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Tight quasi-universality of Reeb graph distances
Ulrich Bauer1, Håvard Bakke Bjerkevik2,3, Benedikt Fluhr4
1Department of Mathematics and Munich Data Science Institute, Technical University of Munich (TUM), Munich, Germany.
Journal of Applied and Computational Topology
|February 17, 2025
Summary
This study proves quasi-universality for Reeb graph distances, including a novel functional contortion distance. This finding holds for contour trees and merge trees, demonstrating broad applicability in topological data analysis.
Area of Science:
- Topological Data Analysis
- Computational Topology
Background:
- Reeb graphs are crucial for analyzing scalar fields.
- Understanding distances between Reeb graphs is key for comparing topological structures.
- Existing distances like interleaving and functional distortion have limitations.
Purpose of the Study:
- To establish tight bi-Lipschitz bounds for Reeb graph distances.
- To introduce and analyze a novel functional contortion distance.
- To investigate universality properties of these distances for contour and merge trees.
Main Methods:
- Establishing bi-Lipschitz bounds for graph distances.
- Defining and analyzing the functional contortion distance.
- Proving universality for contour trees and merge trees.
Main Results:
- Tight bi-Lipschitz bounds for interleaving, functional distortion, and functional contortion distances.
- The functional contortion distance is a novel contribution.
- Strict universality is proven for the functional contortion distance in contour trees.
- The functional contortion distance coincides with the interleaving distance for merge trees.
Conclusions:
- Quasi-universality is certified for multiple Reeb graph distances.
- The functional contortion distance offers a new tool for topological analysis.
- Universality is achieved for contour and merge trees under specific distances.
Keywords:
Contour treesFunctional contortion distanceMerge treesReeb graph distancesReeb graphsUniversal distanceMore Related Videos
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