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Abstract:
We study clustering algorithms based on neighborhood graphs on a random sample of data points. The question we ask is how such a graph should be constructed in order to obtain optimal clustering results. Which type of neighborhood graph should one choose, mutual knearestneighbor or symmetric knearestneighbor? What is the optimal parameter k? In our setting, clusters are defined as connected components of the tlevel set of the underlying probability distribution. Clusters are said to be identified in the neighborhood graph if connected components in the graph correspond to the true underlying clusters. Using techniques from random geometric graph theory, we prove bounds on the probability that clusters are identified successfully, both in a noisefree and in a noisy setting. Those bounds lead to several conclusions. First, k has to be chosen surprisingly high (rather of the order n than of the order logn) to maximize the probability of cluster identification. Secondly, the major difference between the mutual and the symmetric knearestneighbor graph occurs when one attempts to detect the most significant cluster only.