Thresholding Greedy Pursuit for Sparse Recovery Problems

We study here sparse recovery problems in the presence of additive noise. We analyze a thresholding version of the CoSaMP algorithm, named Thresholding Greedy Pursuit (TGP). We demonstrate that an appropriate choice of thresholding parameter, even without the knowledge of sparsity level of the signal and strength of the noise, can result in exact recovery with no false discoveries as the dimension of the data increases to infinity.

[1]  Pranab K. Dutta,et al.  Sparse recovery based compressive sensing algorithms for diffuse optical tomography , 2020 .

[2]  H. Zou.  The Adaptive Lasso and Its Oracle Properties , 2006 .

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

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

[5]  Alexei Novikov,et al.  The Noise Collector for sparse recovery in high dimensions , 2019, Proceedings of the National Academy of Sciences.

[6]  Richard G. Baraniuk,et al.  Exact signal recovery from sparsely corrupted measurements through the Pursuit of Justice , 2009, 2009 Conference Record of the Forty-Third Asilomar Conference on Signals, Systems and Computers.

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

[8]  Stephen J. Dilworth,et al.  Explicit constructions of RIP matrices and related problems , 2010, ArXiv.

[9]  Elizabeth B. Klerman,et al.  On-line EEG Denoising using correlated sparse recovery , 2016, 2016 10th International Symposium on Medical Information and Communication Technology (ISMICT).

[10]  A. Belloni,et al.  Square-Root Lasso: Pivotal Recovery of Sparse Signals via Conic Programming , 2010, 1009.5689.

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

[12]  Jean-Luc Starck,et al.  Sparse Solution of Underdetermined Systems of Linear Equations by Stagewise Orthogonal Matching Pursuit , 2012, IEEE Transactions on Information Theory.

[13]  Frank de Hoog,et al.  Orthogonal Matching Pursuit With Thresholding and its Application in Compressive Sensing , 2013, IEEE Transactions on Signal Processing.

[14]  Jie Yang,et al.  A machine learning paradigm based on sparse signal representation , 2013 .

[15]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[16]  Alexei Novikov,et al.  Imaging with highly incomplete and corrupted data , 2019, Inverse Problems.

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

[18]  Martin J. Wainwright,et al.  A Practical Scheme and Fast Algorithm to Tune the Lasso With Optimality Guarantees , 2016, J. Mach. Learn. Res..

[19]  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.

[20]  V. Koltchinskii,et al.  High Dimensional Probability , 2006, math/0612726.

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

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

[23]  Peter Jung,et al.  Robust Nonnegative Sparse Recovery and the Nullspace Property of 0/1 Measurements , 2016, IEEE Transactions on Information Theory.

[24]  T. Blumensath,et al.  Theory and Applications , 2011 .

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

[26]  Michael Elad,et al.  A generalized uncertainty principle and sparse representation in pairs of bases , 2002, IEEE Trans. Inf. Theory.

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

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

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

[30]  J. Claerbout,et al.  Robust Modeling With Erratic Data , 1973 .

[31]  Daniel J. McDonald,et al.  A study on tuning parameter selection for the high-dimensional lasso , 2016, Journal of Statistical Computation and Simulation.

[32]  Arkadi Nemirovski,et al.  On sparse representation in pairs of bases , 2003, IEEE Trans. Inf. Theory.

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

[34]  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.

[35]  Jean-Jacques Fuchs,et al.  Recovery of exact sparse representations in the presence of bounded noise , 2005, IEEE Transactions on Information Theory.

[36]  J WainwrightMartin.  Sharp thresholds for high-dimensional and noisy sparsity recovery using l1-constrained quadratic programming (Lasso) , 2009 .

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

[38]  Jun Yang,et al.  A Review of Sparse Recovery Algorithms , 2019, IEEE Access.

[39]  D. Donoho,et al.  Basis pursuit , 1994, Proceedings of 1994 28th Asilomar Conference on Signals, Systems and Computers.

[40]  Anna C. Gilbert,et al.  Improved time bounds for near-optimal sparse Fourier representations , 2005, SPIE Optics + Photonics.

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