It has been proved in Zhang et al. (2019) that the first-order optimality condition is satisfied.

Summary It has recently been proved that simple non-convex algorithms can recover a low-rank matrix from linear measurements, provided that the measurement operator satisfies a Restricted Isometry Property, with a good enough constant. Indeed, in this case, the natural objective function associated with the problem has no second-order critical point other than the solution. In this article, the authors ask how good the constant must be so that this property holds. They propose a convex program that finds measurement operators satisfying RIP, but for which the objective function has "bad" second-order critical points. They analyze this program in the case where the matrix to be recovered has rank 1, and deduce from this analysis that, for any x,z, there exists a RIP operator such that x is a bad second-order critical point when the true unknown is zz*; they upper bound the RIP constant.