A Revenue Function for Comparison-Based Hierarchical Clustering
This addresses a gap in comparison-based learning for hierarchical clustering, enabling meaningful evaluation without ground-truth or explicit similarities, though it appears incremental as it builds on prior work on linkage methods and Dasgupta's cost.
The paper tackles the problem of evaluating hierarchical clustering dendrograms when only triplet comparisons are available, proposing a new revenue function to measure their goodness and showing it relates to existing similarity-based costs, with theoretical results on approximate recovery and practical algorithms that are empirically compared to existing methods.
Comparison-based learning addresses the problem of learning when, instead of explicit features or pairwise similarities, one only has access to comparisons of the form: \emph{Object $A$ is more similar to $B$ than to $C$.} Recently, it has been shown that, in Hierarchical Clustering, single and complete linkage can be directly implemented using only such comparisons while several algorithms have been proposed to emulate the behaviour of average linkage. Hence, finding hierarchies (or dendrograms) using only comparisons is a well understood problem. However, evaluating their meaningfulness when no ground-truth nor explicit similarities are available remains an open question. In this paper, we bridge this gap by proposing a new revenue function that allows one to measure the goodness of dendrograms using only comparisons. We show that this function is closely related to Dasgupta's cost for hierarchical clustering that uses pairwise similarities. On the theoretical side, we use the proposed revenue function to resolve the open problem of whether one can approximately recover a latent hierarchy using few triplet comparisons. On the practical side, we present principled algorithms for comparison-based hierarchical clustering based on the maximisation of the revenue and we empirically compare them with existing methods.