Solving Systems of Phaseless Equations via Riemannian Optimization with Optimal Sampling Complexity (Journal of Computational Mathematics, 2024)
A Riemannian gradient descent algorithm and a truncated variant are presented to solvesystems of phaseless equations $|Ax|^2=y.$ The algorithms are developed by exploiting theinherent low rank structure of the problem based on the embedded manifold of rank-1positive semidefinite matrices. Theoretical recovery guarantee has been established for thetruncated variant, showing that the algorithm is able to achieve successful recovery whenthe number of equations is proportional to the number of unknowns. Two key ingredients inthe analysis are the restricted well conditioned property and the restricted weak correlationproperty of the associated truncated linear operator. Empirical evaluations show that ouralgorithms are competitive with other state-of-the-art first order nonconvex approacheswith provable guarantees.
Keywords: Phaseless equations; Riemannian gradient descent; Manifold of rank-1 and positive semidefinite matrices; Optimal sampling complexity.