# Quantization of compressive samples with stable and robust recovery

@article{Saab2015QuantizationOC, title={Quantization of compressive samples with stable and robust recovery}, author={Rayan Saab and Rongrong Wang and {\"O}zg{\"u}r Yilmaz}, journal={ArXiv}, year={2015}, volume={abs/1504.00087} }

In this paper we study the quantization stage that is implicit in any compressed sensing signal acquisition paradigm. We propose using Sigma-Delta quantization and a subsequent reconstruction scheme based on convex optimization. We prove that the reconstruction error due to quantization decays polynomially in the number of measurements. Our results apply to arbitrary signals, including compressible ones, and account for measurement noise. Additionally, they hold for sub-Gaussian (including… Expand

#### Paper Mentions

#### 36 Citations

Quantization for Low-Rank Matrix Recovery

- Computer Science, Mathematics
- ArXiv
- 2017

A full generalization of analogous results, obtained in the classical setup of bandlimited function acquisition, and more recently, in the finite frame and compressed sensing setups to the case of low-rank matrices sampled with sub-Gaussian linear operators, is provided. Expand

Quantized compressed sensing for random circulant matrices

- Mathematics
- Applied and Computational Harmonic Analysis
- 2019

Abstract We provide the first analysis of a non-trivial quantization scheme for compressed sensing with structured measurements. We consider compressed sensing matrices consisting of rows selected… Expand

Quantized compressed sensing for partial random circulant matrices

- Mathematics, Computer Science
- 2017 International Conference on Sampling Theory and Applications (SampTA)
- 2017

This analysis studies compressed sensing matrices consisting of rows selected at random, without replacement, from a circulant matrix generated by a random subgaussian vector, and shows that the part of the reconstruction error due to quantization decays polynomially in the number of measurements. Expand

On one-stage recovery for Σ∆-quantized compressed sensing

- Computer Science
- 2019 13th International conference on Sampling Theory and Applications (SampTA)
- 2019

This work focuses on one of the approaches that yield efficient quantizers for CS: Σ∆ quantization, followed by a one-stage tractable reconstruction method, which was developed in [20] with theoretical error guarantees in the case of sub-Gaussian matrices. Expand

Deterministic compressed sensing and quantization

- Computer Science, Engineering
- SPIE Optical Engineering + Applications
- 2015

This note focuses on quantization in CS with chirp sensing matrices and present quantization approaches and numerical experiments. Expand

On one-stage recovery for $\Sigma \Delta$-quantized compressed sensing

- Computer Science, Mathematics
- 2019

This work focuses on one of the approaches that yield efficient quantizers for CS: $\Sigma \Delta$ quantization, followed by a one-stage tractable reconstruction method, which was developed by Saab et al. with theoretical error guarantees in the case of sub-Gaussian matrices. Expand

Memoryless scalar quantization for random frames

- Mathematics, Computer Science
- ArXiv
- 2018

This work rigorously establishes sharp non-asymptotic error bounds without using the WNH that explain the observed decay rate, and extends this approach to the compressed sensing setting, obtaining rigorous error bounds that agree with empirical observations. Expand

One-Bit Compressive Sensing of Dictionary-Sparse Signals

- Computer Science, Mathematics
- ArXiv
- 2016

This work analyzes several different algorithms---based on convex programming and on hard thresholding---and shows that, under natural assumptions on the sensing matrix (satisfied by Gaussian matrices), these algorithms can efficiently recover analysis-dictionary-sparse signals in the one-bit model. Expand

Quantization for spectral super-resolution

- Computer Science, Mathematics
- ArXiv
- 2021

The oversampling ratio λ is defined as the largest integer such that ⌊M/λ⌋ − 1 ≥ 4/∆, where M denotes the number of Fourier measurements and ∆ is the minimum separation distance associated with the atomic measure to be resolved. Expand

Quantized Compressed Sensing: A Survey

- Computer Science
- Applied and Numerical Harmonic Analysis
- 2019

An overview of the state-of-the-art rigorous reconstruction results that have been obtained for three popular quantization models: one-bit quantization, uniform scalarquantization, and noise-shaping methods is given. Expand

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