Structured random measurements in signal processing

Felix Krahmer, Holger Rauhut

Research output: Contribution to journalReview articlepeer-review

17 Scopus citations

Abstract

Compressed sensing and its extensions have recently triggered interest in randomized signal acquisition. A key finding is that random measurements provide sparse signal reconstruction guarantees for efficient and stable algorithms with a minimal number of samples. While this was first shown for (unstructured) Gaussian random measurement matrices, applications require certain structure of the measurements leading to structured random measurement matrices. Near optimal recovery guarantees for such structured measurements have been developed over the past years in a variety of contexts. This article surveys the theory in three scenarios: compressed sensing (sparse recovery), low rank matrix recovery, and phaseless estimation. The random measurement matrices to be considered include random partial Fourier matrices, partial random circulant matrices (subsampled convolutions), matrix completion, and phase estimation from magnitudes of Fourier type measurements. The article concludes with a brief discussion of the mathematical techniques for the analysis of such structured random measurements.

Original languageEnglish
Pages (from-to)217-238
Number of pages22
JournalGAMM Mitteilungen
Volume37
Issue number2
DOIs
StatePublished - 1 Nov 2014
Externally publishedYes

Keywords

  • Compressed sensing
  • matrix completion
  • phase retrieval
  • structured random measurements

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