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
$\boldsymbol{\ell_{p}}$
null space property (
$\boldsymbol{\ell_{p}}$
-NSP) and restricted isometry property (RIP) are two important frames for sparse signal recovery. New sufficient conditions in terms of
$\boldsymbol{\ell_{p}}$
-NSP and RIP are respectively developed in this paper. Firstly, we characterize the
$\boldsymbol{\ell_{p}}$
robust null space property (
$\boldsymbol{\ell_{p}}$
-RNSP) concerning two high-order restricted isometry constants. Then we derive an upper bound of
$\boldsymbol{\ell_{p}}$
-NSC
$\boldsymbol{\rho(p,t,A,k)}$
for the exact recovery of
$\boldsymbol{k}$
-sparse signals via
$\boldsymbol{\ell_{p}}$
minimization. Secondly, we establish an upper bound of RIC
$\boldsymbol{\delta_{tK_{0}}}$
based on an adjustable parameter
$\boldsymbol{t}$
and the sparsity level
$\boldsymbol{K_{0}}$
via constrained
$\boldsymbol{\ell_{p}}$
minimization. The induced high-order RIP condition dependent on the sparsity level
$\boldsymbol{K_{0}}$
is substantially milder compared with the state-of-the-art results. Thirdly, we present new results for the stable recovery of approximately
$\boldsymbol{k}$
-sparse signals in
$\ell_{2}$
bounded noise setting. Moreover, numerical experiments demonstrate the advantage of the obtained results for sparse recovery.