Stability Analysis for a Class of Sparse Optimization Problems
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The sparse optimization problems arise in many areas of science and engineering, such as compressed sensing, image processing, statistical and machine learning. The ℓ_0-minimization problem is one of such optimization problems, which is typically used to deal with signal recovery. The ℓ_1-minimization method is one of the plausible approaches for solving the ℓ_0-minimization problems, and thus the stability of such a numerical method is vital for signal recovery. In this paper, we establish a stability result for the ℓ_1-minimization problems associated with a general class of ℓ_0-minimization problems. To this goal, we introduce the concept of restricted weak range space property (RSP) of a transposed sensing matrix, which is a generalized version of the weak RSP of the transposed sensing matrix introduced in [Zhao et al., Math. Oper. Res., 44(2019), 175-193]. The stability result established in this paper includes several existing ones as special cases.
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