  Multi-aspect user preferences are attracting wider attention in recommender
systems, as they enable more detailed understanding of users' evaluations of
items. Previous studies show that incorporating multi-aspect preferences can
greatly improve the performance and explainability of recommendation. However,
as recommendation is essentially a ranking problem, there is no principled
solution for ranking multiple aspects collectively to enhance the
recommendation.
  In this work, we derive a multi-aspect ranking criterion. To maintain the
dependency among different aspects, we propose to use a vectorized
representation of multi-aspect ratings and develop a probabilistic multivariate
tensor factorization framework (PMTF). The framework naturally leads to a
probabilistic multi-aspect ranking criterion, which generalizes the
single-aspect ranking to a multivariate fashion. Experiment results on a large
multi-aspect review rating dataset confirmed the effectiveness of our solution.
