  We study the non-convex matrix factorization approach to matrix completion
via Riemannian geometry. Based on an optimization formulation over a
Grassmannian manifold, we characterize the landscape based on the notion of
principal angles between subspaces. For the fully observed case, our results
show that there is a region in which the cost is geodesically convex, and
outside of which all critical points are strictly saddle. We empirically study
the partially observed case based on our findings.
