  Beyond accuracy, there are a variety of aspects to the quality of recommender
systems, such as diversity, fairness, and robustness. We argue that many of the
prevalent problems in recommender systems are partly due to low-dimensionality
of user and item embeddings, particularly when dot-product models, such as
matrix factorization, are used.
  In this study, we showcase empirical evidence suggesting the necessity of
sufficient dimensionality for user/item embeddings to achieve diverse, fair,
and robust recommendation. We then present theoretical analyses of the
expressive power of dot-product models. Our theoretical results demonstrate
that the number of possible rankings expressible under dot-product models is
exponentially bounded by the dimension of item factors. We empirically found
that the low-dimensionality contributes to a popularity bias, widening the gap
between the rank positions of popular and long-tail items; we also give a
theoretical justification for this phenomenon.
