AI 中文总结
本文针对编码衍射图案的结构化相位恢复问题，证明了PhaseLift可行性规划在标准随机掩模模型下达到最优掩模复杂度与总采样率，证明采用结合自适应掩模分配和维度无关截断阈值的改进高尔夫方案构造近似对偶证书。
AI 中文摘要
从编码衍射图案（即通过一组掩模调制信号后得到的傅里叶强度）中恢复复值信号，是衍射成像及相关应用中出现的一类基础结构化相位恢复问题。尽管该问题具有重要的实际意义，但针对这一结构化框架的理论分析仍然较为匮乏。在标准随机掩模模型下，可通过计算可行的恢复方法达到的最优采样率一直是悬而未决的问题。\n 本文确立了PhaseLift可行性规划的最优采样率。更确切地说，PhaseLift可从$\u27e8O(\u27e8log n)$个随机掩模中精确恢复未知信号$\u27e8pmb{x}_0\u27e8in\u27e8mathbb{C}^n$（仅存在全局相位差异），且失败概率呈多项式级衰减。由于在擦除掩模集合下，识别特定信号必须用到$Ω(\u27e8log n)$个掩模，因此我们的结果达到了最优的掩模复杂度。等价地，PhaseLift达到了$m=\u27e8O( n\u27e8log n)$个标量强度测量的最优总采样率。其证明基于通过改进的高尔夫方案构造的近似对偶证书，该方案结合了自适应掩模分配与维度无关的截断阈值。
英文摘要
Recovering a complex-valued signal from coded diffraction patterns, namely the Fourier intensities obtained after modulating the signal with a collection of masks, is a fundamental structured phase retrieval problem arising in diffraction imaging and related applications. Despite its practical importance, the theoretical analysis of this structured framework remains scarce. In the standard random mask model, the optimal sampling rate achievable by computationally tractable recovery methods has remained open. In this paper, we establish the optimal sampling rate for the PhaseLift feasibility program. More precisely, PhaseLift achieves exact recovery of an unknown signal $\pmb{x}_0\in\mathbb{C}^n$, up to a global phase, from $\mathcal{O}(\log n)$ random masks, with polynomially decaying failure probability. Since $Ω(\log n)$ masks are necessary to identify certain signals under the erasure mask ensemble, our result thereby achieves the optimal mask complexity. Equivalently, PhaseLift attains the optimal total sampling rate of $m=\mathcal{O}( n\log n)$ scalar intensity measurements. The proof is based on an approximate dual certificate construction via a refined golfing scheme that combines adaptive mask allocation with a dimension-independent truncation threshold.
Comments32 pages, 4 figures
