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Nonconvex matrix recovery is known to contain no spurious local minima undera restricted isometry property (RIP) with a sufficiently small RIP constant$\delta$. If $\delta$ is too large, however, then counterexamples containingspurious local minima are known to exist. In this paper, we introduce a prooftechnique that is capable of establishing sharp thresholds on $\delta$ toguarantee the inexistence of spurious local minima. Using the technique, weprove 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 anyarbitrary initial point (such as a random point). We also prove a localrecovery result: given an initial point $x_{0}$ satisfying$f(x_{0})\le(1-\delta)^{2}f(0)$, any descent algorithm that converges tosecond-order optimality guarantees exact recovery.

Sharp Restricted Isometry Bounds for the Inexistence of Spurious Local Minima in Nonconvex Matrix Recovery