Clustering from Sparse Pairwise Measurements

We consider the problem of grouping items into clusters based on few random pairwise comparisons between the items. We introduce three closely related algorithms for this task: a belief propagation algorithm approximating the Bayes optimal solution, and two spectral algorithms based on the non-backtracking and Bethe Hessian operators. For the case of two symmetric clusters, we conjecture that these algorithms are asymptotically optimal in that they detect the clusters as soon as it is information theoretically possible to do so. We substantiate this claim for one of the spectral approaches we introduce.