Stratification Learning through Homology Inference

We develop a topological approach to stratification learning. Given point cloud data drawn from a stratified space, 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 different strata, and we prove a result which guarantees the correctness of our clustering, given certain topological conditions. We later give bounds on the minimum number of sample points required to infer, with probability, which points belong to the same strata. Finally, we give an explicit algorithm for the clustering and apply it to some simulated data.