Morse Theory for the k-NN Distance Function.

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.