  Matrix completion is the problem of recovering a low rank matrix by observing
a small fraction of its entries. A series of recent works [KOM12,JNS13,HW14]
have proposed fast non-convex optimization based iterative algorithms to solve
this problem. However, the sample complexity in all these results is
sub-optimal in its dependence on the rank, condition number and the desired
accuracy.
  In this paper, we present a fast iterative algorithm that solves the matrix
completion problem by observing $O(nr^5 \log^3 n)$ entries, which is
independent of the condition number and the desired accuracy. The run time of
our algorithm is $O(nr^7\log^3 n\log 1/\epsilon)$ which is near linear in the
dimension of the matrix. To the best of our knowledge, this is the first near
linear time algorithm for exact matrix completion with finite sample complexity
(i.e. independent of $\epsilon$).
  Our algorithm is based on a well known projected gradient descent method,
where the projection is onto the (non-convex) set of low rank matrices. There
are two key ideas in our result: 1) our argument is based on a $\ell_{\infty}$
norm potential function (as opposed to the spectral norm) and provides a novel
way to obtain perturbation bounds for it. 2) we prove and use a natural
extension of the Davis-Kahan theorem to obtain perturbation bounds on the best
low rank approximation of matrices with good eigen-gap. Both of these ideas may
be of independent interest.
