A Novel Riemannian Optimization Approach and Algorithm for Solving the Phase Retrieval Problem

Several imaging applications require constructing the phase of a complex signal given observations of its amplitude. In most applications, a subset of phaseless measurements, say the discrete Fourier transform of the signal, form an orthonormal basis that can be exploited to speed up the recovery. This paper suggests a novel Riemannian optimization approach for solving the Fourier phase retrieval problem by studying and exploiting the geometry of the problem to reduce the ambient dimension and derive extremely fast and accurate algorithms. The phase retrieval problem is reformulated as a constrained problem and a novel Riemannian manifold, referred to as the fixed-norms manifold, is introduced to represent all feasible solutions. The first-order geometry of the Riemannian manifold is derived in closed-form which allows the design of highly efficient optimization algorithms. Numerical simulations indicate that the proposed approach outperforms conventional optimization-based methods both in accuracy and in convergence speed.