  Matrix completion is a basic machine learning problem that has wide
applications, especially in collaborative filtering and recommender systems.
Simple non-convex optimization algorithms are popular and effective in
practice. Despite recent progress in proving various non-convex algorithms
converge from a good initial point, it remains unclear why random or arbitrary
initialization suffices in practice. We prove that the commonly used non-convex
objective function for \textit{positive semidefinite} matrix completion has no
spurious local minima --- all local minima must also be global. Therefore, many
popular optimization algorithms such as (stochastic) gradient descent can
provably solve positive semidefinite matrix completion with \textit{arbitrary}
initialization in polynomial time. The result can be generalized to the setting
when the observed entries contain noise. We believe that our main proof
strategy can be useful for understanding geometric properties of other
statistical problems involving partial or noisy observations.
