Joint Clustering and Ranking from Heterogeneous Pairwise Comparisons

In this paper, the joint clustering and ranking problem is studied. Assume that there are items that belong to several communities and are compared in a pairwise manner repeatedly. The task is to identify the community each item belongs to and to give a consistent preference ordering of the items in each community. A statistical model called "Cluster-BTL" is proposed to model the difference between inter-community and intra-community comparisons. Such a difference is called "heterogeneity" throughout this paper, and we are interested in how the heterogeneity level affects the joint clustering and ranking problem. In this work, we characterize the fundamental limit on the heterogeneity level and the number of comparisons for reliable joint clustering and ranking. The converse part is proved by a Fano-type argument. For the achievability part, a polynomial-time algorithm that combines the Borda counting algorithm and the SDP based method is developed.