Phase retrieval and matrix sensing via benign and overparametrized nonconvex optimization

We study a nonconvex optimization algorithmic approach to phase retrieval and the more general problem of semidefinite low-rank matrix sensing. Specifically, we analyze the nonconvex landscape of a quartic Burer-Monteiro factored least-squares optimization problem. We develop a new analysis framework, taking advantage of the semidefinite problem structure, to understand the properties of second-order critical points -- specifically, whether they (approximately) recover the ground truth matrix. We show that it can be helpful to (mildly) overparametrize the problem, that is, to optimize over matrices of higher rank than the ground truth. We then apply this framework to several well-studied problem instances: in addition to recovering existing state-of-the-art phase retrieval landscape guarantees (without overparametrization), we show that overparametrizing by a factor at most logarithmic in the dimension allows recovery with optimal statistical sample complexity and error for the problems of (1) phase retrieval with sub-Gaussian measurements and (2) more general semidefinite matrix sensing with rank-1 Gaussian measurements. Previously, such statistical results had been shown only for estimators based on semidefinite programming. More generally, our analysis is partially based on the powerful method of convex dual certificates, suggesting that it could be applied to a much wider class of problems.