  We introduce a consistent estimator for the homology (an algebraic structure
representing connected components and cycles) of level sets of both density and
regression functions. Our method is based on kernel estimation. We apply this
procedure to two problems: (1) inferring the homology structure of manifolds
from noisy observations, (2) inferring the persistent homology (a multi-scale
extension of homology) of either density or regression functions. We prove
consistency for both of these problems. In addition to the theoretical results,
we demonstrate these methods on simulated data for binary regression and
clustering applications.
