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

  • Authors:

  • Yi Shen;Song Li

  • Affiliations:

  • Department of Mathematics, Zhejiang University, Hangzhou, People's Republic of China 310027 and Department of Mathematics and Science, Zhejiang Sci-Tech University, Zhejiang Province, People's Rep ...;Department of Mathematics, Zhejiang University, Hangzhou, People's Republic of China 310027

  • Venue:
  • Advances in Computational Mathematics
  • Year:
  • 2012

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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 showed in Chartrand and Staneva (Inverse Problems 24:1---14, 2009) that the l p minimization with 0驴p驴l 1 minimization. They proved that l p minimization with 0驴p驴S-sparse signals x驴驴驴驴 N from fewer Gaussian random measurements for some smaller p with probability exceeding $$ 1 - 1 \Bigg/ {N\choose S}. $$ The first aim of this paper is to show that above result is right for the case of Gaussian random measurements with probability exceeding 1驴驴驴2e 驴驴驴c(p)M , where M is the numbers of rows of Gaussian random 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 Δ p are stable in the sense that they are (2,p) instance optimal for a large class of encoder for 0驴p驴