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