  In recent years rank aggregation has received significant attention from the
machine learning community. The goal of such a problem is to combine the
(partially revealed) preferences over objects of a large population into a
single, relatively consistent ordering of those objects. However, in many
cases, we might not want a single ranking and instead opt for individual
rankings. We study a version of the problem known as collaborative ranking. In
this problem we assume that individual users provide us with pairwise
preferences (for example purchasing one item over another). From those
preferences we wish to obtain rankings on items that the users have not had an
opportunity to explore. The results here have a very interesting connection to
the standard matrix completion problem. We provide a theoretical justification
for a nuclear norm regularized optimization procedure, and provide
high-dimensional scaling results that show how the error in estimating user
preferences behaves as the number of observations increase.
