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

PROJECT TITLE :

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

ABSTRACT:

This letter focuses on the estimation of low-complexity signals after they are observed through uniformly quantized compressive observations. Among such signals, we tend to think about 1-D sparse vectors, low-rank matrices, or compressible signals that are well approximated by one of those 2 models. During 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 when all parameters however are fastened. Our proof is connected to recent bounds on the proximity of vectors or matrices when (i) those belong to a collection 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 tend to give some in depth numerical observations that make sure the theoretical certain as is increased, displaying even faster error decay than predicted. The same phenomenon is observed within the special, however vital case of one-bit CS.

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