Nonconvex landscapes in phase retrieval and semidefinite low-rank matrix sensing with overparametrization

We study a nonconvex 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 factorized 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 overparametrize the problem, that is, to optimize over matrices of higher rank than the ground truth. We then apply this framework to several example problems: 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 for the well-known problems of (1) phase retrieval with sub-Gaussian measurements and (2) more general semidefinite matrix sensing with rank-1 Gaussian measurements. More generally, our analysis (optionally) uses the popular method of convex dual certificates, suggesting that our analysis could be applied to a much wider class of problems.