This paper focuses on the compressed sensing $\ell_1-\ell_2-$minimization model and develops new bounds on cumulative coherence $\mu_1(s)$. We point out that if cumulative coherence $\mu_1(s)$ satisfies (2) or (11), then the sparse signal can stably recover in noise model and exactly recover in free noise by $\ell_1-\ell_2$-minimization model.

This paper focuses on the compressed sensing $\ell_{1-2}$-minimization problem and develops new bounds on cumulative coherence $\mu_1(s)$. We point out that if cumulative coherence $\mu_1(s-1)$ and $\mu_1(2s-1)$ satisfy $(\ref{eq:EqNo2})$, or cumulative coherence $ \mu_1(2s-1)$ satisfies $(\ref{eq:EqNo11})$ then the sparse signal can via $\ell_{1-2}$-minimization problem stably recover in noise model and exact recovery in free noise model.