On the Convergence of Non-Convex Phase Retrieval With Denoising Priors

Abstract

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, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i.e.</i> , 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.