Analyses of the tail-ℓ2 minimization for fast and enhanced sparse selections

We investigate the effectiveness and efficiency of the iterative tail-${\ell }_2$ minimization (tail-${\ell }_2$-min) technique for its sparse selection capabilities. We conduct profile analyses on the tail-${\ell }_2$-min, establishing the equivalence of the tail-${\ell }_2$-min problem to a two-stage profile ${\ell }_2$ formulation, both featuring analytical solutions. The tail null space property (NSP) of sensing matrix $A$ is shown to be equivalent to the NSP of the newly defined profile matrix $\tilde{A}$. Besides the error bound analysis for the tail-${\ell }_2$-min under the typical tail-NSP condition, a novel error bound of the tail-${\ell }_2$-min formulation is also established without relying on NSP or restricted isometry property (RIP) assumptions. It merely contains tractable coefficients of $A$, and offers insights into successful recovery, with the observation of the convergent iterative procedure. Numerical studies and the applications to image reconstruction demonstrate the superiority and fast convergence of the tail-${\ell }_2$ sparse solution over state-of-the-art sparse selection methodologies. The sparsity level of a signal that the tail-${\ell }_2$ profile algorithm guarantees the recovery is around 41% higher than that of the basis pursuit algorithm. The analytical solutions of the tail-${\ell }_2$ method at each iteration also ensure that the tail-${\ell }_2$ sparse recovery process is notably fast, especially for high dimensions and high sparsity levels.