On topology estimation of submanifolds in Riemannian manifolds by random points

TL;DR

This paper demonstrates that sampling enough random points near a compact submanifold in a Riemannian manifold allows for high-confidence topology recovery, given certain curvature conditions.

Contribution

It introduces a method to recover the topology of submanifolds in Riemannian manifolds from random samples under curvature constraints.

Findings

Topology can be recovered with high confidence from random samples.

Sampling density depends on curvature bounds.

Method applies to compact submanifolds in Riemannian manifolds.

Abstract

We show that, by sampling a sufficiently large number of random points in a neighborhood of a compact submanifold M of a Riemannian manifold N, one can recover the topology of M with high confidence. This holds under the assumptions on the curvatures of M and N.