PROMP: A sparse recovery approach to lattice-valued signals

Abstract Applications such as wireless communications require efficient sensing techniques of signals with the a priori knowledge of those being lattice-valued. In this paper, we study the impact of this prior information on compressed sensing methodologies, and introduce and analyze PROMP (“PReprojected Orthogonal Matching Pursuit”) as a novel algorithmic approach for sparse recovery of lattice-valued signals. More precisely, we first show that the straightforward approach to project the solution of Basis Pursuit onto a prespecified lattice does not improve the performance of Basis Pursuit in this situation. We then introduce PROMP as a novel sparse recovery algorithm for lattice-valued signals which has very low computational complexity, alongside a detailed mathematical analysis of its performance and stability under noise. Finally, we present numerical experiments which show that PROMP outperforms standard sparse recovery approaches in the lattice-valued signal regime.

[1]  Lloyd R. Welch,et al.  Lower bounds on the maximum cross correlation of signals (Corresp.) , 1974, IEEE Trans. Inf. Theory.

[2]  Y. C. Pati,et al.  Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition , 1993, Proceedings of 27th Asilomar Conference on Signals, Systems and Computers.

[3]  I. Daubechies,et al.  Iteratively reweighted least squares minimization for sparse recovery , 2008, 0807.0575.

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

[5]  Volkan Cevher,et al.  Model-Based Compressive Sensing , 2008, IEEE Transactions on Information Theory.

[6]  Deanna Needell,et al.  CoSaMP: Iterative signal recovery from incomplete and inaccurate samples , 2008, ArXiv.

[7]  M. Ledoux.  The concentration of measure phenomenon , 2001 .

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

[9]  Feng Wu,et al.  Analysis on Rate-Distortion Performance of Compressive Sensing for Binary Sparse Source , 2009, 2009 Data Compression Conference.

[10]  STEVEN P. LALLEY.  CONCENTRATION INEQUALITIES , 2013 .

[11]  Hassan Mansour,et al.  Recovering Compressively Sampled Signals Using Partial Support Information , 2010, IEEE Transactions on Information Theory.

[12]  Erik G. Larsson,et al.  Spectrum Sensing for Cognitive Radio : State-of-the-Art and Recent Advances , 2012, IEEE Signal Processing Magazine.

[13]  Georgios B. Giannakis,et al.  Exploiting Sparse User Activity in Multiuser Detection , 2011 .

[14]  Yonina C. Eldar,et al.  Introduction to Compressed Sensing , 2022 .

[15]  David L. Donoho,et al.  Counting the Faces of Randomly-Projected Hypercubes and Orthants, with Applications , 2008, Discret. Comput. Geom..

[16]  Geert Leus,et al.  Detection of sparse signals under finite-alphabet constraints , 2009, 2009 IEEE International Conference on Acoustics, Speech and Signal Processing.

[17]  Stark C. Draper,et al.  Compressed sensing over finite fields , 2009, 2009 IEEE International Symposium on Information Theory.

[18]  Jean-Jacques Fuchs,et al.  Spread representations , 2011, 2011 Conference Record of the Forty Fifth Asilomar Conference on Signals, Systems and Computers (ASILOMAR).

[19]  Israa Sh. Tawfic,et al.  Least Support Orthogonal Matching Pursuit Algorithm With Prior Information , 2014 .

[20]  Gitta Kutyniok,et al.  1 . 2 Sparsity : A Reasonable Assumption ? , 2012 .

[21]  Kenneth E. Barner,et al.  Iterative algorithms for compressed sensing with partially known support , 2010, 2010 IEEE International Conference on Acoustics, Speech and Signal Processing.

[22]  Nazanin Rahnavard,et al.  BCS: Compressive sensing for binary sparse signals , 2012, MILCOM 2012 - 2012 IEEE Military Communications Conference.

[23]  Mihailo Stojnic,et al.  Recovery thresholds for ℓ1 optimization in binary compressed sensing , 2010, 2010 IEEE International Symposium on Information Theory.

[24]  Benjamin Recht,et al.  Probability of unique integer solution to a system of linear equations , 2011, Eur. J. Oper. Res..

[25]  Joel A. Tropp,et al.  Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit , 2007, IEEE Transactions on Information Theory.

[26]  Tho Le-Ngoc,et al.  A low-complexity generalized sphere decoding approach for underdetermined linear communication systems: performance and complexity evaluation , 2009, IEEE Transactions on Communications.

[27]  G. C. Shephard,et al.  Convex Polytopes , 1969, The Mathematical Gazette.

[28]  Robert F. H. Fischer,et al.  MMSE-based version of OMP for recovery of discrete-valued sparse signals , 2016 .

[29]  T. Figiel,et al.  The dimension of almost spherical sections of convex bodies , 1976 .

[30]  Joel A. Tropp,et al.  Greed is good: algorithmic results for sparse approximation , 2004, IEEE Transactions on Information Theory.

[31]  Alexander Vardy,et al.  Closest point search in lattices , 2002, IEEE Trans. Inf. Theory.

[32]  Yonina C. Eldar,et al.  Spatial Compressive Sensing for MIMO Radar , 2013, IEEE Transactions on Signal Processing.

[33]  S. Szarek,et al.  Chapter 8 - Local Operator Theory, Random Matrices and Banach Spaces , 2001 .

[34]  Babak Hassibi,et al.  On the Reconstruction of Block-Sparse Signals With an Optimal Number of Measurements , 2008, IEEE Transactions on Signal Processing.

[35]  Karim Abed-Meraim,et al.  A generalized lattice decoder for asymmetrical space-time communication architecture , 2000, 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.00CH37100).

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

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

[38]  D. Donoho,et al.  Sparse nonnegative solution of underdetermined linear equations by linear programming. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[39]  Weiyu Xu,et al.  Weighted ℓ1 minimization for sparse recovery with prior information , 2009, 2009 IEEE International Symposium on Information Theory.

[40]  Harold R. Parks,et al.  Geometric Integration Theory , 2008 .

[41]  M. Rudelson,et al.  Smallest singular value of random matrices and geometry of random polytopes , 2005 .

[42]  Axel Flinth,et al.  Optimal Choice of Weights for Sparse Recovery With Prior Information , 2015, IEEE Transactions on Information Theory.

[43]  Robert F. H. Fischer,et al.  Discrete Sparse Signals: Compressed Sensing by Combining OMP and the Sphere Decoder , 2013, ArXiv.

[44]  Inderjit S. Dhillon,et al.  Orthogonal Matching Pursuit with Replacement , 2011, NIPS.

[45]  Babak Hassibi,et al.  Recovery threshold for optimal weight ℓ1 minimization , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.

[46]  Zongkai Yang,et al.  A new approach for fast generalized sphere decoding in MIMO systems , 2005, IEEE Signal Process. Lett..

[47]  E. Candès,et al.  Error correction via linear programming , 2005, FOCS 2005.