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Morse Theory for the k-NN Distance Function
Published 19 Mar 2024 in cs.CG, math.AT, and math.CO | (2403.12792v1)
Abstract: We study the $k$-th nearest neighbor distance function from a finite point-set in $\mathbb{R}d$. We provide a Morse theoretic framework to analyze the sub-level set topology. In particular, we present a simple combinatorial-geometric characterization for critical points and their indices, along with detailed information about the possible changes in homology at the critical levels. We conclude by computing the expected number of critical points for a homogeneous Poisson process. Our results deliver significant insights and tools for the analysis of persistent homology in order-$k$ Delaunay mosaics, and random $k$-fold coverage.
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