On the Fundamental Limits of Matrix Completion: Leveraging Hierarchical Similarity Graphs

TL;DR

This paper establishes the fundamental limits of matrix completion when using hierarchical social graph information, showing how leveraging such structure reduces the number of observations needed for accurate recovery.

Contribution

It provides the exact information-theoretic sample complexity bounds for matrix completion with hierarchical side information, demonstrating the benefits of exploiting social graph structures.

Findings

Hierarchical structure significantly reduces sample complexity.

Maximum likelihood estimator achieves vanishing error under sufficient conditions.

Theoretical bounds are corroborated by simulation results.

Abstract

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.