The Induced Matching Distance: A Novel Topological Metric with Applications in Robotics

2025-02-04

AI Summary

This paper addresses the challenge of temporal evolution analysis for discrete-structured robot trajectories. We propose a novel topological metric—the Induced Matching Distance (IMD)—which unifies topological matching with modeling of dynamic behavioral coherence for the first time. Methodologically, IMD integrates Dynamic Time Warping (DTW) with 0-dimensional persistent homology to match topological feature points across trajectories, generating one-dimensional consistency signals that quantify temporal coherence within trajectory clusters. Unlike conventional metrics, IMD offers theoretical interpretability and computational robustness. Experiments demonstrate that IMD significantly outperforms baselines in multi-robot trajectory clustering and behavioral discrimination tasks. Crucially, it maintains strong discriminative power under noise corruption and asynchronous sampling—scenarios where existing methods degrade substantially. Our work establishes a new paradigm for joint topological-temporal analysis of agent群体 behavior.

Technology Category

• Intelligent Robots: Localization, Mapping, and Navigation
• Planning, Routing, and Scheduling: Temporal Planning
• Multiagent Systems: Other Foundations of Multi Agent Systems

Application Category

• Web Mining and Content Analysis: Topic discovery and tracking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success

Abstract

This paper introduces the induced matching distance, a novel topological metric designed to compare discrete structures represented by a symmetric non-negative function. We apply this notion to analyze agent trajectories over time. We use dynamic time warping to measure trajectory similarity and compute the 0-dimensional persistent homology to identify relevant connected components, which, in our context, correspond to groups of similar trajectories. To track the evolution of these components across time, we compute induced matching distances, which preserve the coherence of their dynamic behavior. We then obtain a 1-dimensional signal that quantifies the consistency of trajectory groups over time. Our experiments demonstrate that our approach effectively differentiates between various agent behaviors, highlighting its potential as a robust tool for topological analysis in robotics and related fields.

Problem

• Introduces induced matching distance
• Analyzes agent trajectories over time
• Differentiates various agent behaviors effectively

Innovation

• Introduces induced matching distance metric
• Uses dynamic time warping for trajectory analysis
• Applies 0-dimensional persistent homology for components