Topological consistency via kernel estimation

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.