几乎处处复相位检索的最小测量次数
The Minimum Number of Measurements for Almost-Everywhere Complex Phase Retrieval
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中文总结 AI 辅助
该研究确定，在$\mathbb C^d$中进行几乎处处复相位检索时，所需的最小测量次数为$2d$，且$2d-1$次测量无法实现几乎所有信号的唯一恢复。
中文摘要 AI 辅助
设$d\geq2$，$\bm{f}_1,\ldots,\bm{f}_m\in\mathbb C^d$。我们证明，若$m\leq2d-1$，则强度测量映射无法唯一恢复$\mathbb C^d$中几乎所有信号（不计全局相位因子）。结合已知的$2d$次测量的通用充分性，该结果确定$\mathbb C^d$中几乎处处相位检索所需的最小测量次数恰好为$2d$。
英文摘要
Let $d\geq 2$ and let $\bf{f}_1,\ldots,\bf{f}_m\in\mathbb C^d$. We prove that if $m\leq 2d-1$, then the intensity measurement map \[ \bf{x}\longmapsto \bigl( |\langle \bf{x},\bf{f}_1\rangle|^2, \ldots, |\langle \bf{x},\bf{f}_m\rangle|^2 \bigr) \] fails to recover almost every signal in $\mathbb C^d$ uniquely up to a global phase factor. Combined with the known generic sufficiency of $2d$ measurements, our result establishes that the minimum number of measurements required for almost-everywhere phase retrieval in $\mathbb C^d$ is exactly $2d$. This resolves an open problem in phase retrieval by determining the exact measurement threshold for almost-everywhere phase retrieval in ${\mathbb C}^d$.
