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The Optimal Sample Complexity of Matrix Completion with Hierarchical Similarity Graphs

We study a matrix completion problem that leverages a hierarchical structure of social similarity graphs as side information in the context of recommender systems. We assume that users are categorized into clusters, each of which comprises sub-clusters (or what we call “groups”). We consider a low-rank matrix model for the rating matrix, and a hierarchical stochastic block model that well respects practically-relevant social graphs. Under this setting, we characterize the information-theoretic limit on the number of observed matrix entries (i.e., optimal sample complexity) as a function of the quality of graph side information (to be detailed) by proving sharp upper and lower bounds on the sample complexity. Furthermore, we develop a matrix completion algorithm and empirically demonstrate via extensive experiments that the proposed algorithm achieves the optimal sample complexity.