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We prove that it is possible for nonconvex low-rank matrix recovery tocontain no spurious local minima when the rank of the unknown ground truth$r^{\star}<r$ is strictly less than the search rank $r$, and yet for the claimto be false when $r^{\star}=r$. Under the restricted isometry property (RIP),we prove, for the general overparameterized regime with $r^{\star}\le r$, thatan RIP constant of $\delta<1/(1+\sqrt{r^{\star}/r})$ is sufficient for theinexistence of spurious local minima, and that$\delta<1/(1+1/\sqrt{r-r^{\star}+1})$ is necessary due to existence ofcounterexamples. Without an explicit control over $r^{\star}\le r$, an RIPconstant of $\delta<1/2$ is both necessary and sufficient for the exactrecovery of a rank-$r$ ground truth. But if the ground truth is known a priorito have $r^{\star}=1$, then the sharp RIP threshold for exact recovery isimproved to $\delta<1/(1+1/\sqrt{r})$.

Sharp Global Guarantees for Nonconvex Low-Rank Matrix Recovery in the Overparameterized Regime