  Given a sufficiently large amount of labeled data, the non-convex low-rank
matrix recovery problem contains no spurious local minima, so a local
optimization algorithm is guaranteed to converge to a global minimum starting
from any initial guess. However, the actual amount of data needed by this
theoretical guarantee is very pessimistic, as it must prevent spurious local
minima from existing anywhere, including at adversarial locations. In contrast,
prior work based on good initial guesses have more realistic data requirements,
because they allow spurious local minima to exist outside of a neighborhood of
the solution. In this paper, we quantify the relationship between the quality
of the initial guess and the corresponding reduction in data requirements.
Using the restricted isometry constant as a surrogate for sample complexity, we
compute a sharp threshold number of samples needed to prevent each specific
point on the optimization landscape from becoming a spurious local minimum.
Optimizing the threshold over regions of the landscape, we see that for initial
points around the ground truth, a linear improvement in the quality of the
initial guess amounts to a constant factor improvement in the sample
complexity.
