Upper Bound of Null Space Constant [Formula Omitted] and High-Order Restricted Isometry Constant [Formula Omitted] for Sparse Recovery via [Formula Omitted] Minimization

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