论文标题
从本地稳定和有条件连接的测量结果稳定的相位检索
Stable Phase Retrieval from Locally Stable and Conditionally Connected Measurements
论文作者
论文摘要
本文与我们命名为“本地稳定且有条件连接”（LSCC）测量方案的相位检索模型的稳定相位检索有关。对于每个信号$ f $，我们将由LSCC测量方案定义的相应加权图$ g_f $关联，并表明信号$ f $的相位可检索性取决于$ g_f $的连接性。然后，我们通过两个度量来表征信号$ f $的相位检索稳定性，这些度量通常在图理论中用于量化图形连接性：真实有价值信号的$ g_f $的cheeger常数，以及用于复杂价值信号的$ g_f $的代数连接性。 我们使用结果来研究可以作为LSCC测量方案施加的两个相检索模型的稳定性，并专注于理解哪些信号可以避免“维度的诅咒”。我们讨论的第一个模型是用于本地支持的测量值的有限维模型，例如窗口傅立叶变换。对于“没有大孔”的信号，我们显示稳定性常数仅在维度中显示出轻度的多项式生长，与统一稳定性常数倾向于遭受的指数增长形成鲜明对比；更准确地说，在$ r^d $中，常数成比例地生长到$ d^{1/2} $，而在$ c^d $中，它的生长成比例地生长到$ d $。我们还不能降低复杂情况下常数的生长，这表明复杂的相位检索比实际相检索要困难得多。我们认为的第二个模型是在主偏移不变空间中的无限维相检索问题。我们表明，尽管该模型具有无限的维度，但单调指数衰减的信号将具有有限的稳定性常数。相反，如果信号的衰减是多项式，我们结果提供的稳定性将是无限的。
This paper is concerned with stable phase retrieval for a family of phase retrieval models we name "locally stable and conditionally connected" (LSCC) measurement schemes. For every signal $f$, we associate a corresponding 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 then characterize the phase retrieval stability of the signal $f$ by two measures that are commonly used in graph theory to quantify graph connectivity: the Cheeger constant of $G_f$ for real valued signals, and the algebraic connectivity of $G_f$ for complex valued signals. We use our results to study the stability of two phase retrieval models that can be cast as LSCC measurement schemes, and focus on understanding for which signals the "curse of dimensionality" can be avoided. The first model we discuss is a finite-dimensional model for locally supported measurements such as the windowed Fourier transform. For signals "without large holes", we show the stability constant exhibits only a mild polynomial growth in the dimension, in stark contrast with the exponential growth which uniform stability constants tend to suffer from; more precisely, in $R^d$ the constant grows proportionally to $d^{1/2}$, while in $C^d$ it grows proportionally to $d$. We also show the growth of the constant in the complex case cannot be reduced, suggesting that complex phase retrieval is substantially more difficult than real phase retrieval. The second model we consider is an infinite-dimensional phase retrieval problem in a principal shift invariant space. We show that despite the infinite dimensionality of this model, signals with monotone exponential decay will have a finite stability constant. In contrast, the stability bound provided by our results will be infinite if the signal's decay is polynomial.
