Clustering from Sparse Pairwise Measurements

TL;DR

This paper introduces algorithms for clustering items using sparse pairwise comparisons, including belief propagation and spectral methods, and demonstrates their near-optimal performance in detecting clusters.

Contribution

It proposes three novel algorithms for clustering from sparse measurements and provides theoretical and empirical evidence of their optimality in certain cases.

Findings

Algorithms detect clusters at the information-theoretic limit.

Spectral algorithms perform near-optimally for two symmetric clusters.

Belief propagation approximates the Bayes optimal solution.

Abstract

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.