Nonconvex matrix recovery is known to contain no spurious local minima under a restricted isometry property (RIP) with a sufficiently small RIP constant $delta$. If $delta$ is too large, however, then counterexamples containing spurious local minima are known to exist. In this paper, we introduce a proof technique that is capable of establishing sharp thresholds on $delta$ to guarantee the inexistence of spurious local minima. Using the technique, we prove that in the case of a rank-1 ground truth, an RIP constant of $delta<1/2$ is both necessary and sufficient for exact recovery from any arbitrary initial point (such as a random point). We also prove a local recovery result: given an initial point $x_{0}$ satisfying $f(x_{0})le(1-delta)^{2}f(0)$, any descent algorithm that converges to second-order optimality guarantees exact recovery.
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