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.

本文研究泊松噪声模型中相位恢复问题，提出了一种基于 Frank-Wolfe 算法和核范数约束的凸规划算法，并证明了其全局收敛速度为 O (1/t)，算法对相位恢复问题具有更好的收敛保证。

Frank-Wolfe 算法处理非 Lipschitz 连续梯度目标：可伸缩泊松相位恢复

Frank-Wolfe Works for Non-Lipschitz Continuous Gradient Objectives:  Scalable Poisson Phase Retrieval

The observations in many applications consist of counts of discrete events,
such as photons hitting a detector, which cannot be effectively modeled using
an additive bounded or Gaussian noise model, and instead require a Poisson
noise model. As a result, accurate reconstruction of a spatially or temporally
distributed phenomenon (f*) from Poisson data (y) cannot be effectively
accomplished by minimizing a conventional penalized least-squares objective
function. The problem addressed in this paper is the estimation of f* from y in
an inverse problem setting, where (a) the number of unknowns may potentially be
larger than the number of observations and (b) f* admits a sparse
approximation. The optimization formulation considered in this paper uses a
penalized negative Poisson log-likelihood objective function with nonnegativity
constraints (since Poisson intensities are naturally nonnegative). In
particular, the proposed approach incorporates key ideas of using separable
quadratic approximations to the objective function at each iteration and
penalization terms related to l1 norms of coefficient vectors, total variation
seminorms, and partition-based multiscale estimation methods.

本文探讨了如何从 Poisson 数据中准确地重建分布式现象，运用了带惩罚项的负 Poisson 对数似然目标函数，并结合采用分离二次近似和 l1 范数、总变差半范数以及基于分区的多尺度估计方法等关键思想。