The Lovasz-Bregman Divergence and connections to rank aggregation, clustering, and web ranking

We extend the recently introduced theory of Lovasz-Bregman (LB) divergences (Iyer & Bilmes, 2012) in several ways. We show that they represent a distortion between a 'score' and an 'ordering', thus providing a new view of rank aggregation and order based clustering with interesting connections to web ranking. We show how the LB divergences have a number of properties akin to many permutation based metrics, and in fact have as special cases forms very similar to the Kendall-$τ$ metric. We also show how the LB divergences subsume a number of commonly used ranking measures in information retrieval, like the NDCG and AUC. Unlike the traditional permutation based metrics, however, the LB divergence naturally captures a notion of "confidence" in the orderings, thus providing a new representation to applications involving aggregating scores as opposed to just orderings. We show how a number of recently used web ranking models are forms of Lovasz-Bregman rank aggregation and also observe that a natural form of Mallow's model using the LB divergence has been used as conditional ranking models for the 'Learning to Rank' problem.