Asymptotics of hierarchical clustering for growing dimension

Modern day science presents many challenges to data analysts. Advances in data collection provide very large (number of observations and number of dimensions) data sets. In many areas of data analysis an informative task is to find natural separations of data into homogeneous groups, i.e. clusters. In this paper we study the asymptotic behavior of hierarchical clustering in situations where both sample size and dimension grow to infinity. We derive explicit signal vs noise boundaries between different types of clustering behaviors. We also show that the clustering behavior within the boundaries is the same across a wide spectrum of asymptotic settings.