提出分析降维相位恢复算法性能的新理论，发现样本量临界点存在相变现象。
Phase retrieval with rank $d$ measurements -- \emph{descending} algorithms phase transitions
- 基于随机对偶理论，建立降维相位恢复的分析框架
- 发现成功恢复所需最小测量数随维度呈相变行为
- 小规模实验验证理论预测高度一致，适用于真实/复数相位恢复
配套论文[118]发展了基于随机对偶理论（RDT）的分析方法，用于统计刻画降维相位恢复算法（dPR）的性能（包含梯度下降及广泛使用的Wirtinger流等变体）。本文推广该框架，应用于秩d的正定相位恢复（PR）测量（d=1和d=2分别对应实数与复数相位恢复的模拟）。特别地，我们发现确保dPR成功的最小样本复杂度比（测量数除以未知信号维度）表现出相变（PT）现象。针对原始和提升型RDT，我们确定了相变位置。为补充理论结果，我们实现了一种对数障碍梯度下降变体，在问题规模约100的小维度场景下，模拟得到的相变点与理论预测高度吻合。
原文摘要 · Abstract (English)
Companion paper [118] developed a powerful \emph{Random duality theory} (RDT) based analytical program to statistically characterize performance of \emph{descending} phase retrieval algorithms (dPR) (these include all variants of gradient descents and among them widely popular Wirtinger flows). We here generalize the program and show how it can be utilized to handle rank $d$ positive definite phase retrieval (PR) measurements (with special cases $d=1$ and $d=2$ serving as emulations of the real and complex phase retrievals, respectively). In particular, we observe that the minimal sample complexity ratio (number of measurements scaled by the dimension of the unknown signal) which ensures dPR's success exhibits a phase transition (PT) phenomenon. For both plain and lifted RDT we determine phase transitions locations. To complement theoretical results we implement a log barrier gradient descent variant and observe that, even in small dimensional scenarios (with problem sizes on the order of 100), the simulated phase transitions are in an excellent agreement with the theoretical predictions.
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