Classifying weak phase retrieval

TL;DR

This paper thoroughly investigates weak phase retrieval, establishing key equivalences, resolving longstanding open problems, and delineating the limitations of frames capable of weak phase retrieval in real vector spaces.

Contribution

It provides a complete characterization of weak phase retrieval, proves non-density of certain frame families, and shows limitations when frames contain canonical basis vectors.

Findings

Weak phase retrieval frames are not dense in the set of all frames for m ≥ 2n-2.

Frames with canonical basis vectors cannot perform weak phase retrieval.

The results are optimal, with numerous examples illustrating the boundaries of weak phase retrieval.

Abstract

We will give several surprising equivalences and consequences of weak phase retrieval. These results give a complete understanding of the difference between weak phase retrieval and phase retrieval. We also answer two longstanding open problems on weak phase retrieval: (1) We show that the families of weak phase retrievable frames $\{x_{i}\}_{i=1}^{m}$ in $\mathbb{R}^n$ are not dense in the family of $m$-element sets of vectors in $\mathbb{R}^n$ for all $m\ge 2n-2$; (2) We show that any frame $\{x_i\}_{i=1}^{2n-2}$ containing one or more canonical basis vectors in $\mathbb{R}^n$ cannot do weak phase retrieval. We provide numerous examples to show that the obtained results are best possible.