Phase retrieval

TL;DR

This paper advances the understanding of phase retrieval by classifying conditions for phase retrieval using projections, determining minimal vector counts for frames, and analyzing subspace partitions, with numerous examples illustrating theoretical limits.

Contribution

It introduces new theorems classifying phase retrieval by projections, computes minimal vectors for frames to enable phase retrieval, and analyzes subspace partitions near the phase retrieval threshold.

Findings

Classifies phase retrieval by projections using sequences of vectors.

Determines the minimal number of vectors needed to enable phase retrieval.

Shows that near the threshold, partitions of orthonormal bases span hyperplanes.

Abstract

We answer a number of open problems concerning phase retrieval and phase retrieval by projections. In particular, one main theorem classifies phase retrieval by projections via collections of sequences of vectors allowing norm retrieval. Another key result computes the minimal number of vectors needed to add to a frame in order for it to possess the complement property and hence allow phase retrieval. In furthering this idea, in a third main theorem we show that when a collection of subspaces is one subspace short from allowing phase retrieval, then any partition of orthonormal bases from these subspaces into two sets which fail to span, then each spans a hyperplane. We offer many more results in this area as well as provide a large number of examples showing the limitations of the theory.