25 research outputs found
The Topology of Probability Distributions on Manifolds
Let be a set of random points in , generated from a probability
measure on a -dimensional manifold . In this paper we study
the homology of -- the union of -dimensional balls of radius
around , as , and . In addition we study the critical
points of -- the distance function from the set . These two objects
are known to be related via Morse theory. We present limit theorems for the
Betti numbers of , as well as for number of critical points of index
for . Depending on how fast decays to zero as grows, these two
objects exhibit different types of limiting behavior. In one particular case
(), we show that the Betti numbers of perfectly
recover the Betti numbers of the original manifold , a result which is of
significant interest in topological manifold learning