The Topology of Probability Distributions on Manifolds

TL;DR

This paper investigates the topological properties of point cloud data sampled from manifolds, providing limit theorems for Betti numbers and critical points, with implications for topological manifold learning.

Contribution

It introduces new limit theorems for the homology and critical points of unions of balls around random points on manifolds, linking these to the original manifold's topology.

Findings

Betti numbers of unions of balls recover manifold topology under certain conditions

Limit theorems describe the behavior of critical points as sample size grows

Different decay rates of radius lead to different topological limits

Abstract

Let $P$ be a set of $n$ random points in $R^d$, generated from a probability measure on a $m$-dimensional manifold $M \subset R^d$. In this paper we study the homology of $U(P,r)$ -- the union of $d$-dimensional balls of radius $r$ around $P$, as $n \to \infty$, and $r \to 0$. In addition we study the critical points of $d_P$ -- the distance function from the set $P$. These two objects are known to be related via Morse theory. We present limit theorems for the Betti numbers of $U(P,r)$, as well as for number of critical points of index $k$ for $d_P$. Depending on how fast $r$ decays to zero as $n$ grows, these two objects exhibit different types of limiting behavior. In one particular case ($n r^m > C \log n$), we show that the Betti numbers of $U(P,r)$ perfectly recover the Betti numbers of the original manifold $M$, a result which is of significant interest in topological manifold learning.