Weak phase retrieval

Abstract

Phase retrieval and phaseless reconstruction for Hilbert space frames is a very active area of research. Recently, it was shown that these concepts are equivalent. In this paper, we make a detailed study of a weakening of these concepts to weak phase retrieval and weak phaseless reconstruction. We will give several necessary and/or sufficient conditions for frames to have these weak properties. We will prove three surprising results: (1) Weak phaseless reconstruction is equivalent to phaseless reconstruction. That is, it never was weak; (2) weak phase retrieval is not equivalent to weak phaseless reconstruction; (3) weak phase retrieval requires at least 2m − 2 vectors in an m-dimensional Hilbert space. We also give several examples illustrating the relationship between these concepts.