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.

[1]  S. Frick,et al.  Compressed Sensing , 2014, Computer Vision, A Reference Guide.

[2]  Rayan Saab,et al.  Sobolev Duals for Random Frames and ΣΔ Quantization of Compressed Sensing Measurements , 2013, Found. Comput. Math..

[3]  Vivek K Goyal.  Quantized Overcomplete Expansions : Analysis , Synthesis and Algorithms , 1995 .

[4]  Pablo A. Parrilo,et al.  The Convex Geometry of Linear Inverse Problems , 2010, Foundations of Computational Mathematics.

[5]  Laurent Jacques,et al.  Robust 1-Bit Compressive Sensing via Binary Stable Embeddings of Sparse Vectors , 2011, IEEE Transactions on Information Theory.

[6]  Ruby J Pai.  Nonadaptive lossy encoding of sparse signals , 2006 .

[7]  Yaniv Plan,et al.  One-bit compressed sensing with non-Gaussian measurements , 2012, ArXiv.

[8]  Vivek K. Goyal,et al.  Frame permutation quantization , 2010, 2010 44th Annual Conference on Information Sciences and Systems (CISS).

[9]  Laurent Jacques,et al.  Dequantizing Compressed Sensing: When Oversampling and Non-Gaussian Constraints Combine , 2009, IEEE Transactions on Information Theory.

[10]  Vivek K. Goyal,et al.  Quantized Overcomplete Expansions in IRN: Analysis, Synthesis, and Algorithms , 1998, IEEE Trans. Inf. Theory.

[11]  Dustin G. Mixon,et al.  Compressive classification and the rare eclipse problem , 2014, ArXiv.

[12]  Patrick L. Combettes,et al.  Proximal Splitting Methods in Signal Processing , 2009, Fixed-Point Algorithms for Inverse Problems in Science and Engineering.

[13]  Olgica Milenkovic,et al.  Distortion-rate functions for quantized compressive sensing , 2009, 2009 IEEE Information Theory Workshop on Networking and Information Theory.

[14]  Vivek K. Goyal,et al.  Optimal quantization of random measurements in compressed sensing , 2009, 2009 IEEE International Symposium on Information Theory.

[15]  Laurent Jacques,et al.  Small width, low distortions: quasi-isometric embeddings with quantized sub-Gaussian random projections , 2015, ArXiv.

[16]  Emmanuel J. Candès,et al.  Tight Oracle Inequalities for Low-Rank Matrix Recovery From a Minimal Number of Noisy Random Measurements , 2011, IEEE Transactions on Information Theory.

[17]  Laurent Jacques,et al.  Quantization and Compressive Sensing , 2014, ArXiv.

[18]  Sundeep Rangan,et al.  Message-Passing De-Quantization With Applications to Compressed Sensing , 2012, IEEE Transactions on Signal Processing.

[19]  Petros Boufounos,et al.  Universal Rate-Efficient Scalar Quantization , 2010, IEEE Transactions on Information Theory.

[20]  E. Candès,et al.  Stable signal recovery from incomplete and inaccurate measurements , 2005, math/0503066.

[21]  Laurent Jacques,et al.  A Quantized Johnson–Lindenstrauss Lemma: The Finding of Buffon’s Needle , 2013, IEEE Transactions on Information Theory.

[22]  Laurent Jacques,et al.  Error Decay of (Almost) Consistent Signal Estimations From Quantized Gaussian Random Projections , 2014, IEEE Transactions on Information Theory.

[23]  Mikael Skoglund,et al.  Analysis-by-Synthesis Quantization for Compressed Sensing Measurements , 2013, IEEE Transactions on Signal Processing.

[24]  Olgica Milenkovic,et al.  Information Theoretical and Algorithmic Approaches to Quantized Compressive Sensing , 2011, IEEE Transactions on Communications.

[25]  Laurent Jacques,et al.  Error Decay of (almost) Consistent Signal Estimations from Quantized Random Gaussian Projections , 2014, ArXiv.

[26]  Stephen P. Boyd,et al.  Compressed Sensing With Quantized Measurements , 2010, IEEE Signal Processing Letters.

[27]  Rayan Saab,et al.  One-Bit Compressive Sensing With Norm Estimation , 2014, IEEE Transactions on Information Theory.

[28]  Michael A. Saunders,et al.  Atomic Decomposition by Basis Pursuit , 1998, SIAM J. Sci. Comput..

[29]  Pierre Vandergheynst,et al.  UNLocBoX A matlab convex optimization toolbox using proximal splitting methods , 2014, ArXiv.

[30]  Holger Rauhut,et al.  On the gap between RIP-properties and sparse recovery conditions , 2015, ArXiv.

[31]  Michael I. Jordan,et al.  Computational and statistical tradeoffs via convex relaxation , 2012, Proceedings of the National Academy of Sciences.

[32]  Richard G. Baraniuk,et al.  1-Bit compressive sensing , 2008, 2008 42nd Annual Conference on Information Sciences and Systems.

[33]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[34]  Michael I. Jordan,et al.  Computational and Sample Tradeoffs via Convex Relaxation , 2012 .

[35]  Pierre Vandergheynst,et al.  Hyperspectral image compressed sensing via low-rank and joint-sparse matrix recovery , 2012, 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[36]  Holger Rauhut,et al.  A Mathematical Introduction to Compressive Sensing , 2013, Applied and Numerical Harmonic Analysis.

[37]  Yaniv Plan,et al.  Robust 1-bit Compressed Sensing and Sparse Logistic Regression: A Convex Programming Approach , 2012, IEEE Transactions on Information Theory.

[38]  Yaniv Plan,et al.  One‐Bit Compressed Sensing by Linear Programming , 2011, ArXiv.

[39]  David L. Neuhoff,et al.  Quantization , 2022, IEEE Trans. Inf. Theory.

[40]  Emmanuel J. Candès,et al.  Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies? , 2004, IEEE Transactions on Information Theory.