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

TL;DR

This work introduces the induced matching distance, a topology-based metric for comparing discrete structures defined by symmetric non-negative functions, and applies it to analyze agent trajectories through $0$-dimensional persistence of the Vietoris–Rips complex. By coupling bijection-induced block functions with barcodes and the triplet merge tree representation, it yields a continuous, label-aware distance that quantifies how trajectory components evolve and correspond over time. In robotics experiments with a corridor-navigation scenario, the induced matching signal derived from successive windows of trajectories enables clear discrimination among navigation strategies, achieving $97.5\%$ accuracy on a neural classifier. The approach provides a robust tool for topological analysis in robotics and offers avenues for extending to higher-dimensional persistence and broader applications.

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.

Figures (5)

• Figure 1: Point clouds $X_0$ (left) and $X_1$ (right) with the same points but different labels.
• Figure 2: The matching that produces the $q$-Wasserstein distance (left) and the matching that produces the induced matching distance (right) between $\mathop{\mathrm{B}}\nolimits(X_0)$ and $\mathop{\mathrm{B}}\nolimits(X_1)$.
• Figure 3: The first 10 seconds of the trajectories followed by the robots in the corridor scenario are depicted, corresponding to the behaviors SF (left), ORCA (center), and HL (right).
• Figure 4: Induced matching signals for the 600 simulations in the corridor scenario. Bold lines show the median signal for each behavior, and shaded bands represent the interquartile ranges.
• Figure 5: The confusion matrices for the ResNet classifier on the induced matching signals.

Theorems & Definitions (4)

• Lemma 1.1
• Lemma 1.2
• Lemma 1.3
• Theorem 1.4