  Accurately recovering images from phaseless measurements is a challenging and
long-standing problem. In this work, we present "deepECpr," which combines
expectation-consistent (EC) approximation with deep denoising networks to
surpass state-of-the-art phase-retrieval methods in both speed and accuracy. In
addition to applying EC in a non-traditional manner, deepECpr includes a novel
stochastic damping scheme that is inspired by recent diffusion methods. Like
existing phase-retrieval methods based on plug-and-play priors, regularization
by denoising, or diffusion, deepECpr iterates a denoising stage with a
measurement-exploitation stage. But unlike existing methods, deepECpr requires
far fewer denoiser calls. We compare deepECpr to the state-of-the-art prDeep
(Metzler et al., 2018), Deep-ITA (Wang et al., 2020), and Diffusion Posterior
Sampling (Chung et al., 2023) methods for noisy phase-retrieval of color,
natural, and unnatural grayscale images on oversampled-Fourier and
coded-diffraction-pattern measurements and find improvements in both PSNR and
SSIM with 5x fewer denoiser calls.
