  In this paper, we investigate phase retrieval algorithm for the single
particle X-ray imaging data. We present a variance-reduced randomized Kaczmarz
(VR-RK) algorithm for phase retrieval. The VR-RK algorithm is inspired by the
randomized Kaczmarz method and the Stochastic Variance Reduce Gradient Descent
(SVRG) algorithm. Numerical experiments show that the VR-RK algorithm has a
faster convergence rate than randomized Kaczmarz algorithm and the iterative
projection phase retrieval methods, such as the hybrid input output (HIO) and
the relaxed averaged alternating reflections (RAAR) methods. The VR-RK
algorithm can recover the phases with higher accuracy, and is robust at the
presence of noise. Experimental results on the scattering data from individual
particles show that the VR-RK algorithm can recover phases and improve the
single particle image identification.
