Solving Systems of Phaseless Equations: Riemannian Optimization and Global Geometry
Jianfeng Cai Hong Kong University of Science and Technology Mathematics
We consider the problem of solving systems of phaseless equations ||^2=y_i, i=1,…,m and x in R^n is unknown. One application of great importance is the phase retrieval problem, which provides promising and indispensable tools in a wide spectrum of techniques including X-ray crystallography, diffraction imaging, microscopy, and quantum mechanics. (1) We will present a Riemannian gradient descent algorithm and a truncated variant. The algorithms are developed by exploiting the inherent low rank structure of the problem based on the embedded manifold of rank-1 positive semidefinite matrices. Under a random Gaussian model, theoretical recovery guarantee has been established for the truncated variant, showing that the algorithm is able to achieve successful recovery when the number of equations is proportional to the number of unknowns. (2) We will present the global geometry of a new non-convex objective function for solving the phaseless equations. The new objective function is constructed via a least squares fitting to the phaseless equations with an activation function. We prove that this new objective function with m=O(n) has a nice landscape --- there is no spurious local minima. Therefore, any algorithm finding a local minimum will give a solution of the phaseless equation.