On solving phase retrieval problems via neural networks: a unified framework with provable guarantees

Phase retrieval seeks to reconstruct a complex-valued signal using only intensity measurements, and it underpins applications such as X-ray crystallography, coherent diffraction imaging, and microscopy. Classical projection-based solvers remain indispensable, yet they are notoriously sensitive to initialization, frequently stall around local minima, and rarely come with rigorous convergence guarantees. Recent neural approaches offer expressive learned priors but too often sacrifice interpretability. We propose the proximal gradient unfolded network, a deep architecture obtained by interpreting every layer as one iteration of a proximal gradient method with learnable step sizes and a data-driven proximal operator. The resulting framework retains the transparency of classical algorithms while enabling powerful signal-adapted regularization. We establish new convergence theorems showing that, under realistic smoothness and contractivity assumptions, the unfolded network enjoys a provable linear convergence rate. Extensive experiments on handwritten digits, human faces, and macromolecular projections reveal that the proposed method consistently lowers the normalized mean squared error relative to both traditional iterative solvers and prior deep models. The combination of principled algorithmic structure, theoretical guarantees, and strong empirical evidence highlights the potential of optimization-aware neural architectures for challenging inverse problems.