Adaptive Hierarchical Clustering Using Ordinal Queries

In many applications of clustering (for example, ontologies or clusterings of animal or plant species), hierarchical clusterings are more descriptive than a flat clustering. A hierarchical clustering over n elements is represented by a rooted binary tree with n leaves, each corresponding to one element. The subtrees rooted at interior nodes capture the clusters. In this paper, we study active learning of a hierarchical clustering using only ordinal queries. An ordinal query consists of a set of three elements, and the response to a query reveals the two elements (among the three elements in the query) which are "closer" to each other than to the third one. We say that elements x and x' are closer to each other than x '' if there exists a cluster containing x and x', but not x ''. When all the query responses are correct, there is a deterministic algorithm that learns the underlying hierarchical clustering using at most n log(2) n adaptive ordinal queries. We generalize this algorithm to be robust in a model in which each query response is correct independently with probability p > 1/2, and adversarially incorrect with probability 1 - p. We show that in the presence of noise, our algorithm outputs the correct hierarchical clustering with probability at least 1 - delta, using O(n log n + n log(1/delta)) adaptive ordinal queries. For our results, adaptivity is crucial: we prove that even in the absence of noise, every non-adaptive algorithm requires Omega(n(3)) ordinal queries in the worst case.