  We consider the problem of learning the true ordering of a set of
alternatives from largely incomplete and noisy rankings. We introduce a natural
generalization of both the classical Mallows model of ranking distributions and
the extensively studied model of noisy pairwise comparisons. Our selective
Mallows model outputs a noisy ranking on any given subset of alternatives,
based on an underlying Mallows distribution. Assuming a sequence of subsets
where each pair of alternatives appears frequently enough, we obtain strong
asymptotically tight upper and lower bounds on the sample complexity of
learning the underlying complete ranking and the (identities and the) ranking
of the top-k alternatives from selective Mallows rankings. Moreover, building
on the work of (Braverman and Mossel, 2009), we show how to efficiently compute
the maximum likelihood complete ranking from selective Mallows rankings.
