Recovery of sparsest signals via $\ell^q$-minimization

In this paper, it is proved that every $s$-sparse vector ${\bf x}\in {\mathbb R}^n$ can be exactly recovered from the measurement vector ${\bf z}={\bf A} {\bf x}\in {\mathbb R}^m$ via some $\ell^q$-minimization with $0< q\le 1$, as soon as each $s$-sparse vector ${\bf x}\in {\mathbb R}^n$ is uniquely determined by the measurement ${\bf z}$.