Quantization of compressive samples with stable and robust recovery

TL;DR

This paper analyzes the quantization process in compressed sensing, proposing Sigma-Delta quantization with convex optimization for robust, stable signal recovery, achieving polynomial and root-exponential error decay under various conditions.

Contribution

It introduces a novel quantization and reconstruction scheme that guarantees stable recovery with error decay rates, extending to various measurement types and quantization depths.

Findings

Reconstruction error decays polynomially with the number of measurements.

Error decays root-exponentially for sparse signals with optimized quantization order.

Results hold for sub-Gaussian measurements and both high and low bit-depth quantizers.

Abstract

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 Gaussian and Bernoulli) random compressed sensing measurements, as well as for both high bit-depth and coarse quantizers, and they extend to 1-bit quantization. In the noise-free case, when the signal is strictly sparse we prove that by optimizing the order of the quantization scheme one can obtain root-exponential decay in the reconstruction error due to quantization.