Stochastic Vector Approximate Message Passing with applications to phase retrieval

Phase retrieval refers to the problem of recovering a high-dimensional vector $\boldsymbol{x} \in \mathbb{C}^N$ from the magnitude of its linear transform $\boldsymbol{z} = A \boldsymbol{x}$, 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 $\boldsymbol{z}^{(1)} = A^{(1)} \boldsymbol{x},..., \boldsymbol{z}^{(L)} = A^{(L)}\boldsymbol{x}$ 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.