Explicit Frames for Deterministic Phase Retrieval via PhaseLift

TL;DR

This paper presents explicit measurement frames and linear measurements that enable deterministic phase retrieval and low-rank matrix recovery in complex spaces, improving understanding of measurement design for phase retrieval.

Contribution

It provides explicit frames with minimal cardinality for phase retrieval and explicit linear measurements for low-rank matrix recovery, advancing deterministic measurement construction.

Findings

Explicit frame of size 5n-6 for phase retrieval.

Linear measurements with 4r(n-r)+n-2r outcomes for low-rank matrices.

Deterministic measurement schemes for phase retrieval and low-rank recovery.

Abstract

We explicitly give a frame of cardinality $5n-6$ such that every signal in $\mathbb{C}^n$ can be recovered up to a phase from its associated intensity measurements via the PhaseLift approach. Furthermore, we give explicit linear measurements with $4r(n-r)+n-2r$ outcomes that enable the recovery of every positive semidefinite $n\times n$ matrix of rank at most $r$.