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Morse Theory for the k-NN Distance Function.

Yohai Reani1, Omer Bobrowski2

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Summary
This summary is machine-generated.

This study introduces a Morse theory framework to analyze the topology of k-th nearest neighbor distances in point sets. It provides tools for understanding persistent homology in complex data structures.

Keywords:
Applied topologyDistance functionMorse theoryk-nearest neighbor

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Area of Science:

  • Computational Geometry
  • Topology
  • Data Analysis

Background:

  • The k-th nearest neighbor distance function is crucial for analyzing point set topology.
  • Understanding the sub-level set topology is essential for data analysis and visualization.

Purpose of the Study:

  • To develop a Morse theoretic framework for analyzing the topology of k-th nearest neighbor distance functions.
  • To provide combinatorial-geometric characterizations of critical points and their indices.
  • To compute the expected number of critical points for random point processes.

Main Methods:

  • Morse theory applied to the k-th nearest neighbor distance function.
  • Combinatorial and geometric analysis of critical points and homology.
  • Expected Betti number computation for Poisson point processes.

Main Results:

  • A framework for analyzing sub-level set topology using Morse theory.
  • Combinatorial-geometric characterization of critical points and their indices.
  • Detailed information on homology changes at critical levels.
  • Computation of the expected number of critical points for homogeneous Poisson processes.

Conclusions:

  • The developed Morse theoretic framework offers significant insights into persistent homology.
  • The results provide valuable tools for analyzing order-k Delaunay mosaics and random k-fold coverage.
  • This work bridges topological data analysis with geometric and probabilistic methods.