实相位检索中的高斯临界阈值不稳定性
Gaussian Critical-Threshold Instability in Real Phase Retrieval
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中文总结 AI 辅助
本文证明实高斯相位检索在临界注入阈值处的不稳定性自然尺度，给出 Balan--Wang 稳定性参数的精确渐近行为，并揭示其与 Lipschitz 常数及最小奇异值的等价关系。
中文摘要 AI 辅助
我们确立了实高斯相位检索在临界注入阈值处不稳定性的自然尺度。设 $A$ 为具有独立标准高斯元素的 $(2M-1)\times M$ 矩阵，并设 $K_M=\binom{2M-1}{M}$。对于每个满足 $\log w_M=o(M)$ 的 $w_M\to\infty$，我们证明 $\mathbb{P}\{(w_M\sqrt{M}K_M)^{-1}\le\omega(A)\le w_M/(\sqrt{M}K_M)\}\to1$，其中 $\omega(A)$ 是 Balan--Wang 稳定性参数。因此，$-M^{-1}\log\omega(A)\to\log4$ 依概率成立。对于该阈值处的满支撑矩阵，$\omega(A)$ 同时等于 $x\mapsto|Ax|$ 的最优下 Lipschitz 常数以及所有方形行子矩阵的最小最小奇异值。上界来自对重叠子式的二阶矩分析。加权高斯逆尾渐近和逆 Wishart 集中性给出了中心重叠的渐近独立性；矩形硬边界控制其余重叠。匹配的下界来自联合界和方形高斯硬边估计。
英文摘要
We establish the natural scale of instability in real Gaussian phase retrieval at the critical injectivity threshold. Let $A$ be a $(2M-1)\times M$ matrix with independent standard Gaussian entries and let $K_M=\binom{2M-1}{M}$. For every $w_M\to\infty$ with $\log w_M=o(M)$, we prove that $\mathbb{P}\{(w_M\sqrt{M}K_M)^{-1}\leω(A)\le w_M/(\sqrt{M}K_M)\}\to1$, where $ω(A)$ is the Balan--Wang stability parameter. Consequently, $-M^{-1}\logω(A)\to\log4$ in probability. For full-spark matrices at this threshold, $ω(A)$ equals both the optimal lower Lipschitz constant of $x\mapsto|Ax|$ and the minimum least singular value over all square row submatrices. The upper bound follows from a second-moment analysis of overlapping minors. A weighted Gaussian inverse-tail asymptotic and inverse-Wishart concentration yield asymptotic independence for central overlaps; rectangular hard-edge bounds control the remaining overlaps. The matching lower bound follows from a union bound and a square Gaussian hard-edge estimate.
