User rankings from comparisons: Learning permutations in high dimensions

We consider the problem of learning users' preferential orderings for a set of items when only a limited number of pairwise comparisons of items from users is available. This problem is relevant in large collaborative recommender systems where overall rankings of users for objects need to be predicted using partial information from simple pairwise item preferences from chosen users. We consider two natural schemes of obtaining pairwise item orderings random and active (or intelligent) sampling. Under both these schemes, assuming that the users' orderings are constrained in number, we develop efficient, low complexity algorithms that reconstruct all the orderings with provably order-optimal sample complexities. Finally, our algorithms are shown to outperform a matrix completion based approach in terms of sample and computational requirements in numerical experiments.