The topology of probability distributions on manifolds

Let P be a set of n random points in R d , generated from a probability measure on a m -dimensional manifold M ⊂ 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 → ∞ , and r → 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.