Classifying 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.

arXiv admin note: substantial text overlap with arXiv:2110.06868