Phase retrieval by hyperplanes

TL;DR

This paper investigates the relationship between frames and hyperplanes in phase retrieval, showing conditions under which hyperplanes derived from frames do or do not enable phase retrieval, and providing specific examples and bounds.

Contribution

It establishes a connection between frames and hyperplanes for phase retrieval, identifies limitations, and provides explicit examples and bounds in various dimensions.

Findings

A scalable frame does phase retrieval iff its hyperplanes do, in general.

Counterexamples show the equivalence fails in some cases.

At least 2d-2 hyperplanes are needed for phase retrieval in R^d.

Abstract

We show that a scalable frame does phase retrieval if and only if the hyperplanes of its orthogonal complements do phase retrieval. We then show this result fails in general by giving an example of a frame for $\mathbb R^3$ which does phase retrieval but its induced hyperplanes fail phase retrieval. Moreover, we show that such frames always exist in $\mathbb R^d$ for any dimension $d$. We also give an example of a frame in $\mathbb R^3$ which fails phase retrieval but its perps do phase retrieval. We will also see that a family of hyperplanes doing phase retrieval in $\mathbb R^d$ must contain at least $2d-2$ hyperplanes. Finally, we provide an example of six hyperplanes in $\mathbb R^4$ which do phase retrieval.