Consistent Basis Pursuit for Signal and Matrix Estimates in Quantized Compressed Sensing

TL;DR

This paper introduces Consistent Basis Pursuit (CoBP), a novel method for estimating low-complexity signals from quantized compressed sensing measurements, demonstrating theoretical error decay and empirical improvements over existing approaches.

Contribution

The paper proposes CoBP, a new variant of Basis Pursuit Denoise, with proven error bounds and practical algorithms for quantized compressed sensing of sparse vectors and low-rank matrices.

Findings

Reconstruction error decays as M^{-1/4} with increasing measurements.

Numerical results show faster error decay than theoretical predictions.

CoBP performs well even in 1-bit compressed sensing scenarios.

Abstract

This paper focuses on the estimation of low-complexity signals when they are observed through $M$ uniformly quantized compressive observations. Among such signals, we consider 1-D sparse vectors, low-rank matrices, or compressible signals that are well approximated by one of these two models. In this context, we prove the estimation efficiency of a variant of Basis Pursuit Denoise, called Consistent Basis Pursuit (CoBP), enforcing consistency between the observations and the re-observed estimate, while promoting its low-complexity nature. We show that the reconstruction error of CoBP decays like $M^{-1/4}$ when all parameters but $M$ are fixed. Our proof is connected to recent bounds on the proximity of vectors or matrices when (i) those belong to a set of small intrinsic "dimension", as measured by the Gaussian mean width, and (ii) they share the same quantized (dithered) random projections. By solving CoBP with a proximal algorithm, we provide some extensive numerical observations that confirm the theoretical bound as $M$ is increased, displaying even faster error decay than predicted. The same phenomenon is observed in the special, yet important case of 1-bit CS.