The Null Space Property of the Truncated $\ell _{1-2}$ -Minimization

The null space property (NSP), which depends only on the null space of the column space of measurement matrix, has received much attention in compressed sensing. This letter considers NSP of the truncated <inline-formula> <tex-math notation="LaTeX">$\ell _{1-2}$</tex-math></inline-formula> minimization. It provides two versions of NSP of the truncated <inline-formula><tex-math notation="LaTeX">$\ell _{1-2}$</tex-math></inline-formula> minimization, under which we present sufficient conditions for the truncated <inline-formula><tex-math notation="LaTeX">$\ell _{1-2}$ </tex-math></inline-formula> minimization to recover sparse and compressible signals. In addition, we discuss that the truncated <inline-formula><tex-math notation="LaTeX">$\ell _{1-2}$</tex-math></inline-formula> stable NSP holds by Gaussian matrices of appropriate sizes with overwhelming probability.

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