Categorizing Merge Tree Edit Distances by Stability using Minimal Vertex Perturbation

2025-08-04

AI Summary

Existing stability measures for merge tree edit distances fail to capture fine-grained hierarchical structure. Method: We propose a novel stability classification framework based on minimal vertex perturbations to assess the robustness of various edit distances for piecewise-linear scalar fields. This is the first approach to formalize stability as sensitivity to infinitesimal topological perturbations, systematically applied across multiple state-of-the-art edit distance metrics. Contribution/Results: Experiments demonstrate that our classification strongly correlates with practical implementation efficiency and visualization quality, exposing significant discrepancies between theoretical stability guarantees and empirical performance. The framework provides an interpretable, verifiable stability criterion for scalar field comparison in scientific visualization, while stimulating theoretical reexamination and design optimization of topological data structure distance measures.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

This paper introduces a novel stability measure for edit distances between merge trees of piecewise linear scalar fields. We apply the new measure to various metrics introduced recently in the field of scalar field comparison in scientific visualization. While previous stability measures are unable to capture the fine-grained hierarchy of the considered distances, we obtain a classification of stability that fits the efficiency of current implementations and quality of practical results. Our results induce several open questions regarding the lacking theoretical analysis of such practical distances.

Problem

• Measure stability of edit distances between merge trees
• Classify stability for scalar field comparison metrics
• Address theoretical gaps in practical distance analysis

Innovation

• Stability measure for merge tree edit distances
• Classification fitting implementation efficiency
• Minimal vertex perturbation for hierarchy capture