Restricted $p$-isometry property and its application for nonconvex compressive sensing

Abstract

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 $0<p<1.$