受限$p$ - isometry性质及其在非凸压缩传感中的应用
Restricted $p$-isometry property and its application for nonconvex compressive sensing
摘要：
压缩感知是一种利用$l_1$最小化原理从较少的测量数据中恢复稀疏信号的新方法。最近，Chartrand和Staneva在\cite{CS1}中表明，与$l_1$最小化相比，使用$0<p<1$的$l_p$最小化从更少的线性测量中恢复稀疏信号。他们证明，对于概率超过$$1 - 1 / {N\choose S}.$$的一些较小的$p$，使用$0<p<1$的$l_p$最小化从较少的高斯随机测量恢复$S$ -稀疏信号$x\in\RN$。本文的第一个目的是表明上述结果对于概率超过$1-2e^{-c(p)M},$的随机高斯测量的情况是正确的，其中$M$是随机测量的行数，高斯测量和$c(p)$是一个正常数，保证$1-2e^{-c(p)M}>1 - 1 / {N\choose S}$为$p$较小。本文的第二个目的是表明，在某些较弱的条件下，解码器$\triangle_p$是稳定的，在某种意义上，他们是$(2,p)$实例最佳的一大类编码器$0<p<1.$
英文摘要：
Compressed sensing is a new scheme which shows the ability to recover sparse signal from fewer measurements, using $l_1$ minimization. Recently, Chartrand and Staneva shown in \cite{CS1} that the $l_p$ minimization with $0<p<1$ recovers sparse signals from fewer linear measurements than does the $l_1$ minimization. They proved that $l_p$ minimization with $0<p<1$ recovers $S$-sparse signals $x\in\RN$ from fewer Gaussian random measurements for some smaller $p$ with probability exceeding $$1 - 1 / {N\choose S}.$$ The first aim of this paper is to show that above result is right for the case of random,Gaussian measurements with probability exceeding $1-2e^{-c(p)M},$ where $M$ is the numbers of rows of random, Gaussian measurements and $c(p)$ is a positive constant that guarantees $1-2e^{-c(p)M}>1 - 1 / {N\choose S}$ for $p$ smaller. The second purpose of the paper is to show that under certain weaker conditions, decoders $\triangle_p$ are stable in the sense that they are $(2,p)$ instance optimal for a large class of encoder for $0<p<1.$
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
