On the Fundamental Limits of Matrix Completion: Leveraging Hierarchical Similarity Graphs
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 practicallyrelevant social graphs and a lowrank rating matrix model, we characterize the exact informationtheoretic 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 genieaided maximum likelihood estimator. Under each necessary condition, we present examples of a genieaided 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 informationtheoretic limit can be asymptotically achieved.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.05408
 Bibcode:
 2021arXiv210905408A
 Keywords:

 Computer Science  Information Theory;
 Computer Science  Machine Learning;
 Statistics  Machine Learning
 EPrint:
 The first two authors contributed equally to this work. A preliminary version of this work was presented at the 2020 Advances in Neural Information Processing Systems Conference (NeurIPS 2020)