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A randomized Kaczmarz method was recently proposed for phase retrieval, whichhas been shown numerically to exhibit empirical performance over otherstate-of-the-art phase retrieval algorithms both in terms of the samplingcomplexity and in terms of computation time. While the rate of convergence hasbeen studied well in the real case where the signals and measurement vectorsare all real-valued, there is no guarantee for the convergence in the complexcase. In fact, the linear convergence of the randomized Kaczmarz method forphase retrieval in the complex setting is left as a conjecture by Tan andVershynin. 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 complexunit sphere, or equivalent are independent complex Gaussian random vectors,when $m \ge Cn$ for some universal positive constant $C$, the randomizedKaczmarz scheme with a good initialization converges linearly to the targetsolution (up to a global phase) in expectation with high probability. Thisgives a positive answer to that conjecture.

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