Sparse Approximate Reconstruction Decomposed by Two Optimization Problems

Abstract

We propose a novel sparse signal reconstruction method aiming to directly minimize $$\ell _{0}$$ -quasinorm. Based on the smoothed $$\ell _{0}$$ -quasinorm, we show that there exists an unconstrained optimization problem such that both this problem and the basis pursuit problem are subproblems of the $$\ell _{0}$$ minimization problem. Moreover, we can obtain a sparse solution to the $$\ell _{0}$$ minimization by solving these two subproblems. In addition, we establish the relation between solutions to the $$\ell _{0}$$ minimization and the least square solutions of a linear system. Finally, we present some numerical experiments to illustrate our results.