Phase retrieval with prior information can be cast as a nonsmooth and nonconvex optimization problem. We solve the problem by graph projection splitting (GPS), where the two proximity subproblems and the graph projection step can be solved efficiently. With slight modification, we also propose a robust graph projection splitting (RGPS) method to stabilize the iteration for noisy measurements. Contrary to intuition, RGPS outperforms GPS with fewer iterations to locate a satisfying solution even for noiseless case. Based on the connection between GPS and Douglas-Rachford iteration, under mild conditions on the sampling vectors, we analyze the fixed point sets and provide the local convergence of GPS and RGPS applied to noiseless phase retrieval without prior information. For noisy case, we provide the error bound of the reconstruction. Compared to other existing methods, thanks for the splitting approach, GPS and RGPS can efficiently solve phase retrieval with prior information regularization for general sampling vectors which are not necessarily isometric. For Gaussian phase retrieval, compared to existing gradient flow approaches, numerical results show that GPS and RGPS are much less sensitive to the initialization. Thus they markedly improve the phase transition in noiseless case and reconstruction in the presence of noise respectively. GPS shows sharpest phase transition among existing methods including RGPS, while it needs more iterations than RGPS when the number of measurement is large enough. RGPS outperforms GPS in terms of stability for noisy measurements. When applying RGPS to more general non-Gaussian measurements with prior information, such as support, sparsity and TV minimization, RGPS either outperforms state-of-the-art solvers or can be combined with state-of-the-art solvers to improve their reconstruction quality.

中文翻译：

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通过图投影分割解决相位检索

具有先验信息的相位检索可以被视为非光滑和非凸优化问题。我们通过图投影分裂 (GPS) 来解决该问题，其中两个邻近子问题和图投影步骤可以有效地解决。通过稍加修改，我们还提出了一种稳健的图投影分割（RGPS）方法来稳定噪声测量的迭代。与直觉相反，RGPS 的性能优于 GPS，即使在无噪声情况下，迭代次数也更少，可以找到令人满意的解决方案。基于 GPS 和 Douglas-Rachford 迭代之间的联系，在采样向量的温和条件下，我们分析了不动点集，并提供了 GPS 和 RGPS 的局部收敛，用于无先验信息的无噪声相位检索。对于嘈杂的情况，我们提供了重建的误差界限。与其他现有方法相比，由于采用了分裂方法，GPS 和 RGPS 可以通过先验信息正则化有效地解决相位检索问题，用于不一定是等距的一般采样向量。对于高斯相位检索，与现有的梯度流方法相比，数值结果表明 GPS 和 RGPS 对初始化的敏感度要低得多。因此，它们分别显着改善了无噪声情况下的相变和存在噪声时的重建。GPS 在包括 RGPS 在内的现有方法中显示出最尖锐的相变，而当测量数量足够大时，它需要比 RGPS 更多的迭代。RGPS 在噪声测量的稳定性方面优于 GPS。当将 RGPS 应用于具有先验信息（例如支持）的更一般的非高斯测量时，