Compressive Sensing by Shortest Solution Guided Decimation

Compressed sensing, which reconstructs a signal by finding sparse solutions to underdetermined linear systems, is an important problem in many fields of science and engineering. In this work we propose a deterministic and non-parametric algorithm, Shortest-Solution guided Decimation (SSD), to construct support of the sparse solution in an iterative way guided by the shortest Euclidean-length solution of the recursively decimated linear equation under orthogonal projections. The most significant feature of SSD is its insensitivity to correlations in the sampling matrix. Using extensive numerical experiments we show that SSD greatly outperforms L1-based methods, Orthogonal Matching Pursuit, and Approximate Message Passing when the sampling matrix contains strong correlations. This nice property of correlation tolerance makes SSD a versatile and robust tool for different types of real-world signal acquisition tasks.

[1]  Sundeep Rangan,et al.  Vector approximate message passing , 2017, 2017 IEEE International Symposium on Information Theory (ISIT).

[2]  M. Ohzeki,et al.  Sparse modeling approach to analytical continuation of imaginary-time quantum Monte Carlo data. , 2017, Physical review. E.

[3]  Li Ping,et al.  Orthogonal AMP , 2016, IEEE Access.

[4]  David Zhang,et al.  A Survey of Sparse Representation: Algorithms and Applications , 2015, IEEE Access.

[5]  Holger Rauhut,et al.  A Mathematical Introduction to Compressive Sensing , 2013, Applied and Numerical Harmonic Analysis.

[6]  P. Händel,et al.  Projection-based atom selection in orthogonal matching pursuit for compressive sensing , 2012, 2012 National Conference on Communications (NCC).

[7]  Florent Krzakala,et al.  Statistical physics-based reconstruction in compressed sensing , 2011, ArXiv.

[8]  Andrea Montanari,et al.  The dynamics of message passing on dense graphs, with applications to compressed sensing , 2010, 2010 IEEE International Symposium on Information Theory.

[9]  Andrea Montanari,et al.  Message-passing algorithms for compressed sensing , 2009, Proceedings of the National Academy of Sciences.

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

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

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

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

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

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

[16]  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.

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

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

[19]  Zhifeng Zhang,et al.  Adaptive time-frequency decompositions , 1994 .

[20]  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.

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

[22]  Sheng Chen,et al.  Orthogonal least squares methods and their application to non-linear system identification , 1989 .

[23]  R. Palmer,et al.  Solution of 'Solvable model of a spin glass' , 1977 .

[24]  Gitta Kutyniok.  Compressed Sensing , 2012 .

[25]  Wang Liang-jun,et al.  Advances in Theory and Application of Compressed Sensing , 2009 .

[26]  E. Candes,et al.  11-magic : Recovery of sparse signals via convex programming , 2005 .

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