Consistent procedures for cluster tree estimation and pruning

For a density f on mathbb Rd, a it high-density cluster is any connected component of x: f(x) ≥ λ, for some λ > 0. The set of all high-density clusters forms a hierarchy called the it cluster tree of f. We present two procedures for estimating the cluster tree given samples from f. The first is a robust variant of the single linkage algorithm for hierarchical clustering. The second is based on the k-nearest neighbor graph of the samples. We give finite-sample convergence rates for these algorithms which also imply consistency, and we derive lower bounds on the sample complexity of cluster tree estimation. Finally, we study a tree pruning procedure that guarantees, under milder conditions than usual, to remove clusters that are spurious while recovering those that are salient.