Towards stratification learning through homology inference

A topological approach to stratification learning is developed for point cloud data drawn from a stratified space. Given such data, our objective is to infer which points belong to the same strata. First we define a multi-scale notion of a stratified space, giving a stratification for each radius level. We then use methods derived from kernel and cokernel persistent homology to cluster the data points into differ-ent strata, and we prove a result which guarantees the correctness of our clustering, given certain topological conditions; some geometric intuition for these topologi-cal conditions is also provided. Our correctness result is then given a probabilistic flavor: we give bounds on the minimum number of sample points required to in-fer, with probability, which points belong to the same strata. Finally, we give an explicit algorithm for the clustering, prove its correctness, and apply it to some simulated data.