Cluster Trees on Manifolds

In this paper we investigate the problem of estimating the cluster tree for a density f supported on or near a smooth d-dimensional manifold M isometrically embedded in ℝD. We analyze a modified version of a k-nearest neighbor based algorithm recently proposed by Chaudhuri and Dasgupta. The main results of this paper show that under mild assumptions on f and M, we obtain rates of convergence that depend on d only but not on the ambient dimension D. We also show that similar (albeit non-algorithmic) results can be obtained for kernel density estimators. We sketch a construction of a sample complexity lower bound instance for a natural class of manifold oblivious clustering algorithms. We further briefly consider the known manifold case and show that in this case a spatially adaptive algorithm achieves better rates.