Frank-Wolfe Works for Non-Lipschitz Continuous Gradient Objectives:   Scalable Poisson Phase Retrieval
  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. 
Gergely Odor and Yen-Huan Li and Alp Yurtsever and Ya-Ping Hsieh and   Quoc Tran-Dinh and Marwa El Halabi and Volkan Cevher
math.OC stat.AP