Equivalence of ℓ0- and ℓ1-Minimization
The ℓ1-minimization method cannot always solve an ℓ0-minimization problem. In this chapter, we first discuss the condition under which the ℓ0- and ℓ1-minimization problems are equivalent. This condition indicates that the uniqueness of the solution to an ℓ0-minimization problem cannot guarantee that the problem can be solved by ℓ1-minimization, and the multiplicity of the solutions to an ℓ0-minimization problem cannot also prevent the problem from being solved by ℓ1-minimization. It is the RSP of AT at a solution of the ℓ0-minimization problem that determines whether the ℓ0-minimization problem can be solved by ℓ1-minimization. We also discuss the equivalence of ℓ0-minimization and weighted ℓ
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