Robust 1-Bit compressive sensing via binary stable embeddings of sparse vectors

Laurent Jacques, Jason N. Laska, Petros T. Boufounos, Richard G. Baraniuk

Research output: Contribution to journalArticle

303 Scopus citations

Abstract

The compressive sensing (CS) framework aims to ease the burden on analog-to-digital converters (ADCs) by reducing the sampling rate required to acquire and stably recover sparse signals. Practical ADCs not only sample but also quantize each measurement to a finite number of bits; moreover, there is an inverse relationship between the achievable sampling rate and the bit depth. In this paper, we investigate an alternative CS approach that shifts the emphasis from the sampling rate to the number of bits per measurement. In particular, we explore the extreme case of 1-bit CS measurements, which capture just their sign. Our results come in two flavors. First, we consider ideal reconstruction from noiseless 1-bit measurements and provide a lower bound on the best achievable reconstruction error.We also demonstrate that i.i.d. random Gaussian matrices provide measurement mappings that, with overwhelming probability, achieve nearly optimal error decay. Next, we consider reconstruction robustness to measurement errors and noise and introduce the binary -stable embedding property, which characterizes the robustness of the measurement process to sign changes. We show that the same class of matrices that provide almost optimal noiseless performance also enable such a robustmapping.On the practical side, we introduce the binary iterative hard thresholding algorithm for signal reconstruction from 1-bit measurements that offers state-of-the-art performance.

Original languageEnglish (US)
Article number6418031
Pages (from-to)2082-2102
Number of pages21
JournalIEEE Transactions on Information Theory
Volume59
Issue number4
DOIs
StatePublished - Jan 1 2013

Keywords

  • 1-bit compressed sensing
  • Approximation error
  • Compressed sensing
  • Consistent reconstruction
  • Dimensionality reduction
  • Iterative reconstruction
  • Quantization
  • Reconstruction algorithms
  • Signal reconstruction
  • Sparsity

ASJC Scopus subject areas

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

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