On The Exact Recovery Condition of Simultaneous Orthogonal Matching Pursuit

Several exact recovery criteria (ERC) ensuring that orthogonal matching pursuit (OMP) identifies the correct support of sparse signals have been developed in the last few years. These ERC rely on the restricted isometry property (RIP), the associated restricted isometry constant (RIC) and sometimes the restricted orthogonality constant (ROC). In this paper, three of the most recent ERC for OMP are examined. The contribution is to show that these ERC remain valid for a generalization of OMP, entitled simultaneous orthogonal matching pursuit (SOMP), that is capable to process several measurement vectors simultaneously and return a common support estimate for the underlying sparse vectors. The sharpness of the bounds is also briefly discussed in light of previous works focusing on OMP.

[1]  Wei Huang,et al.  The Exact Support Recovery of Sparse Signals With Noise via Orthogonal Matching Pursuit , 2013, IEEE Signal Processing Letters.

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

[3]  Qun Mo,et al.  A Sharp Restricted Isometry Constant Bound of Orthogonal Matching Pursuit , 2015, ArXiv.

[4]  Ren-hong Wang,et al.  Robustness of orthogonal matching pursuit under restricted isometry property , 2014 .

[5]  Joel A. Tropp,et al.  Algorithms for simultaneous sparse approximation. Part I: Greedy pursuit , 2006, Signal Process..

[6]  Frank de Hoog,et al.  Coherence and RIP Analysis for Greedy Algorithms in Compressive Sensing , 2013, ArXiv.

[7]  Jian Wang,et al.  Improved Recovery Bounds of Orthogonal Matching Pursuit using Restricted Isometry Property , 2012, ArXiv.

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

[9]  Michael Elad,et al.  Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ1 minimization , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[10]  Entao Liu,et al.  Orthogonal Super Greedy Algorithm and Applications in Compressed Sensing ∗ , 2010 .

[11]  Joel A. Tropp,et al.  ALGORITHMS FOR SIMULTANEOUS SPARSE APPROXIMATION , 2006 .

[12]  Tong Zhang,et al.  Sparse Recovery With Orthogonal Matching Pursuit Under RIP , 2010, IEEE Transactions on Information Theory.

[13]  Jian Wang,et al.  Near optimal bound of orthogonal matching pursuit using restricted isometric constant , 2012, EURASIP J. Adv. Signal Process..

[14]  Yun Tian,et al.  Perturbation analysis of simultaneous orthogonal matching pursuit , 2015, Signal Process..

[15]  Michael B. Wakin,et al.  Analysis of Orthogonal Matching Pursuit Using the Restricted Isometry Property , 2009, IEEE Transactions on Information Theory.

[16]  Song Li,et al.  Sparse Signals Recovery from Noisy Measurements by Orthogonal Matching Pursuit , 2011, 1105.6177.

[17]  H. Rauhut,et al.  Atoms of All Channels, Unite! Average Case Analysis of Multi-Channel Sparse Recovery Using Greedy Algorithms , 2008 .

[18]  慧 廣瀬.  A Mathematical Introduction to Compressive Sensing , 2015 .

[19]  Wei Dan.  A Sharp RIP Condition for Orthogonal Matching Pursuit , 2015 .

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

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

[22]  S. Mallat,et al.  Adaptive greedy approximations , 1997 .

[23]  Yonina C. Eldar,et al.  Robust Recovery of Signals From a Structured Union of Subspaces , 2008, IEEE Transactions on Information Theory.

[24]  Lie Wang,et al.  Orthogonal Matching Pursuit for Sparse Signal Recovery With Noise , 2011, IEEE Transactions on Information Theory.

[25]  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.

[26]  Jian Wang,et al.  On the Recovery Limit of Sparse Signals Using Orthogonal Matching Pursuit , 2012, IEEE Transactions on Signal Processing.

[27]  Jian Wang,et al.  Exact Recovery of Sparse Signals Using Orthogonal Matching Pursuit: How Many Iterations Do We Need? , 2012, IEEE Transactions on Signal Processing.

[28]  E. Candès.  The restricted isometry property and its implications for compressed sensing , 2008 .

[29]  Lie Wang,et al.  Shifting Inequality and Recovery of Sparse Signals , 2010, IEEE Transactions on Signal Processing.

[30]  Yi Shen,et al.  A Remark on the Restricted Isometry Property in Orthogonal Matching Pursuit , 2012, IEEE Transactions on Information Theory.

[31]  Laurent Jacques,et al.  Simultaneous Orthogonal Matching Pursuit With Noise Stabilization: Theoretical Analysis , 2015, ArXiv.

[32]  Jie Ding,et al.  Robustness of orthogonal matching pursuit for multiple measurement vectors in noisy scenario , 2012, 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).