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.

翻译：除了准确性外，推荐系统的质量还涉及多样性、公平性和鲁棒性等多个方面。我们认为，推荐系统中的许多普遍问题部分源于用户和物品嵌入的低维性，特别是在使用矩阵分解等点积模型时。本研究通过实证证据表明，用户/物品嵌入需要足够的维度才能实现多样化、公平且鲁棒的推荐。随后，我们从理论上分析了点积模型的表达能力。理论结果表明，在点积模型下可表达的可能排序数量受到物品因子维度的指数级限制。我们通过实验发现，低维性会导致流行度偏差，加剧流行物品与长尾物品排名位置之间的差距；同时，我们对此现象给出了理论解释。