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.

翻译：最近为阶段检索建议了一个随机的卡兹马尔兹方法,这个方法在数字上展示了与其他最先进的阶段检索算法相比,在取样复杂程度和计算时间方面表现出经验性表现。虽然在信号和测量矢量都真实估价的真实情况下,对趋同率进行了很好的研究,但在复杂情况下无法保证趋同。事实上,随机的卡兹马尔兹方法在复杂环境下的阶段检索的线性趋同率被Tan和Vershynin作为一种猜测。在本文件中,我们为它提供了第一个理论保证。我们对随机测量 $\ mathb{a ⁇ j\ in\ mathb{C ⁇ n, j=1\\\\\\\\\\\\\\\n}n, j=1\\\ldot,m 美元与复杂的单位范围独立和统一,或相当的美元是独立的高斯随机矢量矢量,而对于某种普遍正值不变的美元,则以千元计算为单位。在随机化的卡兹马兹马兹计划中,我们提供了第一个理论保证。我们展示了它的理论保证。我们对随机性准的初始测量的概率接近目标的答案。