Cluster Trees on Manifolds
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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 R^D. 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.
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