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

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.