A necessary and sufficient condition for sparse vector recovery via ℓ1 − ℓ2 minimization

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In this paper, we focus on ℓ 1 − ℓ 2 minimization model, i.e., investigating the nonconvex model: min ⁡ ‖ x ‖ 1 − ‖ x ‖ 2 s.t. A x = y and provide a null space property of the measurement matrix A such that a vector x can be recovered from Ax via ℓ 1 − ℓ 2 minimization. The ℓ 1 − ℓ 2 minimization model was first proposed by E.Esser, et al (2013) [8] . As a nonconvex model, it is well known that global minimizer and local minimizer are usually inconsistent. In this paper, we present a necessary and sufficient condition for the measurement matrix A such that (1) a vector x can be recovered from Ax via ℓ 1 − ℓ 2 local minimization ( Theorem 4 ); (2) any k -sparse vector x can be recovered from Ax via ℓ 1 − ℓ 2 local minimization ( Theorem 5 ); (3) any k -sparse vector x can be recovered from Ax via ℓ 1 − ℓ 2 global minimization ( Theorem 6 ).