  We consider sparse matrix estimation where the goal is to estimate an
$n\times n$ matrix from noisy observations of a small subset of its entries. We
analyze the estimation error of the popularly utilized 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 fraction of entries observed at random scale
as $\frac{\log^{1+\kappa}(n)}{n}$ for any fixed $\kappa > 0$, the estimation
error with respect to the $\max$-norm decays to $0$ as $n\to\infty$ assuming
the underlying matrix of interest has constant rank $r$. Our result is robust
to model mis-specification in that if the underlying matrix is approximately
rank $r$, then the estimation error decays to the approximate error with
respect to the $\max$-norm. In the process, we establish algorithm's ability to
handle arbitrary bounded noise in the observations.
