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A Riemannian gradient descent algorithm and a truncated variant are presentedto solve systems of phaseless equations $|Ax|^2=y$. The algorithms aredeveloped by exploiting the inherent low rank structure of the problem based onthe embedded manifold of rank-$1$ positive semidefinite matrices. Theoreticalrecovery guarantee has been established for the truncated variant, showing thatthe algorithm is able to achieve successful recovery when the number ofequations is proportional to the number of unknowns. Two key ingredients in theanalysis are the restricted well conditioned property and the restricted weakcorrelation property of the associated truncated linear operator. Empiricalevaluations show that our algorithms are competitive with otherstate-of-the-art first order nonconvex approaches with provable guarantees.

Solving systems of phaseless equations via Riemannian optimization with optimal sampling complexity