  We propose a novel and efficient algorithm for the collaborative preference
completion problem, which involves jointly estimating individualized rankings
for a set of entities over a shared set of items, based on a limited number of
observed affinity values. Our approach exploits the observation that while
preferences are often recorded as numerical scores, the predictive quantity of
interest is the underlying rankings. Thus, attempts to closely match the
recorded scores may lead to overfitting and impair generalization performance.
Instead, we propose an estimator that directly fits the underlying preference
order, combined with nuclear norm constraints to encourage low--rank
parameters. Besides (approximate) correctness of the ranking order, the
proposed estimator makes no generative assumption on the numerical scores of
the observations. One consequence is that the proposed estimator can fit any
consistent partial ranking over a subset of the items represented as a directed
acyclic graph (DAG), generalizing standard techniques that can only fit
preference scores. Despite this generality, for supervision representing total
or blockwise total orders, the computational complexity of our algorithm is
within a $\log$ factor of the standard algorithms for nuclear norm
regularization based estimates for matrix completion. We further show promising
empirical results for a novel and challenging application of collaboratively
ranking of the associations between brain--regions and cognitive neuroscience
terms.
