标题： 鲁棒稀疏相位检索：统计保证、最优性理论与收敛算法 显示英文标题

标题： Robust Sparse Phase Retrieval: Statistical Guarantee, Optimality Theory and Convergent Algorithm

摘要： 相位检索（PR）是信号处理和机器学习中的一个热门研究课题。然而，当测量数据受到噪声或异常值的污染时，其性能会显著下降。为了解决这一局限性，我们提出了一种新颖的鲁棒稀疏相位检索方法，涵盖了实值和复值两种情况。该方法的核心是利用Huber函数来衡量损失，并采用$\ell_{1/2}$-范数正则化以实现特征选择，从而提高PR的鲁棒性。从理论上讲，我们建立了这种鲁棒性的统计保证，并推导出全局极小值的必要最优条件。特别是对于复值情况，我们基于Wirtinger导数提供了一个不动点包含性质。此外，我们通过将梯度下降法整合到重大化最小化（MM）框架中，开发出一种高效的优化算法。严格证明了在温和条件下，整个生成序列不仅收敛，而且具有线性收敛速度，这在以前的研究中尚未被探讨过。在不同类型的噪声下的数值例子验证了我们所提出方法的鲁棒性和有效性。 显示英文摘要

摘要： Phase retrieval (PR) is a popular research topic in signal processing and machine learning. However, its performance degrades significantly when the measurements are corrupted by noise or outliers. To address this limitation, we propose a novel robust sparse PR method that covers both real- and complex-valued cases. The core is to leverage the Huber function to measure the loss and adopt the $\ell_{1/2}$-norm regularization to realize feature selection, thereby improving the robustness of PR. In theory, we establish statistical guarantees for such robustness and derive necessary optimality conditions for global minimizers. Particularly, for the complex-valued case, we provide a fixed point inclusion property inspired by Wirtinger derivatives. Furthermore, we develop an efficient optimization algorithm by integrating the gradient descent method into a majorization-minimization (MM) framework. It is rigorously proved that the whole generated sequence is convergent and also has a linear convergence rate under mild conditions, which has not been investigated before. Numerical examples under different types of noise validate the robustness and effectiveness of our proposed method.