Sampling and homology via bottlenecks

TL;DR

This paper introduces an efficient algorithm for dense sampling of smooth compact varieties using bottlenecks, providing guarantees for homology recovery and including implementation and experimental validation.

Contribution

The paper presents a new algorithm leveraging bottlenecks and local reach to produce dense samples with provable homology recovery guarantees.

Findings

Algorithm successfully produces dense samples of varieties.

Bounds on sample density ensure homology can be recovered.

Numerical experiments validate the approach.

Abstract

In this paper we present an efficient algorithm to produce a provably dense sample of a smooth compact variety. The procedure is partly based on computing $\textit{bottlenecks}$ of the variety. Using geometric information such as the bottlenecks and the $\textit{local reach}$ we also provide bounds on the density of the sample needed in order to guarantee that the homology of the variety can be recovered from the sample. An implementation of the algorithm is provided together with numerical experiments and a computational comparison to the algorithm by Dufresne et. al.