Signal Recovery From Incomplete and Inaccurate Measurements Via Regularized Orthogonal Matching Pursuit

We demonstrate a simple greedy algorithm that can reliably recover a vector <i>v</i> ¿ ¿<sup>d</sup> from incomplete and inaccurate measurements <i>x</i> = ¿<i>v</i> + <i>e</i>. Here, ¿ is a <i>N</i> x <i>d</i> measurement matrix with <i>N</i><<d, and <i>e</i> is an error vector. Our algorithm, Regularized Orthogonal Matching Pursuit (ROMP), seeks to provide the benefits of the two major approaches to sparse recovery. It combines the speed and ease of implementation of the greedy methods with the strong guarantees of the convex programming methods. For any measurement matrix ¿ that satisfies a quantitative restricted isometry principle, ROMP recovers a signal <i>v</i> with <i>O</i>(<i>n</i>) nonzeros from its inaccurate measurements <i>x</i> in at most <i>n</i> iterations, where each iteration amounts to solving a least squares problem. The noise level of the recovery is proportional to ¿{log<i>n</i>} ||<i>e</i>||<sub>2</sub>. In particular, if the error term <i>e</i> vanishes the reconstruction is exact.

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

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

[3]  D. Donoho.  For most large underdetermined systems of linear equations the minimal 𝓁1‐norm solution is also the sparsest solution , 2006 .

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

[5]  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.

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

[7]  Kurt M. Anstreicher,et al.  Linear Programming in O([n3/ln n]L) Operations , 1999, SIAM J. Optim..

[8]  E.J. Candes.  Compressive Sampling , 2022 .

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

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

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

[12]  D. Donoho.  For most large underdetermined systems of equations, the minimal 𝓁1‐norm near‐solution approximates the sparsest near‐solution , 2006 .

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

[14]  Roman Vershynin,et al.  Uncertainty Principles and Vector Quantization , 2006, IEEE Transactions on Information Theory.

[15]  Mike E. Davies,et al.  Iterative Hard Thresholding for Compressed Sensing , 2008, ArXiv.

[16]  E. Berg,et al.  In Pursuit of a Root , 2007 .

[17]  Wotao Yin,et al.  Bregman Iterative Algorithms for \ell1-Minimization with Applications to Compressed Sensing , 2008, SIAM J. Imaging Sci..

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

[19]  J. Tropp.  JUST RELAX: CONVEX PROGRAMMING METHODS FOR SUBSET SELECTION AND SPARSE APPROXIMATION , 2004 .

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

[21]  David L. Donoho,et al.  Neighborly Polytopes And Sparse Solution Of Underdetermined Linear Equations , 2005 .

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

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

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

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