Asymptotic quadratic convergence of the Gauss-Newton method for complex phase retrieval
<|reference_start|>Asymptotic quadratic convergence of the Gauss-Newton method for complex phase retrieval: In this paper, we introduce a Gauss-Newton method for solving the complex phase retrieval problem. In contrast to the real-valued setting, the Gauss-Newton matrix for complex-valued signals is rank-deficient and, thus, non-invertible. To address this, we utilize a Gauss-Newton step that moves orthogonally to certain trivial directions. We establish that this modified Gauss-Newton step has a closed-form solution, which corresponds precisely to the minimal-norm solution of the associated least squares problem. Additionally, using the leave-one-out technique, we demonstrate that $m\ge O( n\log^3 n)$ independent complex Gaussian random measurements ensures that the entire trajectory of the Gauss-Newton iterations remains confined within a specific region of incoherence and contraction with high probability. This finding allows us to establish the asymptotic quadratic convergence rate of the Gauss-Newton method without the need of sample splitting.<|reference_end|>
@article{huang2024asymptotic,
    title={Asymptotic quadratic convergence of the Gauss-Newton method for complex
  phase retrieval},
    author={Meng Huang},
    journal={arXiv preprint arXiv:2406.09903},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.09903},
    primaryClass={math.NA cs.IT cs.NA math.IT}
}