We study the matrix completion problem that leverages hierarchical similarity graphs as side information in the context of recommender systems. Under a hierarchical stochastic block model that well respects practically-relevant social graphs and a low-rank rating matrix model, we characterize the exact information-theoretic limit on the number of observed matrix entries (i.e., optimal sample complexity) by proving sharp upper and lower bounds on the sample complexity. In the achievability proof, we demonstrate that probability of error of the maximum likelihood estimator vanishes for sufficiently large number of users and items, if all sufficient conditions are satisfied. On the other hand, the converse (impossibility) proof is based on the genie-aided maximum likelihood estimator. Under each necessary condition, we present examples of a genie-aided estimator to prove that the probability of error does not vanish for sufficiently large number of users and items. One important consequence of this result is that exploiting the hierarchical structure of social graphs yields a substantial gain in sample complexity relative to the one that simply identifies different groups without resorting to the relational structure across them. More specifically, we analyze the optimal sample complexity and identify different regimes whose characteristics rely on quality metrics of side information of the hierarchical similarity graph. Finally, we present simulation results to corroborate our theoretical findings and show that the characterized information-theoretic limit can be asymptotically achieved.

中文翻译：

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关于矩阵完成的基本限制：利用层次相似图

我们研究了在推荐系统的上下文中利用分层相似图作为辅助信息的矩阵完成问题。在充分尊重实际相关社交图和低秩评级矩阵模型的分层随机块模型下，我们通过证明尖锐的上限和样本复杂度的下限。在可实现性证明中，我们证明，如果满足所有充分条件，最大似然估计器的错误概率对于足够多的用户和项目会消失。另一方面，逆（不可能）证明是基于精灵辅助的最大似然估计器。在每个必要条件下，我们提供了精灵辅助估计器的示例，以证明对于足够多的用户和项目，错误概率不会消失。这一结果的一个重要结果是，相对于简单地识别不同群体而不诉诸于它们之间的关系结构的样本复杂性，利用社交图的层次结构会产生大量的样本复杂性。更具体地说，我们分析了最佳样本复杂度，并确定了不同的制度，其特征依赖于分层相似图的辅助信息的质量度量。最后，我们展示了模拟结果以证实我们的理论发现，并表明可以渐近地实现特征信息理论极限。