This paper considers the problem of completing a rating matrix based on sub-sampled matrix entries as well as observed social graphs and hypergraphs. We show that there exists a \emph{sharp threshold} on the sample probability for the task of exactly completing the rating matrix -- the task is achievable when the sample probability is above the threshold, and is impossible otherwise -- demonstrating a phase transition phenomenon. The threshold can be expressed as a function of the ``quality'' of hypergraphs, enabling us to \emph{quantify} the amount of reduction in sample probability due to the exploitation of hypergraphs. This also highlights the usefulness of hypergraphs in the matrix completion problem. En route to discovering the sharp threshold, we develop a computationally efficient matrix completion algorithm that effectively exploits the observed graphs and hypergraphs. Theoretical analyses show that our algorithm succeeds with high probability as long as the sample probability exceeds the aforementioned threshold, and this theoretical result is further validated by synthetic experiments. Moreover, our experiments on a real social network dataset (with both graphs and hypergraphs) show that our algorithm outperforms other state-of-the-art matrix completion algorithms.

翻译：本文考虑基于子采样矩阵元素以及观测到的社交图和超图来完成评分矩阵的问题。我们证明了在精确完成评分矩阵的任务中存在一个关于采样概率的*尖锐阈值*——当采样概率高于该阈值时任务可实现，否则不可实现——这展示了一个相变现象。该阈值可表示为超图“质量”的函数，使我们能够*量化*利用超图所带来的采样概率降低幅度。这也凸显了超图在矩阵补全问题中的实用性。在发现尖锐阈值的过程中，我们开发了一种计算高效的矩阵补全算法，该算法能有效利用观测到的图和超图。理论分析表明，只要采样概率超过上述阈值，我们的算法就能以高概率成功，这一理论结果也得到了合成实验的验证。此外，在真实社交网络数据集（同时包含图和超图）上的实验表明，我们的算法优于其他现有最优的矩阵补全算法。