Performance analysis of unconstrained ℓp minimization for sparse recovery

Abstract

In view of coherence, this paper firstly presents a coherence-based theoretical guarantee, including a sufficient condition and associated error estimate, for a non-convex unconstrained ${\ell }_p$ ($0<p\le 1$) minimization to robustly reconstruct any non-sparse signal in the noisy situation. In a sense, this result supplements the preceding founded ones for the constrained ${\ell }_p$ minimization. Specially, when $p=1$, our coherence-based condition reduces to the state-of-art sharp one, i.e, $\mu <1/(2s-1)$. It should also be emphasized that for sparse metric models, based on the theory of coherence, our condition has reached consistency with the corresponding constraint situation. According to the established result, the error in the case that the representative constrained ${\ell }_p$ minimization is substituted with unconstrained ${\ell }_p$ minimization is also studied. Additionally, the relationship between coherence and null space property (NSP) is discussed and the derived result claims that coherence could imply NSP. In view of the induced NSP, the recovery theory that guarantees the sparse signal can be recovered via the unconstrained ${\ell }_p$ minimization is established. Based on the synthetic signals and the real-life signals, it is demonstrated by experimental results that compared with state-of-art methods and convex ${\ell }_1$ minimization, the performance of non-convex ${\ell }_p$ minimization is more competitive.