We study a matrix completion problem which 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 hierarchical stochastic block model that well respects practically-relevant social graphs and follows a low-rank rating matrix model. 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. One important consequence of this result is that leveraging the hierarchical structure of similarity 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. Another implication of the result is when the graph information is rich, the optimal sample complexity is proportional to the number of clusters, while it nearly stays constant as the number of groups in a cluster increases. We empirically demonstrate through extensive experiments that the proposed algorithm achieves the optimal sample complexity.

In this paper, we design and analyze MC2G (Matrix Completion with 2 Graphs), an algorithm that performs matrix completion in the presence of social and item similarity graphs. MC2G runs in quasilinear time and is parameter free. It is based on spectral clustering and local refinement steps. For the matrix completion problem which possesses additional block structures in its rows and columns, we derive the expected number of sampled entries required for MC2G to succeed, and further show that it matches an information-theoretic lower bound up to a constant factor for a wide range of parameters. We perform extensive experiments on both synthetic datasets and a semi-real dataset inspired by real graphs. The experimental results show that MC2G outperforms other state-of-the-art matrix completion algorithms.