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

This letter 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.