Analysis of the equivalence relationship between -minimization and -minimization

In signal processing theory, $$-minimization is an important mathematical model. Unfortunately, $$-minimization is actually NP-hard. The most widely studied approach to this NP-hard problem is based on solving $$-minimization ($$). In this paper, we present an analytic expression of $$, which is formulated by the dimension of the matrix $$, the eigenvalue of the matrix $$, and the vector $$, such that every k-sparse vector $$ can be exactly recovered via $$-minimization whenever $$, that is, $$-minimization is equivalent to $$-minimization whenever $$. The superiority of our results is that the analytic expression and each its part can be easily calculated. Finally, we give two examples to confirm the validity of our conclusions.