Applied Mathematics Colloquium **SPECIAL TIME/LOCATION**
Sanjoy Dasgupta is a Professor in the Department of Computer Science and Engineering at UC San Diego. He received his PhD from Berkeley in 2000, and spent two years at AT&T Research Labs before joining UCSD. His area of research is algorithmic statistics, with a focus on unsupervised and minimally supervised learning. He is the author of a textbook, "Algorithms" (with Christos Papadimitriou and Umesh Vazirani), that appeared in 2006.
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.