Analyses of Successful Sparse Signal Recovery via Tail-$\ell_{2}$ Minimization

Abstract

Recovery guarantee analyses of sparse signals by the tail-$\ell_{2}$ minimization approach are presented. Known for the lack of sparse recovery capacity by traditional $\ell_{2}$ minimization, its variation by an iterative tail-$\ell_{2}$ penalty procedure, however, is shown to be exceedingly effective in sparse selections. The analytical close-form solutions of the tail-$\ell_{2}$ formulation also reveal its superb efficiency. This article is focused on the analyses of the successful recovery by the tail-$\ell_{2}$ technique. A necessary and sufficient condition for the uniqueness of the tail-$\ell_{2}$ minimizer is established, which is seen inherently different from that of a similar tail-$\ell_{1}$ minimization problem. The inherent differences lead to further analyses of sufficient conditions for the uniqueness, and a notion of admissible solutions. Successful probability analysis is then carried out based on these conditions. The estimated probability of successful recovery $\mathbb{P}_{T}$ is righteously related to the cardinality of $T^{c}\cap S$, where $T$ is an estimated support of the solution index $S$. The smaller the $|T^{c}\cap S|$ is, the greater the $\mathbb{P}_{T}$ will be, and $\mathbb{P}_{T}$ naturally approaches 1 as $|T^{c}\cap S|$ approaches 0. Numerical experiments sufficiently validate the efficiency and the successful probability of the sparse signal recovery by the tail-$\ell_{2}$ minimization procedure.