Consistent Basis Pursuit for Signal and Matrix Estimates in Quantized Compressed Sensing
(2016) IEEE Signal Processing Letters — Vol. 23, n° 1, p. 25-29 (2016)
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- Authors
Moshtaghpour, Amirafshar 
- 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.
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Citations
Moshtaghpour, A., Jacques, L., Cambareri, V., Degraux, K., & De Vleeschouwer, C. (2016). Consistent Basis Pursuit for Signal and Matrix Estimates in Quantized Compressed Sensing. IEEE Signal Processing Letters, 23(1), 25-29. https://doi.org/10.1109/LSP.2015.2497543 (Original work published 2016)