[Paper Review] Geometric Learning and Topological Inference with Biobotic Networks: Convergence Analysis

Alireza Dirafzoon, Alper Bozkurt+1 more|arXiv (Cornell University)|Jun 30, 2016

Topological and Geometric Data Analysis25 references3 citations

TL;DR

This paper proposes a geometric and topological mapping framework for biobotic networks using only local, coordinate-free interactions among cyborg insects. By constructing an encounter graph and applying manifold learning and persistent homology, the method achieves convergence to geodesic distances and stable topological inference, with landmark placement and encounter density critically influencing geometric and topological accuracy.

ABSTRACT

In this study, we present and analyze a framework for geometric and topological estimation for mapping of unknown environments. We consider agents mimicking motion behaviors of cyborg insects, known as biobots, and exploit coordinate-free local interactions among them to infer geometric and topological information about the environment, under minimal sensing and localization constraints. Local interactions are used to create a graphical representation referred to as the encounter graph. A metric is estimated over the encounter graph of the agents in order to construct a geometric point cloud using manifold learning techniques. Topological data analysis (TDA), in particular persistent homology, is used in order to extract topological features of the space and a classification method is proposed to infer robust features of interest (e.g. existence of obstacles). We examine the asymptotic behavior of the proposed metric in terms of the convergence to the geodesic distances in the underlying manifold of the domain, and provide stability analysis results for the topological persistence. The proposed framework and its convergences and stability analysis are demonstrated through numerical simulations and experiments.

Motivation & Objective

• Address the challenge of mapping unknown, unstructured environments—such as collapsed buildings—where traditional sensing and localization fail.
• Overcome limitations of GPS, vision-based SLAM, and inertial sensors in cluttered, indoor, or underground disaster zones.
• Enable robust environmental mapping using only minimal sensing and localization, leveraging the natural stochastic motion of biobots.
• Simultaneously estimate geometric structure (via metric learning) and topological features (via persistent homology) from sparse, local agent encounters.
• Provide theoretical convergence and stability guarantees for both geometric and topological inference under realistic biobotic network constraints.

Proposed method

• Model biobots as agents with correlated random walk motion, using wireless communication to detect local encounters.
• Construct an encounter graph from pairwise agent interactions, representing the network topology without global coordinates.
• Estimate a geodesic-like metric on the encounter graph using manifold learning techniques to reconstruct a point cloud approximation of the environment.
• Apply persistent homology to extract topological features (e.g., connected components, holes) from the reconstructed point cloud.
• Use a classification method to infer robust topological features such as obstacle presence based on persistence diagrams.
• Theoretical analysis establishes convergence of the estimated metric to the true geodesic distance on the underlying manifold and stability of persistence diagrams under sampling variation.

Experimental results

Research questions

• RQ1 How can a network of biobots with minimal sensing and localization capabilities infer the intrinsic geometry of an unknown environment?
• RQ2 What conditions ensure the convergence of the estimated metric on the encounter graph to the true geodesic distance on the underlying manifold?
• RQ3 How does the number of landmarks and encounter samples affect the accuracy of geometric and topological inference?
• RQ4 Can topological data analysis provide stable and robust inference of environmental topology (e.g., obstacles, connectivity) from sparse, local agent interactions?
• RQ5 What trade-offs exist between geometric accuracy and topological stability in the proposed framework?

Key findings

• The geometric metric estimation error, quantified by the convergence parameter λ₃, decreases monotonically with increasing numbers of encounter samples (Nₘ), showing strong sensitivity to encounter density.
• Topological stability, measured by bottleneck distance d_B(PD_M, PD_E), decreases with increasing landmark count (Nₗ), indicating that landmarks significantly improve topological inference accuracy.
• The bottleneck distance d_B(PD_M, PD_E) is consistently smaller than d_B(PD_M, PD_C), confirming that the estimated persistence diagram from encounters is closer to the true manifold than the one from all agents.
• Increasing Nₗ has a greater impact on topological stability than increasing Nₘ, while Nₘ has a stronger effect on geometric convergence, revealing a key trade-off in resource allocation.
• Theoretical analysis confirms that with dense sampling, connected encounter graph, and sufficient landmarks, the method converges to the true intrinsic geometry and topology of the environment.
• A practical strategy is to allocate 10–20% of agents as landmarks to balance geometric estimation over encounters and topological inference over the encounter set.