RSP-Based Analysis for Sparsest and Least ℓ1-Norm Solutions to Underdetermined Linear Systems

Recently, the worse-case analysis, probabilistic analysis and empirical justification have been employed to address the fundamental question: When does l1-minimization find the sparsest solution to an underdetermined linear system? In this paper, a deterministic analysis, rooted in the classic linear programming theory, is carried out to further address this question. We first identify a necessary and sufficient condition for the uniqueness of least l1-norm solutions to linear systems. From this condition, we deduce that a sparsest solution coincides with the unique least l1-norm solution to a linear system if and only if the so-called range space property (RSP) holds at this solution. This yields a broad understanding of the relationship between l0and l1-minimization problems. Our analysis indicates that the RSP truly lies at the heart of the relationship between these two problems. Through RSP-based analysis, several important questions in this field can be largely addressed. For instance, how to efficiently interpret the gap between the current theory and the actual numerical performance of l1-minimization by a deterministic analysis, and if a linear system has multiple sparsest solutions, when does l1-minimization guarantee to find one of them? Moreover, new matrix properties (such as the RSP of order K and the Weak-RSP of order K) are introduced in this paper, and a new theory for sparse signal recovery based on the RSP of order K is established.