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A necessary and sufficient condition for sparse vector recovery via ℓ1 − ℓ2 minimization
Applied and Computational Harmonic Analysis  (IF3.055),  Pub Date : 2021-10-01, DOI: 10.1016/j.acha.2021.09.003
Ning Bi, Wai-Shing Tang

In this paper, we focus on 12 minimization model, i.e., investigating the nonconvex model:minx1x2s.t.Ax=y and provide a null space property of the measurement matrix A such that a vector x can be recovered from Ax via 12 minimization. The 12 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 12 local minimization (Theorem 4); (2) any k-sparse vector x can be recovered from Ax via 12 local minimization (Theorem 5); (3) any k-sparse vector x can be recovered from Ax via 12 global minimization (Theorem 6).