  In this paper, we propose a listwise approach for constructing user-specific
rankings in recommendation systems in a collaborative fashion. We contrast the
listwise approach to previous pointwise and pairwise approaches, which are
based on treating either each rating or each pairwise comparison as an
independent instance respectively. By extending the work of (Cao et al. 2007),
we cast listwise collaborative ranking as maximum likelihood under a
permutation model which applies probability mass to permutations based on a low
rank latent score matrix. We present a novel algorithm called SQL-Rank, which
can accommodate ties and missing data and can run in linear time. We develop a
theoretical framework for analyzing listwise ranking methods based on a novel
representation theory for the permutation model. Applying this framework to
collaborative ranking, we derive asymptotic statistical rates as the number of
users and items grow together. We conclude by demonstrating that our SQL-Rank
method often outperforms current state-of-the-art algorithms for implicit
feedback such as Weighted-MF and BPR and achieve favorable results when
compared to explicit feedback algorithms such as matrix factorization and
collaborative ranking.
