Learning Ordinal Information Under Bipartite Stochastic Block Models

A problem of learning ordinal information from noisy data is studied under bipartite stochastic block models with stochastically identical and ordinally homogeneous settings. Two clustering-based algorithms are proposed for the two settings. In the stochastically identical setting, a value-oriented plug-in approach is developed to recover rankings by estimating parameters through a reduction from maximizing likelihood to minimizing clustering cost. In the ordinally homogeneous setting, a ranking-oriented empirical risk minimization approach is developed to cluster users based on a ranking distance and aggregate rankings optimally through a reduction to a minimum bipartite matching problem. It is shown that both algorithms converge in polynomial time with the problem size.