CoSaMP: Iterative signal recovery from incomplete and inaccurate samples

Compressive sampling (CoSa) is a new paradigm for developing data sampling technologies. It is based on the principle that many types of vector-space data are compressible, which is a term of art in mathematical signal processing. The key ideas are that randomized dimension reduction preserves the information in a compressible signal and that it is possible to develop hardware devices that implement this dimension reduction efficiently. The main computational challenge in CoSa is to reconstruct a compressible signal from the reduced representation acquired by the sampling device. This extended abstract describes a recent algorithm, called, CoSaMP , that accomplishes the data recovery task. It was the first known method to offer near-optimal guarantees on resource usage.

[1]  Michael A. Saunders,et al.  LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares , 1982, TOMS.

[2]  D. Donoho,et al.  Uncertainty principles and signal recovery , 1989 .

[3]  A. Atkinson.  Subset Selection in Regression , 1992 .

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

[5]  Yurii Nesterov,et al.  Interior-point polynomial algorithms in convex programming , 1994, Siam studies in applied mathematics.

[6]  J. Navarro-Pedreño.  Numerical Methods for Least Squares Problems , 1996 .

[7]  Clifford Stein,et al.  Introduction to Algorithms, 2nd edition. , 2001 .

[8]  Sudipto Guha,et al.  Near-optimal sparse fourier representations via sampling , 2002, STOC '02.

[9]  Sudipto Guha,et al.  Fast, small-space algorithms for approximate histogram maintenance , 2002, STOC '02.

[10]  V. Temlyakov.  Nonlinear Methods of Approximation , 2003, Found. Comput. Math..

[11]  S. Muthukrishnan,et al.  Approximation of functions over redundant dictionaries using coherence , 2003, SODA '03.

[12]  I. Daubechies,et al.  An iterative thresholding algorithm for linear inverse problems with a sparsity constraint , 2003, math/0307152.

[13]  S. Muthukrishnan,et al.  Improved sparse approximation over quasiincoherent dictionaries , 2003, Proceedings 2003 International Conference on Image Processing (Cat. No.03CH37429).

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

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

[16]  Emmanuel J. Candès,et al.  Decoding by linear programming , 2005, IEEE Transactions on Information Theory.

[17]  D. Donoho,et al.  Counting faces of randomly-projected polytopes when the projection radically lowers dimension , 2006, math/0607364.

[18]  Emmanuel J. Candès,et al.  Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , 2004, IEEE Transactions on Information Theory.

[19]  Graham Cormode,et al.  Combinatorial Algorithms for Compressed Sensing , 2006, 2006 40th Annual Conference on Information Sciences and Systems.

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

[21]  David L. Donoho,et al.  Sparse Solution Of Underdetermined Linear Equations By Stagewise Orthogonal Matching Pursuit , 2006 .

[22]  M. Rudelson,et al.  Sparse reconstruction by convex relaxation: Fourier and Gaussian measurements , 2006, 2006 40th Annual Conference on Information Sciences and Systems.

[23]  Joel A. Tropp,et al.  Algorithmic linear dimension reduction in the l_1 norm for sparse vectors , 2006, ArXiv.

[24]  R. Vershynin,et al.  One sketch for all: fast algorithms for compressed sensing , 2007, STOC '07.

[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]  H. Rauhut.  On the Impossibility of Uniform Sparse Reconstruction using Greedy Methods , 2007 .

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

[28]  R. DeVore,et al.  Compressed sensing and best k-term approximation , 2008 .

[29]  Olgica Milenkovic,et al.  Subspace Pursuit for Compressive Sensing: Closing the Gap Between Performance and Complexity , 2008, ArXiv.

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

[31]  Galen Reeves,et al.  Sampling bounds for sparse support recovery in the presence of noise , 2008, 2008 IEEE International Symposium on Information Theory.

[32]  Mark A. Iwen,et al.  A deterministic sub-linear time sparse fourier algorithm via non-adaptive compressed sensing methods , 2007, SODA '08.

[33]  J. Tropp,et al.  CoSaMP: Iterative signal recovery from incomplete and inaccurate samples , 2008, Commun. ACM.

[34]  Deanna Needell,et al.  Uniform Uncertainty Principle and Signal Recovery via Regularized Orthogonal Matching Pursuit , 2007, Found. Comput. Math..

[35]  Olgica Milenkovic,et al.  Subspace Pursuit for Compressive Sensing Signal Reconstruction , 2008, IEEE Transactions on Information Theory.

[36]  Piotr Indyk,et al.  Sparse Recovery Using Sparse Matrices , 2010, Proceedings of the IEEE.

[37]  S. Foucart.  A note on guaranteed sparse recovery via ℓ1-minimization , 2010 .

[38]  Justin K. Romberg,et al.  Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals , 2009, IEEE Transactions on Information Theory.

[39]  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.