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.
Source: IEEE Transactions on Signal Processing - Category: Biomedical Engineering Source Type: research