In this paper, we study the stability of phase retrieval problems via a family of locally stable phase retrieval frame measurements in Banach spaces, which we call “locally stable and conditionally connected” (LSCC) measurement schemes. For any signal f in the Banach space, we associate it with a weighted graph $G_f$, defined by the LSCC measurement scheme, and show that the phase retrievability of the signal f is determined by the connectivity of $G_f$. We quantify the phase retrieval stability of the signal by two common measures of graph connectivity: The Cheeger constant for real-valued signals, and algebraic connectivity for complex-valued signals. We then use our results to study the stability of two phase retrieval models. In the first model, we study a finite-dimensional phase retrieval problem from locally supported measurements such as the windowed Fourier transform. We show that signals “without large holes” are phase retrievable, and that for such signals in ${\mathbb{R}}^d$ the phase retrieval stability constant grows proportionally to $d^{1/2}$, while in ${\mathbb{C}}^d$ it grows proportionally to d. The second model we consider is an infinite-dimensional phase retrieval problem in a shift-invariant space. In infinite-dimension spaces, even phase retrievable signals can have the Cheeger constant being zero, and hence have an infinite stability constant. We give an example of signals with monotone polynomial decay which has the Cheeger constant being zero, and an example with exponential decay which has a strictly positive Cheeger constant.

在本文中，我们通过 Banach 空间中的一系列局部稳定相位检索框架测量来研究相位检索问题的稳定性，我们将其称为“局部稳定和条件连接”（LSCC）测量方案。对于Banach 空间中的任何信号f，我们将其与一个加权图相关联$G_F$，由 LSCC 测量方案定义，并表明信号f的相位可恢复性由$G_F$. 我们通过两种常见的图连通性度量来量化信号的相位检索稳定性：实值信号的 Cheeger 常数和复值信号的代数连通性。然后我们使用我们的结果来研究两阶段检索模型的稳定性。在第一个模型中，我们从局部支持的测量（例如加窗傅立叶变换）研究有限维相位检索问题。我们证明了“没有大洞”的信号是相位可恢复的，对于这样的信号${\mathbb{电阻}}^d$ 相位恢复稳定常数成正比增长 $d^{1/2}$， 而在 ${\mathbb{C}}^d$它与d成比例增长。我们考虑的第二个模型是平移不变空间中的无限维相位检索问题。在无限维空间中，即使是相位可恢复信号也可以具有为零的 Cheeger 常数，因此具有无限的稳定常数。我们给出了一个单调多项式衰减信号的例子，其 Cheeger 常数为零，以及一个指数衰减的例子，其具有严格的正 Cheeger 常数。