A Performance Guarantee for Orthogonal Matching Pursuit Using Mutual Coherence

In this paper, we present a new performance guarantee for the orthogonal matching pursuit (OMP) algorithm. We use mutual coherence as a metric for determining the suitability of an arbitrary overcomplete dictionary for exact recovery. Specifically, a lower bound for the probability of correctly identifying the support of a sparse signal with additive white Gaussian noise and an upper bound for the mean square error is derived. Compared to the previous work, the new bound takes into account the signal parameters such as dynamic range, noise variance, and sparsity. Numerical simulations show significant improvements over previous work and a much closer correlation to empirical results of OMP.

[1]  Soo-Chang Pei,et al.  Fast OMP: Reformulating OMP via iteratively refining ℓ2-norm solutions , 2012, 2012 IEEE Statistical Signal Processing Workshop (SSP).

[2]  Michael Zibulevsky,et al.  Underdetermined blind source separation using sparse representations , 2001, Signal Process..

[3]  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.

[4]  Gene H. Golub,et al.  Matrix computations , 1983 .

[5]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

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

[7]  Chi Hau Chen,et al.  UNDERDETERMINED BLIND SOURCE SEPARATION* , 2022 .

[8]  Michael Elad,et al.  Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries , 2006, IEEE Transactions on Image Processing.

[9]  Stephen J. Wright,et al.  Sparse Reconstruction by Separable Approximation , 2008, IEEE Transactions on Signal Processing.

[10]  KarahanogluNazim Burak,et al.  A* orthogonal matching pursuit , 2012 .

[11]  D. L. Donoho,et al.  Compressed sensing , 2006, IEEE Trans. Inf. Theory.

[12]  Z.-T. Huang,et al.  Direction-of-Arrival Estimation of Wideband Signals via Covariance Matrix Sparse Representation , 2011, IEEE Transactions on Signal Processing.

[13]  Lie Wang,et al.  Stable Recovery of Sparse Signals and an Oracle Inequality , 2010, IEEE Transactions on Information Theory.

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

[15]  M. Emadi,et al.  DOA Estimation of Multi-Reflected Known Signals in Compact Arrays , 2013, IEEE Transactions on Aerospace and Electronic Systems.

[16]  Graeme Pope,et al.  Compressive sensing: a summary of reconstruction algorithms , 2009 .

[17]  Jussi Klemel,et al.  Smoothing of Multivariate Data: Density Estimation and Visualization , 2009 .

[18]  Deanna Needell,et al.  Signal Recovery From Incomplete and Inaccurate Measurements Via Regularized Orthogonal Matching Pursuit , 2007, IEEE Journal of Selected Topics in Signal Processing.

[19]  Jonas Unger,et al.  Compressive Image Reconstruction in Reduced Union of Subspaces , 2015, Comput. Graph. Forum.

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

[21]  Jean-Jacques Fuchs,et al.  On sparse representations in arbitrary redundant bases , 2004, IEEE Transactions on Information Theory.

[22]  Ting Sun,et al.  Single-pixel imaging via compressive sampling , 2008, IEEE Signal Process. Mag..

[23]  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.

[24]  Stéphane Mallat,et al.  Matching pursuits with time-frequency dictionaries , 1993, IEEE Trans. Signal Process..

[25]  Mário A. T. Figueiredo,et al.  Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems , 2007, IEEE Journal of Selected Topics in Signal Processing.

[26]  Feng Liu,et al.  Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries in Wavelet Domain , 2009, 2009 Fifth International Conference on Image and Graphics.

[27]  Babak Hossein Khalaj,et al.  A unified approach to sparse signal processing , 2009, EURASIP Journal on Advances in Signal Processing.

[28]  Michael Elad,et al.  Stable recovery of sparse overcomplete representations in the presence of noise , 2006, IEEE Transactions on Information Theory.

[29]  Yonina C. Eldar,et al.  Coherence-Based Performance Guarantees for Estimating a Sparse Vector Under Random Noise , 2009, IEEE Transactions on Signal Processing.

[30]  Michael P. Friedlander,et al.  Probing the Pareto Frontier for Basis Pursuit Solutions , 2008, SIAM J. Sci. Comput..

[31]  Michael Elad,et al.  Sparse and Redundant Representations - From Theory to Applications in Signal and Image Processing , 2010 .

[32]  Xiaoming Huo,et al.  Uncertainty principles and ideal atomic decomposition , 2001, IEEE Trans. Inf. Theory.

[33]  Joel A. Tropp,et al.  Just relax: convex programming methods for identifying sparse signals in noise , 2006, IEEE Transactions on Information Theory.

[34]  Dmitry M. Malioutov,et al.  A sparse signal reconstruction perspective for source localization with sensor arrays , 2005, IEEE Transactions on Signal Processing.

[35]  G. Bennett.  Probability Inequalities for the Sum of Independent Random Variables , 1962 .

[36]  R. Tibshirani,et al.  Least angle regression , 2004, math/0406456.

[37]  Guillermo Sapiro,et al.  Learning Discriminative Sparse Representations for Modeling, Source Separation, and Mapping of Hyperspectral Imagery , 2011, IEEE Transactions on Geoscience and Remote Sensing.

[38]  Thomas S. Huang,et al.  Image Super-Resolution Via Sparse Representation , 2010, IEEE Transactions on Image Processing.

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

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