Adaptive Hierarchical Clustering Using Ordinal Queries

Abstract

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 $p > \frac{1}{2}$, 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.