标题： 相位恢复中从强度测量的条件数 显示英文标题

标题： The Condition Number in Phase Retrieval from Intensity Measurements

摘要： 本文研究了通过分析非线性映射$\Psi_{\boldsymbol{A}}(\boldsymbol{x}) = \bigl(\lvert \langle {\boldsymbol{a}}_j, \boldsymbol{x} \rangle \rvert^2 \bigr)_{1 \le j \le m}$的条件数来实现相位恢复的稳定性，其中$\boldsymbol{a}_j \in \mathbb{H}^n$是已知的感知向量且满足$\mathbb{H} \in \{\mathbb{R}, \mathbb{C}\}$。 对于每个$p \ge 1$，我们定义条件数$\beta_{\Psi_{\boldsymbol{A}}}^{\ell_p}$为在度量$\mathrm {dist}_\mathbb{H}\left(\boldsymbol{x}, \boldsymbol{y}\right) = \|\boldsymbol{x} \boldsymbol{x}^\ast - \boldsymbol{y} \boldsymbol{y}^\ast\|_*$的情况下，$\Psi_{\boldsymbol{A}}$在$\ell_p$范数下的最优上 Lipschitz 常数与下 Lipschitz 常数的比值。 我们建立了对任何感知矩阵$\boldsymbol{A} \in \mathbb{H}^{m \times d}$的$\beta_{\Psi_{\boldsymbol{A}}}^{\ell_p}$的通用下界，证明了在实数情况下$\beta_{\Psi_{\boldsymbol{A}}}^{\ell_1} \ge \pi/2$和$\beta_{\Psi_{\boldsymbol{A}}}^{\ell_2} \ge \sqrt{3}$，以及在复数情况下$(\mathbb{H} = \mathbb{R})$和$\beta_{\Psi_{\boldsymbol{A}}}^{\ell_p} \ge 2$对$p=1,2$的$(\mathbb{H} = \mathbb{C})$。 这些界限被证明是渐近紧的：一个确定性调和框架$\boldsymbol{E}_m \in \mathbb{R}^{m \times 2}$和高斯随机矩阵$\boldsymbol{A} \in \mathbb{H}^{m \times d}$渐近地达到它们。值得注意的是，当$p=2$时，调和框架$\boldsymbol{E}_m \in \mathbb{R}^{m \times 2}$对所有$m \ge 3$达到最优下界$\sqrt{3}$，从而在$\boldsymbol{A} \in \mathbb{R}^{m \times 2}$内作为最优感知矩阵。 我们的结果提供了对$\beta_{\Psi_{\boldsymbol{A}}}^{\ell_p}$的第一个显式统一下界，并提供了关于相位恢复基本稳定性极限的见解。 显示英文摘要

摘要： This paper investigates the stability of phase retrieval by analyzing the condition number of the nonlinear map $\Psi_{\boldsymbol{A}}(\boldsymbol{x}) = \bigl(\lvert \langle {\boldsymbol{a}}_j, \boldsymbol{x} \rangle \rvert^2 \bigr)_{1 \le j \le m}$, where $\boldsymbol{a}_j \in \mathbb{H}^n$ are known sensing vectors with $\mathbb{H} \in \{\mathbb{R}, \mathbb{C}\}$. For each $p \ge 1$, we define the condition number $\beta_{\Psi_{\boldsymbol{A}}}^{\ell_p}$ as the ratio of optimal upper and lower Lipschitz constants of $\Psi_{\boldsymbol{A}}$ measured in the $\ell_p$ norm, with respect to the metric $\mathrm {dist}_\mathbb{H}\left(\boldsymbol{x}, \boldsymbol{y}\right) = \|\boldsymbol{x} \boldsymbol{x}^\ast - \boldsymbol{y} \boldsymbol{y}^\ast\|_*$. We establish universal lower bounds on $\beta_{\Psi_{\boldsymbol{A}}}^{\ell_p}$ for any sensing matrix $\boldsymbol{A} \in \mathbb{H}^{m \times d}$, proving that $\beta_{\Psi_{\boldsymbol{A}}}^{\ell_1} \ge \pi/2$ and $\beta_{\Psi_{\boldsymbol{A}}}^{\ell_2} \ge \sqrt{3}$ in the real case $(\mathbb{H} = \mathbb{R})$, and $\beta_{\Psi_{\boldsymbol{A}}}^{\ell_p} \ge 2$ for $p=1,2$ in the complex case $(\mathbb{H} = \mathbb{C})$. These bounds are shown to be asymptotically tight: both a deterministic harmonic frame $\boldsymbol{E}_m \in \mathbb{R}^{m \times 2}$ and Gaussian random matrices $\boldsymbol{A} \in \mathbb{H}^{m \times d}$ asymptotically attain them. Notably, the harmonic frame $\boldsymbol{E}_m \in \mathbb{R}^{m \times 2}$ achieves the optimal lower bound $\sqrt{3}$ for all $m \ge 3$ when $p=2$, thus serving as an optimal sensing matrix within $\boldsymbol{A} \in \mathbb{R}^{m \times 2}$. Our results provide the first explicit uniform lower bounds on $\beta_{\Psi_{\boldsymbol{A}}}^{\ell_p}$ and offer insights into the fundamental stability limits of phase retrieval.