Linear convergence of randomized Kaczmarz method for solving complex-valued phaseless equations

A randomized Kaczmarz method was recently proposed for phase retrieval, which has been shown numerically to exhibit empirical performance over other state-of-the-art phase retrieval algorithms both in terms of the sampling complexity and in terms of computation time. While the rate of convergence has been studied well in the real case where the signals and measurement vectors are all real-valued, there is no guarantee for the convergence in the complex case. In fact, the linear convergence of the randomized Kaczmarz method for phase retrieval in the complex setting is left as a conjecture by Tan and Vershynin. In this paper, we provide the first theoretical guarantees for it. We show that for random measurements $\mathbf{a}_j \in \mathbb{C}^n, j=1,\ldots,m$ which are drawn independently and uniformly from the complex unit sphere, or equivalent are independent complex Gaussian random vectors, when $m \ge Cn$ for some universal positive constant $C$, the randomized Kaczmarz scheme with a good initialization converges linearly to the target solution (up to a global phase) in expectation with high probability. This gives a positive answer to that conjecture.