Varying Data with Interleaving Distances](https://mlsanity.com/paper/2107.14373/) (2021)
- [Branch Decomposition-Independent Edit Distances for Merge Trees](https://mlsanity.com/paper/2204.02919/) (2022)
- [Comparative Analysis of Merge Trees using Local Tree Edit Distance](https://mlsanity.com/paper/2111.04382/) (2021)
- [Wasserstein Auto-Encoders of Merge Trees (and Persistence Diagrams)](https://mlsanity.com/paper/2307.02509/) (2023)
- [The edit distance for Reeb graphs of surfaces](https://mlsanity.com/paper/1411.1544/) (2014)
- [The Reeb Graph Edit Distance is Universal](https://mlsanity.com/paper/1801.01866/) (2018)
- [A Domain-Oblivious Approach for Learning Concise Representations of Filtered Topological Spaces for Clustering](https://mlsanity.com/paper/2105.12208/) (2021)
- [Computing a Stable Distance on Merge Trees](https://mlsanity.com/paper/2210.08644/) (2022)
- [Taming Horizontal Instability in Merge Trees: On the Computation of a Comprehensive Deformation-based Edit Distance](https://mlsanity.com/paper/2308.08484/) (2023)

## Built on by (most cited first, 2 of 2)

- [RTD-Lite: Scalable Topological Analysis for Comparing Weighted Graphs in Learning Tasks](https://mlsanity.com/paper/2503.11910/) (2025)
- [MS-COOT: Comparing Morse-Smale Complexes with Co-Optimal Transport](https://mlsanity.com/paper/2606.08258/) (2026)

## Abstract (arXiv)

Merge trees are a valuable tool in the scientific visualization of scalar fields; however, current methods for merge tree comparisons are computationally expensive, primarily due to the exhaustive matching between tree nodes. To address this challenge, we introduce the Merge Tree Neural Network (MTNN), a learned neural network model designed for merge tree comparison. The MTNN enables rapid and high-quality similarity computation. We first demonstrate how to train graph neural networks, which emerged as effective encoders for graphs, in order to produce embeddings of merge trees in vector spaces for efficient similarity comparison. Next, we formulate the novel MTNN model that further improves the similarity comparisons by integrating the tree and node embeddings with a new topological attention mechanism. We demonstrate the effectiveness of our model on real-world data in different domains and examine our model's generalizability across various datasets. Our experimental analysis demonstrates our approach's superiority in accuracy and efficiency. In particular, we speed up the prior state-of-the-art by more than 100× on the benchmark datasets while maintaining an error rate below 0.1%.