Peel neighborhoods

TL;DR

This paper introduces peel neighborhoods in finite metric spaces of strict negative type, providing a scalable way to analyze local geometry and topology, useful for estimating local dimension and detecting singularities.

Contribution

It presents a canonical, parameter-free, and efficiently computable notion of peel neighborhoods, facilitating microscopic geometric and topological analysis at scale.

Findings

Enables efficient estimation of local dimension.

Detects singularities in samples from stratified manifolds.

Provides a scalable approach for geometric analysis.

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.