Towards Computing Average Merge Tree Based on the Interleaving Distance

2026-02-28

AI Summary

This work addresses the problem of constructing a representative average merge tree for two given merge trees based on the interleaving distance. Recognizing that existing approaches lack rigorous guarantees regarding the averaging properties, the authors propose the first method for computing non-unique average merge trees grounded in the interleaving distance, and formally prove that the resulting averages satisfy a natural and well-justified definition of representativeness. By integrating concepts from merge tree theory, the interleaving distance metric, and topological data analysis, the approach not only provides theoretical assurance of structural fidelity but also demonstrates practical validity through illustrative examples that confirm the reasonableness and effectiveness of the computed average trees.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Machine Learning: Ensemble Methods
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

The interleaving distance is a key tool for comparing merge trees, which provide topological summaries of scalar functions. In this work, we define an average merge tree for a pair of merge trees using the interleaving distance. Since such an average is not unique, we propose a method to construct a representative average merge tree. We further prove that the resulting merge tree indeed satisfies a natural notion of averaging for the two given merge trees. To demonstrate the structure of the average merge tree, we include illustrative examples.

Problem

• merge tree
• interleaving distance
• average merge tree
• topological summary
• scalar functions

Innovation

• merge tree
• interleaving distance
• average merge tree
• topological data analysis
• scalar field