Recent work established that rank overparameterization eliminates spurious local minima in nonconvex low-rank matrix recovery under the restricted isometry property (RIP). But this does not fully explain the practical success of overparameterization, because real algorithms can still become trapped at nonstrict saddle points (approximate second-order points with arbitrarily small negative curvature) even when all local minima are global. Moreover, the result does not accommodate for noisy measurements, but it is unclear whether such an extension is even possible, in view of the many discontinuous and unintuitive behaviors already known for the overparameterized regime. In this paper, we introduce a novel proof technique that unifies, simplifies, and strengthens two previously competing approaches -- one based on escape directions and the other based on the inexistence of counterexample -- to provide sharp global guarantees in the noisy overparameterized regime. We show, once local minima have 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 convex approaches. Our results are sharp with respect to the noise level and the solution accuracy, and hold for both the symmetric parameterization $XX^{T}$, as well as the asymmetric parameterization $UV^{T}$ under a balancing regularizer; we demonstrate that the balancing regularizer is indeed necessary.

翻译：近期研究表明，在限制等距性质（RIP）条件下，秩过参数化能够消除非凸低秩矩阵恢复中的伪局部极小值。但这并不能完全解释过参数化在实际应用中的成功，因为即使所有局部极小值都是全局极小值，实际算法仍可能被困在非严格鞍点（具有任意小负曲率的近似二阶点）处。此外，该结果未考虑含噪声的测量情况，但鉴于过参数化机制中已知存在的诸多不连续且反直觉的行为，尚不清楚此类扩展是否可能实现。本文提出了一种新颖的证明技术，统一、简化并强化了先前两种相互竞争的方法——一种基于逃逸方向，另一种基于反例不存在性——从而为噪声过参数化机制提供了尖锐的全局保证。我们证明，当局部极小值通过轻微过参数化转化为全局极小值后，近二阶点能够达到与成本显著更高的凸方法相同的最优极小极大恢复界（相差仅小常数因子）。我们的结果在噪声水平和求解精度方面具有尖锐性，并同时适用于对称参数化$XX^{T}$以及平衡正则化下的非对称参数化$UV^{T}$；我们论证了平衡正则化确实具有必要性。