Projections and Phase retrieval

TL;DR

This paper investigates conditions under which a vector can be reconstructed from the magnitudes of its projections onto subspaces, establishing bounds on the number of subspaces needed for accurate phase retrieval in real vector spaces.

Contribution

It characterizes collections of orthogonal projections enabling vector reconstruction from magnitude data and determines sharp bounds for the number of subspaces required.

Findings

Reconstruction possible with N ≥ 2M - 1 subspaces in M-dimensional space.

Sharp bound established at N = 2^k + 1 for generic collections.

Answers key questions in phase retrieval literature.

Abstract

We characterize collections of orthogonal projections for which it is possible to reconstruct a vector from the magnitudes of the corresponding projections. As a result we are able to show that in an $M$-dimensional real vector space a vector can be reconstructed from the magnitudes of its projections onto a generic collection of $N \geq 2M-1$ subspaces. We also show that this bound is sharp when $N = 2^k +1$. The results of this paper answer a number of questions raised in \cite{CCPW:13}.