Robustness of Partial Sparse Signal Recovery Based on $l_{q}$ Minimization Model

This paper discusses the recovery of partial sparse signal in compressed sensing. Firstly, a $l_{q} (0 < q< 1)$ non-convex optimization model is developed for partial sparse signal recovery under noisy measurement. Secondly, according to the existing partial $l_{q}$ null space property ($l_{q}$-NSP), we propose the partial $l_{q}$ robust null space property ($l_{q}$-RNSP) and the partial $l_{2,q}$ robust null space property ($1_{2,q}$-RNSP), and it is show that both properties are weaker than the partial restricted isometry property (RIP) proposed in the existing reference. Finally, the robustness estimation of the model solution is established based on the partial RNSP.