Phase retrieval stability via notions of graph connectivity

Phase retrieval is the inverse problem of reconstructing a signal from linear measurements, when the phase of the measurements is lost and only the magnitude is known. This problem occurs in many applications including crystallography, optics, and acoustics.

In the talk I will discuss results on invertibility and stability of phase retrieval. I will focus on a recent paradigm which characterizes phase retrieval stability via appropriate notions of graph connectivity, and in particular  our recent results relating real phase retrieval to the Cheeger constant, and complex phase retrieval to the spectral gap of the graph Laplacian. As corollaries we obtain sharp estimates for the dependence of the stability constant on the dimension of the ambient space, and examples of (in)stable signals in infinite dimensions.  

The talk is based on the paper

Stable Phase Retrieval from Locally Stable and Conditionally Connected Measurements

https://arxiv.org/abs/2006.11709