Denoising priors have achieved empirical success in solving non-convex phase retrieval but still lack convergence guarantees in theory. In this paper, we provide a novel insight on the convergence guarantees for a general class of non-convex phase retrieval with denoising priors. Specifically, we propose a Wirtinger flow based framework, named DWF, that allows monotone and contractive iterative optimization. We demonstrate in theory that, under a milder demi-Lipschitz condition on denoisers, the proposed DWF framework converges to the actual signal (up to a global sign) at a geometric rate. The derived convergence guarantees and rates are general to accommodate to complex-valued and real-valued signals in the presence of noise perturbation. Furthermore, we exemplify the proposed framework with two prevailing denoising priors, i.e. , plug-and-play priors (PnP) and regularization by denoising (RED), including specific conditions on denoisers and convergence rates. To our best knowledge, this paper is the first attempt to provide theoretical guarantees of convergence for non-convex phase retrieval with denoising priors. Numerical evaluations demonstrate the theoretical results for analytic denoiser like arithmetic mean filter and deep learning based denoiser DnCNN. Furthermore, extensive experiments under the Gaussian model and coded diffraction pattern show that the proposed framework outperforms existing denoising prior-based methods and evidently reduces the necessary sampling rate for stable reconstruction with a guarantee of convergence in theory.

中文翻译：

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非凸相位检索与去噪先验的收敛性

去噪先验在解决非凸相位检索方面取得了经验上的成功，但在理论上仍然缺乏收敛保证。在本文中，我们提供了关于具有去噪先验的一般类非凸相位检索的收敛保证的新见解。具体来说，我们提出了一个基于 Wirtinger 流的框架，名为 DWF，它允许单调和收缩迭代优化。我们在理论上证明，在降噪器上较温和的半李普希茨条件下，所提出的 DWF 框架以几何速率收敛到实际信号（直至全局符号）。在存在噪声扰动的情况下，导出的收敛保证和速率通常可以适应复值和实值信号。此外，我们用两个流行的去噪先验来举例说明所提出的框架，即，即插即用先验（PnP）和去噪正则化（RED），包括去噪器和收敛速度的具体条件。据我们所知，本文首次尝试为具有去噪先验的非凸相位检索提供收敛的理论保证。数值评估证明了分析降噪器（如算术平均滤波器）和基于深度学习的降噪器 DnCNN 的理论结果。此外，在高斯模型和编码衍射图案下的大量实验表明，所提出的框架优于现有的基于去噪先验的方法，并且在理论上保证收敛的情况下明显降低了稳定重建所需的采样率。