Stochastic Vector Approximate Message Passing with applications to phase retrieval

TL;DR

This paper introduces a stochastic version of the Vector Approximate Message Passing algorithm tailored for phase retrieval, demonstrating faster convergence in high-dimensional inverse problems involving multiple measurements.

Contribution

The paper proposes a novel stochastic VAMP algorithm for phase retrieval, enhancing convergence speed over existing methods.

Findings

Faster convergence of stochastic VAMP in phase retrieval tasks.

Effective handling of multiple sensing matrices in high-dimensional settings.

Improved computational efficiency for Bayesian inverse problems.

Abstract

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.