改进相位恢复的鲁棒梯度下降法，应对重尾噪声和对抗性干扰。
Robust Gradient Descent for Phase Retrieval
- 引入鲁棒梯度下降提升Wirtinger Flow对重尾噪声与对抗污染的适应能力。
- 在输入输出均受扰动下，仍能实现信号稳定恢复，第四阶矩有界条件下表现良好。
- 提出预处理机制，使未知均值噪声场景可转化为可解形式，适用于真实复杂环境。
近期鲁棒统计学习进展主要集中在凸问题（如均值估计或线性回归），而非凸挑战关注较少。相位恢复是典型非凸问题，需从线性测量的幅值中恢复信号，缺乏相位信息。尽管已有多种非凸方法（尤其是Wirtinger Flow算法）适用于无噪声或轻度噪声场景，但在重尾噪声和对抗性污染下的解决方案仍是开放难题。本文研究一种利用鲁棒梯度下降技术改进Wirtinger Flow的方法，使其能同时应对第四阶矩有界的噪声和输入（协变量）及输出（响应）中的对抗性污染。我们考虑两种情形：已知零均值噪声与完全未知噪声。针对后者，提出一种预处理步骤，将问题转化为传统相位恢复不适用的新形式，但仍可通过适配后的零均值噪声算法求解。
原文摘要 · Abstract (English)
Recent progress in robust statistical learning has mainly tackled convex problems, like mean estimation or linear regression, with non-convex challenges receiving less attention. Phase retrieval exemplifies such a non-convex problem, requiring the recovery of a signal from only the magnitudes of its linear measurements, without phase (sign) information. While several non-convex methods, especially those involving the Wirtinger Flow algorithm, have been proposed for noiseless or mild noise settings, developing solutions for heavy-tailed noise and adversarial corruption remains an open challenge. In this paper, we investigate an approach that leverages robust gradient descent techniques to improve the Wirtinger Flow algorithm's ability to simultaneously cope with fourth moment bounded noise and adversarial contamination in both the inputs (covariates) and outputs (responses). We address two scenarios: known zero-mean noise and completely unknown noise. For the latter, we propose a preprocessing step that alters the problem into a new format that does not fit traditional phase retrieval approaches but can still be resolved with a tailored version of the algorithm for the zero-mean noise context.
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