Uniform Exact Reconstruction of Sparse Signals and Low-Rank Matrices From Phase-Only Measurements

Junren Chen*, Michael K. Ng

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

2 Citations (Scopus)

Abstract

In phase-only compressive sensing (PO-CS), our goal is to recover low-complexity signals (e.g., sparse signals, low-rank matrices) from the phase of complex linear measurements. While perfect recovery of signal direction in PO-CS was observed quite early, the exact reconstruction guarantee for a fixed, real signal was recently done by Jacques and Feuillen. However, two questions remain open: the uniform recovery guarantee and exact recovery of complex signal. In this paper, we almost completely address these two open questions. We prove that, all complex sparse signals or low-rank matrices can be uniformly, exactly recovered from a near optimal number of complex Gaussian measurement phases. By recasting PO-CS as a linear compressive sensing problem, the exact recovery follows from restricted isometry property (RIP). Our approach to uniform recovery guarantee is based on covering arguments that involve a delicate control of the (original linear) measurements with overly small magnitude. To work with complex signal, a different sign-product embedding property and a careful rescaling of the sensing matrix are employed. In addition, we show an extension that the uniform recovery is stable under moderate bounded noise. We also propose to add Gaussian dither before capturing the phases to achieve full reconstruction with norm information. Experimental results are reported to corroborate and demonstrate our theoretical results.

Original languageEnglish
Pages (from-to)6739-6764
Number of pages26
JournalIEEE Transactions on Information Theory
Volume69
Issue number10
Early online date10 Jul 2023
DOIs
Publication statusPublished - Oct 2023

Scopus Subject Areas

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences

User-Defined Keywords

  • Compressed sensing
  • low-rankness
  • phase-only measurement
  • sparsity
  • uniform recovery

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