Matrix Completion with Hypergraphs:Sharp Thresholds and Efficient Algorithms
This addresses matrix completion for recommendation systems or social networks by leveraging hypergraphs, offering a theoretical and practical improvement over existing methods, though it is incremental in combining hypergraphs with known matrix completion frameworks.
The paper tackles the problem of completing a rating matrix using sub-sampled entries and observed social graphs and hypergraphs, showing a sharp threshold for exact completion with a phase transition phenomenon and quantifying sample reduction due to hypergraphs, while developing an efficient algorithm that outperforms state-of-the-art methods in real-world experiments.
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.