# American Institute of Mathematical Sciences

2014, 4(1): 25-38. doi: 10.3934/naco.2014.4.25

## Partial $S$-goodness for partially sparse signal recovery

 1 Department of Applied Mathematics, Beijing Jiaotong University, Beijing 100044, China, China 2 Department of Anesthesia, Dongzhimen Hospital, Beijing University of Chinese Medicine, No.5 Haiyuncang, Dongcheng District, Beijing 100700, China

Received  May 2013 Revised  October 2013 Published  December 2013

In this paper, we will consider the problem of partially sparse signal recovery (PSSR), which is the signal recovery from a certain number of linear measurements when its part is known to be sparse. We establish and characterize partial $s$-goodness for a sensing matrix in PSSR. We show that the partial $s$-goodness condition is equivalent to the partial null space property (NSP), and is weaker than partial restricted isometry property. Moreover, this provides a verifiable approach for partial NSP via partial $s$-goodness constants. We also give exact and stable partially $s$-sparse recovery via the partial $l_1$-norm minimization under mild assumptions.
Citation: Lingchen Kong, Naihua Xiu, Guokai Liu. Partial $S$-goodness for partially sparse signal recovery. Numerical Algebra, Control and Optimization, 2014, 4 (1) : 25-38. doi: 10.3934/naco.2014.4.25
##### References:
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##### References:
 [1] A. Bandeira, K. Scheinberg and L. N. Vicente, Computation of sparse low degree interpolating polynomials and their application to derivative-free optimization, Math. Program., 134 (2012), 223-257. doi: 10.1007/s10107-012-0578-z. [2] A. Bandeira, K. Scheinberg and L. N. Vicente, On partially sparse recovery, Tech. Rep., 2011. [3] A. M. Bruckstein, D. L. Donoho and M. Elad, From sparse solutions of systems of equations to sparse modeling of signals and images, SIAM Review, 51 (2009), 34-81. doi: 10.1137/060657704. [4] E. J. Candés, J. Romberg and T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform. Theory, 52 (2006), 489-509. doi: 10.1109/TIT.2005.862083. [5] E. J. Candés and T. Tao, Decoding by linear programming, IEEE Trans. Inform. Theory, 51 (2005), 4203-4215. doi: 10.1109/TIT.2005.858979. [6] E. J. Candés, M. B. Wakin and S. P. Boyd, Enhancing sparsity by reweighted l1 minimization, J Fourier Anal Appl., 14 (2008), 877-905. doi: 10.1007/s00041-008-9045-x. [7] D. L. Donoho, Compressed sensing, IEEE Trans. Inform. Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582. [8] L. Jacques, A short note on compressed sensing with partially known signal support, Signal Processing, 90 (2010), 3308-3312. [9] A. Juditsky and A. S. Nemirovski, On verifiable sufficient conditions for sparse signal recovery via l1 minimization, Math. Program., 127 (2011), 57-88. doi: 10.1007/s10107-010-0417-z. [10] A. Juditsky, F. Karzan and A. S. Nemirovski, Verifiable conditions of l1-recovery of sparse signals with sign restrictions, Math. Program., 127 (2011), 89-122. doi: 10.1007/s10107-010-0418-y. [11] M. A. Khajehnejad, W. Xu, A. S. Avestimehr and B. Hassibi, Analyzing weighted minimization for sparse recovery with nonuniform sparse models, IEEE Trans. Signal Process., 59 (2011), 1985-2001. doi: 10.1109/TSP.2011.2107904. [12] A. Majumdar and R. K. Ward, An algorithm for sparse MRI reconstruction by Schatten p-norm minimization, Magnetic Resonance Imaging, 29 (2011), 408-417. [13] N. Vaswani and W. Lu, Modifed-CS: modifying compressive sensing for problems with partially known support, IEEE Trans. Signal Process., 58 (2010), 4595-4607. doi: 10.1109/TSP.2010.2051150.
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