Deep Expectation-Consistent Approximation for Phase Retrieval

The expectation consistent (EC) approximation framework is a state-of-the-art approach for solving (generalized) linear inverse problems with high-dimensional random forward operators and i.i.d. signal priors. In image inverse problems, however, both the forward operator and image pixels are structured, which plagues traditional EC implementations. In this work, we propose a novel incarnation of EC that exploits deep neural networks to handle structured operators and signals. For phase-retrieval, we propose a simplified variant called “deepECpr” that reduces to iterative denoising. In experiments recovering natural images from phaseless, shot-noise corrupted, coded-diffraction-pattern measurements, we observe accuracy surpassing the state-of-the-art prDeep (Metzler et al., 2018) and Diffusion Posterior Sampling (Chung et al., 2023) approaches with two-orders-of-magnitude complexity reduction.