Morse Theory for the k-NN Distance Function

TL;DR

This paper develops a Morse-theoretic framework for the $k$-NN distance function $d^{(k)}_{\,\mathcal{P}}$ in $\mathbb{R}^d$ to analyze the topology of its sublevel sets $B_r^{(k)}({\mathcal P})$. It provides a combinatorial-geometric characterization of critical points: a point $c$ is critical iff $c\in\sigma(\mathcal{P}_c^{\partial})$, with index $\mu_c=N_c-k$, and non-degeneracy, yielding homology changes across the critical level governed by $\Delta_c=\binom{N_c^{\partial}-1}{\mu_c}$ and distributed between degrees $\mu_c$ and $\mu_c-1$. The work extends Morse theory to piecewise-smooth settings via the Clarke subdifferential and the Agrachev framework, and introduces an auxiliary complex $K_c$ to compute local homology changes, linking critical points to steps in the order-$k$ Delaunay filtration and to persistent homology of $B_r^{(k)}({\mathcal P})$. Additionally, it derives a Poisson-process formula for the expected number of critical points of each index, enabling analysis of random $k$-fold coverage and homological connectivity in stochastic topology.

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.

Figures (3)

• Figure 1: Critical points of $d_{{\cal{P}}}^{(k)}$ in ${\mathbb{R}}^2$, for $k=2$. The points $x_1,x_2,x_3$, and $y$ are in ${\cal{P}}$, and the point $c$ represents the critical point. Top left:${\cal{P}}_c^{\partial }=\{x_1,x_2\}$, ${\cal{P}}_c^{\mathcal{I}}=\emptyset$, and $\mu_c=0$. This critical point adds a new generator to $H_0$ (new component). Bottom left:${\cal{P}}_c^{\partial }=\{x_1,x_2\}$, ${\cal{P}}_c^{\mathcal{I}}=\{y\}$, and $\mu_c=1$. This critical point kills a generator in $H_0$ (components merge). Top right:${\cal{P}}_c^{\partial }=\{x_1,x_2,x_3\}$, ${\cal{P}}_c^{\mathcal{I}}=\emptyset$, and $\mu_c=1$. This critical point kills two generators in $H_0$ (three components merge into one). Bottom right:${\cal{P}}_c^{\partial }=\{x_1,x_2,x_3\}$, ${\cal{P}}_c^{\mathcal{I}}=\{y\}$, and $\mu_c=2$. This critical point kills an existing $1$-cycle.
• Figure 2: The effect of a critical point on the homology. The point $c\in{\mathbb{R}}^2$ is a critical point of $d_{{\cal{P}}}^{(2)}$ of index $\mu_c=1$, where ${\cal{P}}=\{x_1,x_2,x_3,y_1,\ldots,y_4\}$. In this case, we have $N_c^\partial=3$, and therefore, $\Delta_c = \binom{2}{1} = 2$. Indeed, we observe exactly two changes in the homology of the sub-level sets (purple shaded regions), once $c$ is reached. One change is the generation of a new $1$-cycle on the right side (the red dashed cycle). Another change is the elimination of the connected component ($0$-cycle) on the left side.
• Figure 3: The auxiliary complex used in the proof of Lemma \ref{['lem:Uc_open']}. Left: In both figures $c$ is a critical point of $d_{{\cal{P}}}^{(2)}$, and the purple regions are the $2$-fold cover, at radius $r$ that is slightly smaller than $r_c$. Right: The corresponding auxiliary complex $K_c$ (in green). Top: The critical point $c$ is of index $\mu_c=1$. The sets ${\cal{N}}_1,{\cal{N}}_2,{\cal{N}}_3$ are equal to $\{1,2\},\{1,3\},\{2,3\}$, respectively. Thus, the sets $\bar{{\cal{N}}}_1,\bar{{\cal{N}}}_2,\bar{{\cal{N}}}_3$ that span $K_c$, are equal to $\{3\},\{2\},\{1\}$, respectively. (b) The critical point $c$ is of index $\mu_c=2$. The sets ${\cal{N}}_1,{\cal{N}}_2,{\cal{N}}_3$ are equal to $\{1\},\{2\},\{3\}$. Thus, the sets $\bar{{\cal{N}}}_1,\bar{{\cal{N}}}_2,\bar{{\cal{N}}}_3$ that span $K_c$, are equal to $\{2,3\},\{1,3\},\{1,2\}$, respectively.

Theorems & Definitions (18)

• Theorem 1
• Theorem 2
• Remark 3
• Remark 4
• Definition 5: Definition 1.1 in agrachev1997morse
• Definition 6: Definition 2.2 in agrachev1997morse
• Theorem 7: Theorem 4.2 in agrachev1997morse
• Lemma 8
• Lemma 9
• Remark 10