Sparse signal recovery by ${\mathrm{\ell }}_q$ minimization under restricted isometry property

In the context of compressed sensing, the nonconvex ${\mathrm{\ell }}_q$ minimization with $0<q<1$ has been studied in recent years. In this paper, by generalizing the sharp bound for ${\mathrm{\ell }}_1$ minimization of Cai and Zhang, we show that the condition ${\delta }_{(s^q+1)k}<{\displaystyle \frac{1}{\sqrt{s^{q-2}+1}}}$ in terms of \emph{restricted isometry constant (RIC)} can guarantee the exact recovery of $k$-sparse signals in noiseless case and the stable recovery of approximately $k$-sparse signals in noisy case by ${\mathrm{\ell }}_q$ minimization. This result is more general than the sharp bound for ${\mathrm{\ell }}_1$ minimization when the order of RIC is greater than $2k$ and illustrates the fact that a better approximation to ${\mathrm{\ell }}_0$ minimization is provided by ${\mathrm{\ell }}_q$ minimization than that provided by ${\mathrm{\ell }}_1$ minimization.