Invariance of the spark, NSP order and RIP order under elementary transformations of matrices

The theory of compressed sensing tells us that recovering all k-sparse signals requires a sensing matrix to satisfy that its spark is greater than 2k, or its order of the null space property (NSP) or the restricted isometry property (RIP) is 2k or above. If we perform elementary row or column operations on the sensing matrix, what are the changes of its spark, NSP order and RIP order? In this paper, we study this problem and discover that these three quantitative indexes of sensing matrices all possess invariance under all matrix elementary transformations except column-addition ones. Putting this result in form of matrix products, we get the types of matrices which multiply a sensing matrix and make the products still have the same properties of sparse recovery as the sensing matrix. According to these types of matrices, we made an interesting discovery that sensing matrices with deterministic constructions do not possess the property universality which belongs to sensing matrices with random constructions.

[1]  Massimo Fornasier,et al.  Theoretical Foundations and Numerical Methods for Sparse Recovery , 2010, Radon Series on Computational and Applied Mathematics.

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

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

[4]  Gitta Kutyniok,et al.  Theory and applications of compressed sensing , 2012, 1203.3815.

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

[6]  S. Mendelson,et al.  Uniform Uncertainty Principle for Bernoulli and Subgaussian Ensembles , 2006, math/0608665.

[7]  Michael Elad,et al.  From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images , 2009, SIAM Rev..

[8]  S. Frick,et al.  Compressed Sensing , 2014, Computer Vision, A Reference Guide.

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

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

[11]  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.

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

[13]  Massimo Fornasier,et al.  Compressive Sensing , 2015, Handbook of Mathematical Methods in Imaging.

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

[15]  Otmar Scherzer,et al.  Handbook of Mathematical Methods in Imaging , 2015, Handbook of Mathematical Methods in Imaging.

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

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

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

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