Active Clustering: Robust and Efficient Hierarchical Clustering using Adaptively Selected Similarities

Hierarchical clustering based on pairwise similarities is a common tool used in a broad range of scientific applications. However, in many problems it may be expensive to obtain or compute similarities between the items to be clustered. This paper investigates the possibility of hierarchical clustering of $N$ items based on a small subset of pairwise similarities, significantly less than the complete set of $N(N-1)/2$ similarities. First, we show that, if the intracluster similarities exceed intercluster similarities, then it is possible to correctly determine the hierarchical clustering from as few as $3N \log N$ similarities. We demonstrate this order of magnitude saving in the number of pairwise similarities necessitates sequentially selecting which similarities to obtain in an adaptive fashion, rather than picking them at random. Finally, we propose an active clustering method that is robust to a limited fraction of anomalous similarities, and show how even in the presence of these noisy similarity values we can resolve the hierarchical clustering using only $O(N \log^2 N)$ pairwise similarities.