Upper Bound of Null Space Constant $\rho(p,t,A,k)$ and High-Order Restricted Isometry Constant $\delta_{tk}$ for Sparse Recovery via $\ell_{p}$ Minimization

The $\ell_p$ null space property ( $\ell_p$ -NSP) and restricted isometry property (RIP) are two important frames for sparse signal recovery. New sufficient conditions in terms of $\ell_p$ -NSP and RIP are respectively developed in this paper. Firstly, we characterize the $\ell_p$ robust null space property ( $\ell_p$ -RNSP) concerning two high-order restricted isometry constants. Then we derive an upper bound of $\ell_p$ -NSC $\rho(p,t, A,k)$ for the exact recovery of $k$ -sparse signals via $\ell_p$ minimization. Secondly, we establish an upper bound of RIC $\delta_{tK_0}$ based on an adjustable parameter $t$ and sparsity level $K_0$ via constrained $\ell_p$ minimization. The induced high-order RIP condition dependent on the sparsity level $K_0$ is substantially milder compared with the state-of-the-art results. Thirdly, we present new results for the stable recovery of approximately $k$ -sparse signals in $\ell_2$ bounded noise setting. Moreover, numerical experiments demonstrate the advantage of the obtained results for sparse recovery.