Stable phase retrieval from locally stable and conditionally connected measurements

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 R^d the phase retrieval stability constant grows proportionally to d^1/2, while in 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.