Stochastic Vector Approximate Message Passing with Applications to Phase Retrieval

Phase retrieval refers to the problem of recovering a high-dimensional vector ${\mathbf{x}} \in {\mathbb{C}^N}$ from the magnitude of its linear transform z = Ax, observed through a noisy channel. To improve the ill-posed nature of the inverse problem, it is a common practice to observe the magnitude of linear measurements z(1) = A(1)x,…, z(L) = A(L)x using multiple sensing matrices A(1),…, A(L), with ptychographic imaging being a remarkable example of such strategies. Inspired by existing algorithms for ptychographic reconstruction, we introduce stochasticity to Vector Approximate Message Passing (VAMP), a computationally efficient algorithm applicable to a wide range of Bayesian inverse problems. By testing our approach in the setup of phase retrieval, we show the superior convergence speed of the proposed algorithm.