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Applied Mathematics-A Journal of Chinese Universities  2018, Vol. 33 Issue (1): 1-24    DOI: 10.1007/s11766-018-3430-2
    
Sparse recovery in probability via $l_q$-minimization with Weibull random matrices for 0 < $q$ ≤ 1
GAO Yi1,2, PENG Ji-gen1,3, YUE Shi-gang4
1 School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China. Email: jgpeng@xjtu.edu.cn
2 School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China. Email: gaoyimh@163.com
3 School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China. Email: jgpeng@xjtu.edu.cn
4 School of Computer Science, University of Lincoln, Lincoln LN6 7TS, UK. Email: syue@lincoln.ac.uk
Sparse recovery in probability via $l_q$-minimization with Weibull random matrices for 0 < $q$ ≤ 1
GAO Yi1,2, PENG Ji-gen1,3, YUE Shi-gang4
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China. Email: jgpeng@xjtu.edu.cn
2 School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China. Email: gaoyimh@163.com
3 School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China. Email: jgpeng@xjtu.edu.cn
4 School of Computer Science, University of Lincoln, Lincoln LN6 7TS, UK. Email: syue@lincoln.ac.uk
 全文: PDF 
摘要: Although Gaussian random matrices play an important role of measurement matrices in compressed sensing, one hopes that there exist other random matrices which can also be used to serve as the measurement matrices. Hence, Weibull random matrices induce extensive interest. In this paper, we first propose the $l_{2,q}$ robust null space property that can weaken the $D$-RIP, and show that Weibull random matrices satisfy the $l_{2,q}$ robust null space property with high probability. Besides, we prove that Weibull random matrices also possess the $l_q$ quotient property with high probability. Finally, with the combination of the above mentioned properties, we give two important approximation characteristics of the solutions to the $l_q$-minimization with Weibull random matrices, one is on the stability estimate when the measurement noise $e \in \mathbb{R}^n$ needs a priori $\|e\|_2\leq \epsilon$, the other is on the robustness estimate without needing to estimate the bound of $\|e\|_2$. The results indicate that the performance of Weibull random matrices is similar to that of Gaussian random matrices in sparse recovery.
关键词: compressed sensing$l_q$-minimizationWeibull matricesnull space propertyquotient property    
Abstract: Although Gaussian random matrices play an important role of measurement matrices in compressed sensing, one hopes that there exist other random matrices which can also be used to serve as the measurement matrices. Hence, Weibull random matrices induce extensive interest. In this paper, we first propose the $l_{2,q}$ robust null space property that can weaken the $D$-RIP, and show that Weibull random matrices satisfy the $l_{2,q}$ robust null space property with high probability. Besides, we prove that Weibull random matrices also possess the $l_q$ quotient property with high probability. Finally, with the combination of the above mentioned properties, we give two important approximation characteristics of the solutions to the $l_q$-minimization with Weibull random matrices, one is on the stability estimate when the measurement noise $e \in \mathbb{R}^n$ needs a priori $\|e\|_2\leq \epsilon$, the other is on the robustness estimate without needing to estimate the bound of $\|e\|_2$. The results indicate that the performance of Weibull random matrices is similar to that of Gaussian random matrices in sparse recovery.
Key words: compressed sensing    $l_q$-minimization    Weibull matrices    null space property    quotient property
收稿日期: 2016-01-01 出版日期: 2018-03-27
CLC:  15A52  
基金资助: Supported by the National Natural Science Foundation of China (11761003, 11771347, 91730306, 41390454), the Natural Science Foundation of Ningxia (NZ17097), and in part by the Horizon2020 project STEP2DYNA(691154).
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引用本文:

GAO Yi, PENG Ji-gen, YUE Shi-gang. Sparse recovery in probability via $l_q$-minimization with Weibull random matrices for 0 < $q$ ≤ 1[J]. Applied Mathematics-A Journal of Chinese Universities, 2018, 33(1): 1-24.

GAO Yi, PENG Ji-gen, YUE Shi-gang. Sparse recovery in probability via $l_q$-minimization with Weibull random matrices for 0 < $q$ ≤ 1. Applied Mathematics-A Journal of Chinese Universities, 2018, 33(1): 1-24.

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http://www.zjujournals.com/amjcub/CN/10.1007/s11766-018-3430-2        http://www.zjujournals.com/amjcub/CN/Y2018/V33/I1/1

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