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Peel neighborhoods

Steve Huntsman

Abstract

We introduce the canonical, parameter-free, and efficiently computable notion of peel neighborhoods in a finite metric space of strict negative type. Using a soft threshold to upper bound their radius or cardinality allows peel neighborhoods to be computed at scale, enabling useful microscopic descriptions of geometry and topology. As an example of their utility, peel neighborhoods enable efficient and performant estimates of local dimension and detections of singularities in samples from stratified manifolds.

Peel neighborhoods

Abstract

We introduce the canonical, parameter-free, and efficiently computable notion of peel neighborhoods in a finite metric space of strict negative type. Using a soft threshold to upper bound their radius or cardinality allows peel neighborhoods to be computed at scale, enabling useful microscopic descriptions of geometry and topology. As an example of their utility, peel neighborhoods enable efficient and performant estimates of local dimension and detections of singularities in samples from stratified manifolds.

Paper Structure

This paper contains 17 sections, 13 theorems, 53 equations, 36 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Weightings and diversity
  3. The peeling theorem
  4. Some useful results
  5. Peel neighborhoods
  6. A useful model and a Poisson process experiment
  7. Experiments
  8. Comparison with fixed-radius and $k$-nearest neighborhoods
  9. Samples from annuli
  10. Peel neighborhood scaling and relation to peels
  11. Local dimension estimation in sampled stratified manifolds
  12. Gradient norms of dimension estimates
  13. Comparison of dimension estimates on peel neighborhoods with the volume growth transform
  14. Conclusions
  15. Behavior of radii of thresholded peel neighborhoods
  16. ...and 2 more sections

Key Result

Theorem 2.1

Let $Z_{jk} = \exp(-t d_{jk})$ for $t > 0$ and $d$ strict negative type. If the weighting $w$ that solves $Zw = 1$ is positive, then $w$ is proportional to the diversity-maximizing distribution for all $q$.

Figures (36)

  • Figure 1: The peel distribution, shown with overlaid colored and variably-sized disks circled in red, of a sample of 4782 IID uniform points from a complicated region in $\mathbb{R}^2$, which are shown in black. The support of the peel distribution is just 10 points located at extremities.
  • Figure 1: Left: unthresholded (for thresholding, see below) peel neighborhoods of 1000 IID uniform points in $B_1 \subset \mathbb{R}^2$, indicated by shaded disks and line segments from basepoints to the point at distance $\rho(x)$. Neighborhoods and segments for points in the overall peel are shown in pink and red, while others are shown are gray and black. Right: $\rho(x)$ is larger for points in the overall peel, which comprise a notion of boundary. Not shown: imposing a practical default radial threshold discussed below affects just 58 of 1000 neighborhoods, near the upper envelope of the plotted points.
  • Figure 1: Top left: the graph with edges given by peel neighborhoods on a uniform sample of approximately $1000$ points from a sphere $S^2 \subset \mathbb{R}^3$, plus $11$ equispaced points slightly to the left, embedded in 10 dimensions with small Gaussian noise added. Edges are shaded according to their average in the third dimension. Top right: the similar graph with edges given by $k$-nearest neighbors with the minimal $k$ such that the graph has a single connected component (here, $k = 5$). Middle: the relative efficiencies per edge and per length of radial graphs versus peel neighborhood graphs, along with the relative numbers of connected components, all as functions of radius, with means and standard deviations over 100 trials indicated. Bottom: as in the middle panels, but for $k$NN graphs. Initial values of $k$ for which the $k$NN graph is fully connected are indicated by the vertical patch of $\pm$ a standard deviation about the mean.
  • Figure 1: Average radii of thresholded peel neighborhoods (with standard deviations shown) in correspondence with Figure \ref{['fig:peelNeighborhoodsHoles']}. Each panel is over $[0,1] \times [0,1]$.
  • Figure 1: Left: in black, a sample of 1000 IID uniform points from the fundamental domain of the Bolza surface in the Poincaré disk. Colors from red to blue indicate translates of the sample to the 48 edge- and vertex-adjacent domains, outlined in black. These translations suffice to compute distances on the Bolza surface via the usual metric on the Poincaré disk. Right: the undirected graph on the sample points obtained with peel neighborhoods along the lines of prior figures. The periodic boundary conditions are evident through long graph edges.
  • ...and 31 more figures

Theorems & Definitions (26)

  • Theorem 2.1
  • Theorem 3.1: peeling theorem; huntsman2025peeling
  • Lemma 3.2
  • Proof 1
  • Lemma 3.3
  • Proof 2
  • Proposition 3.4
  • Proof 3
  • Proposition 3.5
  • Proof 4
  • ...and 16 more