Phase Retrieval By Projections

TL;DR

This paper investigates the problem of phase retrieval using projections onto subspaces, providing characterizations of injective families, a construction for minimal projections needed, and discussing open questions in the field.

Contribution

It offers new characterizations of subspace families that enable injective measurements and proves phase retrieval is possible with 2M-1 projections of arbitrary rank in M-dimensional space.

Findings

Injective measurement conditions characterized for subspace families

Phase retrieval achievable with 2M-1 projections of arbitrary rank

Open problems identified for further research in subspace phase retrieval

Abstract

The problem of recovering a vector from the absolute values of its inner products against a family of measurement vectors has been well studied in mathematics and engineering. A generalization of this phase retrieval problem also exists in engineering: recovering a vector from measurements consisting of norms of its orthogonal projections onto a family of subspaces. There exist semidefinite programming algorithms to solve this problem, but much remains unknown for this more general case. Can families of subspaces for which such measurements are injective be completely classified? What is the minimal number of subspaces required to have injectivity? How closely does this problem compare to the usual phase retrieval problem with families of measurement vectors? In this paper, we answer or make incremental steps toward these questions. We provide several characterizations of subspaces which yield injective measurements, and through a concrete construction, we prove the surprising result that phase retrieval can be achieved with $2M-1$ projections of arbitrary rank in $\HH_M$. Finally we present several open problems as we discuss issues unique to the phase retrieval problem with subspaces.