Matrix Completion with Hierarchical Graph Side Information
This work addresses matrix completion problems in domains like social networks or recommendation systems by leveraging hierarchical graph structures, offering a substantial gain in sample complexity compared to non-hierarchical approaches.
The paper tackles matrix completion using hierarchical graph side information, developing a parameter-free algorithm that achieves optimal sample complexity under a hierarchical stochastic block model and low-rank rating matrix, with experiments showing significant performance improvements over existing methods.
We consider a matrix completion problem that exploits social or item similarity graphs as side information. We develop a universal, parameter-free, and computationally efficient algorithm that starts with hierarchical graph clustering and then iteratively refines estimates both on graph clustering and matrix ratings. Under a hierarchical stochastic block model that well respects practically-relevant social graphs and a low-rank rating matrix model (to be detailed), we demonstrate that our algorithm achieves the information-theoretic limit on the number of observed matrix entries (i.e., optimal sample complexity) that is derived by maximum likelihood estimation together with a lower-bound impossibility result. One 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. We conduct extensive experiments both on synthetic and real-world datasets to corroborate our theoretical results as well as to demonstrate significant performance improvements over other matrix completion algorithms that leverage graph side information.