Weak phase retrieval and phaseless reconstruction

TL;DR

This paper investigates weakened versions of phase retrieval and phaseless reconstruction in Hilbert spaces, revealing surprising equivalences and necessary conditions for weak phase retrieval.

Contribution

It introduces the concepts of weak phase retrieval and weak phaseless reconstruction, establishing their properties and relationships, including key equivalences and minimal vector requirements.

Findings

Weak phaseless reconstruction is equivalent to phaseless reconstruction.

Weak phase retrieval is not equivalent to weak phaseless reconstruction.

Weak phase retrieval requires at least 2m-2 vectors in an m-dimensional space.

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. I.e. 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 gives several examples illustrating the relationship between these concepts.