Iterative Collaborative Filtering for Sparse Matrix Estimation

We consider sparse matrix estimation where the goal is to estimate an n-by-n matrix from noisy observations of a small subset of its entries. We analyze the estimation error of the popularly used collaborative filtering algorithm for the sparse regime. Specifically, we propose a novel iterative variant of the algorithm, adapted to handle the setting of sparse observations. We establish that as long as the number of entries observed at random scale logarithmically larger than linear in n, the estimation error with respect to the entry-wise max norm decays to zero as n goes to infinity, assuming the underlying matrix of interest has constant rank r. Our result is robust to model misspecification in that if the underlying matrix is approximately rank r, then the estimation error decays to the approximation error with respect to the $\max$-norm. In the process, we establish the algorithm’s ability to handle arbitrary bounded noise in the observations.

Funding: This work was supported in part by Microsoft Research New England; the National Science Foundation’s Division of Computing and Communication Foundations [Grant 1948256], Division of Computer and Network Systems [Grant 1955997], and Division of Civil, Mechanical and Manufacturing Innovation [Grants 1462158 and 1634259]; and the Directorate for Computer and Information Science and Engineering [TRIPODS Phase I Project]. C. L. Yu is also supported by an Intel Rising Stars Award.