  Phase retrieval has been an attractive but difficult problem rising from
physical science, and there has been a gap between state-of-the-art theoretical
convergence analyses and the corresponding efficient retrieval methods.
Firstly, these analyses all assume that the sensing vectors and the iterative
updates are independent, which only fits the ideal model with infinite
measurements but not the reality, where data are limited and have to be reused.
Secondly, the empirical results of some efficient methods, such as the
randomized Kaczmarz method, show linear convergence, which is beyond existing
theoretical explanations considering its randomness and reuse of data. In this
work, we study for the first time, without the independence assumption, the
convergence behavior of the randomized Kaczmarz method for phase retrieval.
Specifically, beginning from taking expectation of the squared estimation error
with respect to the index of measurement by fixing the sensing vector and the
error in the previous step, we discard the independence assumption, rigorously
derive the upper and lower bounds of the reduction of the mean squared error,
and prove the linear convergence. This work fills the gap between a fast
converging algorithm and its theoretical understanding. The proposed
methodology may contribute to the study of other iterative algorithms for phase
retrieval and other problems in the broad area of signal processing and machine
learning.
