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Recent work established that rank overparameterization eliminates spuriouslocal minima in nonconvex low-rank matrix recovery under the restrictedisometry property (RIP). But this does not fully explain the practical successof overparameterization, because real algorithms can still become trapped atnonstrict saddle points (approximate second-order points with arbitrarily smallnegative curvature) even when all local minima are global. Moreover, the resultdoes not accommodate for noisy measurements, but it is unclear whether such anextension is even possible, in view of the many discontinuous and unintuitivebehaviors already known for the overparameterized regime. In this paper, weintroduce a novel proof technique that unifies, simplifies, and strengthens twopreviously competing approaches -- one based on escape directions and the otherbased on the inexistence of counterexample -- to provide sharp globalguarantees in the noisy overparameterized regime. We show, once local minimahave been converted into global minima through slight overparameterization,that near-second-order points achieve the same minimax-optimal recovery bounds(up to small constant factors) as significantly more expensive convexapproaches. Our results are sharp with respect to the noise level and thesolution accuracy, and hold for both the symmetric parameterization $XX^{T}$,as well as the asymmetric parameterization $UV^{T}$ under a balancingregularizer; we demonstrate that the balancing regularizer is indeed necessary.

Sharp Global Guarantees for Nonconvex Low-rank Recovery in the Noisy Overparameterized Regime