Shifting Inequality and Recovery of Sparse Signals

In this paper, we present a concise and coherent analysis of the constrained ¿1 minimization method for stable recovering of high-dimensional sparse signals both in the noiseless case and noisy case. The analysis is surprisingly simple and elementary, while leads to strong results. In particular, it is shown that the sparse recovery problem can be solved via ¿1 minimization under weaker conditions than what is known in the literature. A key technical tool is an elementary inequality, called Shifting Inequality, which, for a given nonnegative decreasing sequence, bounds the ¿2 norm of a subsequence in terms of the ¿1 norm of another subsequence by shifting the elements to the upper end.

[1]  S. Foucart,et al.  Sparsest solutions of underdetermined linear systems via ℓq-minimization for 0 , 2009 .

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

[3]  M. Lustig,et al.  Compressed Sensing MRI , 2008, IEEE Signal Processing Magazine.

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

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

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

[7]  T. Tony Cai,et al.  ON BLOCK THRESHOLDING IN WAVELET REGRESSION: ADAPTIVITY, BLOCK SIZE, AND THRESHOLD LEVEL , 2002 .

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

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

[10]  Vahid Tarokh,et al.  A Frame Construction and a Universal Distortion Bound for Sparse Representations , 2008, IEEE Transactions on Signal Processing.

[11]  Stephen J. Wright,et al.  Toeplitz-Structured Compressed Sensing Matrices , 2007, 2007 IEEE/SP 14th Workshop on Statistical Signal Processing.

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

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

[14]  Rémi Gribonval,et al.  Sparse representations in unions of bases , 2003, IEEE Trans. Inf. Theory.

[15]  Emmanuel J. Cand.  The Restricted Isometry Property and Its Implications for Compressed Sensing , 2008 .

[16]  R. DeVore,et al.  A Simple Proof of the Restricted Isometry Property for Random Matrices , 2008 .

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

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

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

[20]  Jun Zhang,et al.  On Recovery of Sparse Signals Via $\ell _{1}$ Minimization , 2008, IEEE Transactions on Information Theory.

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

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

[23]  Terence Tao,et al.  The Dantzig selector: Statistical estimation when P is much larger than n , 2005, math/0506081.

[24]  Jun Zhang,et al.  On Recovery of Sparse Signals via ℓ1 Minimization , 2008, ArXiv.

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

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

[27]  D. Madigan,et al.  [Least Angle Regression]: Discussion , 2004 .

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