Applied Mathematics Colloquium **SPECIAL TIME/LOCATION**
Cluster Trees, Near-Neighbor Graphs, and Continuum Percolation
What information does the clustering of a finite data set reveal about the underlying distribution from which the data were sampled? This basic question has proved elusive even for the most widely-used clustering procedures. One natural criterion is to seek clusters that converge (as the data set grows) to regions of high density. When all possible density levels are considered, this is a hierarchical clustering problem where the sought limit is called the "cluster tree". We give a simple algorithm for estimating this tree that implicitly constructs a multiscale hierarchy of near-neighbor graphs on the data points. We show that the procedure is consistent, answering an open problem of Hartigan. We also obtain rates of convergence, using a percolation argument that gives insight into how near-neighbor graphs should be constructed.