  In this paper, we introduce a powerful technique based on Leave-one-out
analysis to the study of low-rank matrix completion problems. Using this
technique, we develop a general approach for obtaining fine-grained, entrywise
bounds for iterative stochastic procedures in the presence of probabilistic
dependency. We demonstrate the power of this approach in analyzing two of the
most important algorithms for matrix completion: (i) the non-convex approach
based on Projected Gradient Descent (PGD) for a rank-constrained formulation,
also known as the Singular Value Projection algorithm, and (ii) the convex
relaxation approach based on nuclear norm minimization (NNM).
  Using this approach, we establish the first convergence guarantee for the
original form of PGD without regularization or sample splitting}, and in
particular shows that it converges linearly in the infinity norm. For NNM, we
use this approach to study a fictitious iterative procedure that arises in the
dual analysis. Our results show that \NNM recovers an $ d $-by-$ d $ rank-$ r $
matrix with $\mathcal{O}(\mu r \log(\mu r) d \log d )$ observed entries. This
bound has optimal dependence on the matrix dimension and is independent of the
condition number. To the best of our knowledge, this is the first sample
complexity result for a tractable matrix completion algorithm that satisfies
these two properties simultaneously.
