The topology of probability distributions on manifolds

Abstract

Let P be a set of n random points in Rd, generated from a probability measure on a m-dimensional manifold M⊂Rd. In this paper we study the homology of U(P,r)—the union of d-dimensional balls of radius r around P, n→∞, and r→0. In addition we study the critical points of dP—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 dP. Depending on how fast r decays to zero as n grows, these two objects exhibit different types of limiting behavior. In one particular case (nrm≥Clogn), 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.