  We study a phase retrieval problem in the Poisson noise model. Motivated by
the PhaseLift approach, we approximate the maximum-likelihood estimator by
solving a convex program with a nuclear norm constraint. While the Frank-Wolfe
algorithm, together with the Lanczos method, can efficiently deal with nuclear
norm constraints, our objective function does not have a Lipschitz continuous
gradient, and hence existing convergence guarantees for the Frank-Wolfe
algorithm do not apply. In this paper, we show that the Frank-Wolfe algorithm
works for the Poisson phase retrieval problem, and has a global convergence
rate of O(1/t), where t is the iteration counter. We provide rigorous
theoretical guarantee and illustrating numerical results.
